📖 Tier 1: Prepare & Study Guide
✓ 100% Solved with Rationales
Statistics Solved Questions Bank & Study Material (2026) - Apex Rankers
Statistics & Data Science | Topic-Wise Comprehensive Solved Pool
600 Total Solved Questions
~900 mins Estimated Reading Time
9 Subject Areas / Chapters
Select Topic Area / Chapter:
Click any section below to switch questions
Bayesian Inference, Time Series & Disaster Modeling
Showing 25 of 90 (28%)
Jump:
🎯 Practice This Area
🎉 All 90 Questions Loaded!
Descriptive Statistics & Probability
Showing 25 of 60 (42%)
Jump:
🎯 Practice This Area
🎉 All 60 Questions Loaded!
Distributions & Sampling Theory
Showing 25 of 50 (50%)
Jump:
🎯 Practice This Area
🎉 All 50 Questions Loaded!
Hypothesis Testing & Non-Parametric Methods
Showing 25 of 90 (28%)
Jump:
🎯 Practice This Area
🎉 All 90 Questions Loaded!
Hypothesis Testing, Regression & ANOVA
Showing 25 of 36 (69%)
Jump:
🎯 Practice This Area
🎉 All 36 Questions Loaded!
Linear & Multiple Regression, Correlation & ANOVA
Showing 25 of 90 (28%)
Jump:
🎯 Practice This Area
🎉 All 90 Questions Loaded!
Probability Distributions & Random Variables
Showing 25 of 90 (28%)
Jump:
🎯 Practice This Area
🎉 All 90 Questions Loaded!
Probability Distributions & Sampling
Showing 4 of 4 (100%)
🎉 All 4 Questions Loaded!
Sampling Theory, Estimation & Statistical Inference
Showing 25 of 90 (28%)
Jump:
🎯 Practice This Area
🎉 All 90 Questions Loaded!
Q. 1
Statistics
Difficulty: easy
(1 Mark)
In Bayesian statistical inference, what is Bayes' Theorem for continuous parameter $\theta$ and data $\mathbf{x}$ expressed as?
A
$\pi(\theta \mid \mathbf{x}) = \frac{f(\mathbf{x} \mid \theta) \pi(\theta)}{\int f(\mathbf{x} \mid \theta) \pi(\theta) d\theta} \propto L(\theta; \mathbf{x}) \times \pi(\theta)$
✓ Correct
B
$\pi(\theta \mid \mathbf{x}) = f(\mathbf{x} \mid \theta) + \pi(\theta)$
C
$\pi(\theta \mid \mathbf{x}) = \frac{\pi(\theta)}{f(\mathbf{x} \mid \theta)}$
D
$\pi(\theta \mid \mathbf{x}) = \int f(\mathbf{x} \mid \theta) d\theta$
💡
Step-by-Step Explanation & Concept Rationale
Bayes' Theorem updates the prior probability distribution $\pi(\theta)$ with the observed data likelihood $L(\theta; \mathbf{x})$ to yield the posterior distribution $\pi(\theta \mid \mathbf{x}) \propto \text{Likelihood} \times \text{Prior}$.
Q. 2
Statistics
Difficulty: easy
(1 Mark)
What is a 'Conjugate Prior' in Bayesian probability theory?
A
A prior distribution that belongs to the same probability distribution family as the resulting posterior distribution
✓ Correct
B
A prior distribution that is always uniform on $[0, 1]$
C
A prior distribution that produces zero variance
D
A prior distribution chosen by a panel of judges
💡
Step-by-Step Explanation & Concept Rationale
A prior is conjugate to the likelihood if the posterior distribution $\pi(\theta \mid \mathbf{x})$ belongs to the exact same parametric family as the prior $\pi(\theta)$ (e.g., Beta prior is conjugate to Binomial likelihood, yielding a Beta posterior).
Q. 3
Statistics
Difficulty: medium
(1 Mark)
If a Binomial likelihood $X \sim B(n, p)$ has a conjugate $\text{Beta}(\alpha, \beta)$ prior on $p$, what is the resulting posterior distribution $\pi(p \mid x)$?
A
$\text{Beta}(\alpha + x, \beta + n - x)$
✓ Correct
B
$\text{Beta}(\alpha + n, \beta + x)$
C
$\text{Gamma}(\alpha + x, \beta + n)$
D
$\text{Normal}(\alpha, \beta)$
💡
Step-by-Step Explanation & Concept Rationale
Multiplying the Beta prior $p^{\alpha-1}(1-p)^{\beta-1}$ by the Binomial likelihood $\binom{n}{x}p^x(1-p)^{n-x}$ yields $p^{(\alpha+x)-1}(1-p)^{(\beta+n-x)-1}$, which is exactly $\text{Beta}(\alpha + x, \beta + n - x)$.
Q. 4
Statistics
Difficulty: medium
(1 Mark)
What is the conjugate prior distribution for the mean $\mu$ of a Normal distribution $N(\mu, \sigma^2)$ when variance $\sigma^2$ is known?
A
A Normal distribution $N(\mu_0, \sigma_0^2)$
✓ Correct
B
A Gamma distribution
C
A Uniform distribution
D
A Chi-Square distribution
💡
Step-by-Step Explanation & Concept Rationale
The Normal distribution is conjugate to a Normal likelihood for known variance: if prior is $N(\mu_0, \sigma_0^2)$, the posterior distribution of $\mu$ is also strictly Normal with precision equal to the sum of prior and data precisions.
Q. 5
Statistics
Difficulty: medium
(1 Mark)
What is the conjugate prior for the parameter $\lambda$ of a Poisson distribution?
A
Gamma distribution $\text{Gamma}(\alpha, \beta)$
✓ Correct
B
Beta distribution
C
Normal distribution
D
Exponential distribution
💡
Step-by-Step Explanation & Concept Rationale
The Gamma distribution $\text{Gamma}(\alpha, \beta)$ is the conjugate prior for the Poisson rate $\lambda$. With $n$ observations summing to $\sum x_i$, the posterior is $\text{Gamma}(\alpha + \sum x_i, \beta + n)$.
Q. 6
Statistics
Difficulty: hard
(1 Mark)
What is Jeffreys' Prior in Bayesian statistics, and what is its defining invariance property?
A
$\pi(\theta) \propto \sqrt{\det I(\theta)}$ (where $I(\theta)$ is Fisher Information); it is invariant under 1-to-1 reparameterization $\phi = g(\theta)$
✓ Correct
B
$\pi(\theta) = 1$ for all parameters
C
$\pi(\theta) = \frac{1}{\theta^2}$
D
$\pi(\theta) = e^{-\theta}$
💡
Step-by-Step Explanation & Concept Rationale
Harold Jeffreys (1946) derived non-informative priors proportional to the square root of the Fisher Information determinant, ensuring that transforming parameters (e.g., from variance to standard deviation) preserves mathematical consistency.
Q. 7
Statistics
Difficulty: easy
(1 Mark)
What is the Bayesian Point Estimator under Squared Error Loss $L(\theta, \hat{\theta}) = (\theta - \hat{\theta})^2$?
A
The Posterior Mean $E[\theta \mid \mathbf{x}]$
✓ Correct
B
The Posterior Median
C
The Posterior Mode (MAP)
D
The Prior Mean
💡
Step-by-Step Explanation & Concept Rationale
Minimizing expected squared error loss $E[(\theta - \hat{\theta})^2 \mid \mathbf{x}]$ yields the conditional expectation (Posterior Mean) $\hat{\theta} = \int \theta \pi(\theta \mid \mathbf{x}) d\theta$.
Q. 8
Statistics
Difficulty: medium
(1 Mark)
What is the Bayesian Point Estimator under Absolute Error Loss $L(\theta, \hat{\theta}) = |\theta - \hat{\theta}|$?
A
The Posterior Median of $\pi(\theta \mid \mathbf{x})$
✓ Correct
B
The Posterior Mean
C
The Posterior Mode (MAP)
D
The Maximum Likelihood Estimator
💡
Step-by-Step Explanation & Concept Rationale
Minimizing expected absolute error loss $E[|\theta - \hat{\theta}| \mid \mathbf{x}]$ yields the 50th percentile (Posterior Median) of the posterior distribution.
Q. 9
Statistics
Difficulty: medium
(1 Mark)
What is the Maximum A Posteriori (MAP) estimator?
A
The mode of the posterior distribution: $\hat{\theta}_{MAP} = \arg\max_\theta \pi(\theta \mid \mathbf{x}) = \arg\max_\theta [\ln L(\theta; \mathbf{x}) + \ln \pi(\theta)]$
✓ Correct
B
The average of the prior and likelihood
C
The maximum value in the sample dataset
D
The median of the prior distribution
💡
Step-by-Step Explanation & Concept Rationale
The MAP estimator is the parameter value that maximizes the posterior density function, corresponding to optimal Bayesian decision under 0-1 (all-or-nothing) loss.
Q. 10
Statistics
Difficulty: hard
(1 Mark)
What is a 'Highest Posterior Density' (HPD) Credible Interval in Bayesian analysis?
A
A $100(1-\alpha)\%$ credible region where every parameter point inside the interval has higher posterior density than any point outside, achieving the shortest possible interval width for that coverage
✓ Correct
B
An interval where the two tail areas are exactly equal
C
A frequentist confidence interval calculated on sample means
D
An interval containing 95% of the raw data points
💡
Step-by-Step Explanation & Concept Rationale
An HPD interval is the optimal Bayesian credible set: it encompasses $1-\alpha$ posterior probability while ensuring every point inside has density higher than any point outside, resulting in the minimum interval length.
Q. 11
Statistics
Difficulty: medium
(1 Mark)
How does a Bayesian $95\%$ Credible Interval fundamentally differ from a Frequentist $95\%$ Confidence Interval in philosophical interpretation?
A
A Bayesian credible interval means there is a $95\%$ probability that the unknown parameter $\theta$ lies within the fixed interval $[a, b]$ given the observed data; a frequentist CI treats $\theta$ as fixed and the interval bounds as random under hypothetical repeated sampling
✓ Correct
B
There is no difference; they are mathematically identical in all cases
C
Frequentist intervals can only be computed for coin tosses
D
Bayesian intervals do not use data
💡
Step-by-Step Explanation & Concept Rationale
Bayesian inference treats the parameter $\theta$ as a random variable with a posterior probability distribution conditioned on the observed data, enabling direct probabilistic statements about the parameter itself.
Q. 12
Statistics
Difficulty: hard
(1 Mark)
What is the Metropolis-Hastings (MH) algorithm in Markov Chain Monte Carlo (MCMC)?
A
An MCMC algorithm that generates samples from a target posterior distribution $\pi(\theta)$ by proposing candidate moves $\theta^* \sim q(\theta^* \mid \theta^{(t)})$ and accepting them with probability $\alpha = \min\left(1, \frac{\pi(\theta^*) q(\theta^{(t)} \mid \theta^*)}{\pi(\theta^{(t)}) q(\theta^* \mid \theta^{(t)})}\right)$
✓ Correct
B
A method for sorting numbers in ascending order
C
A deterministic numerical integration algorithm
D
A linear regression algorithm
💡
Step-by-Step Explanation & Concept Rationale
Metropolis-Hastings generates a stationary Markov chain whose equilibrium distribution is exactly the target posterior $\pi(\theta)$, enabling Bayesian sampling without computing the intractable normalizing denominator $\int L(\theta)\pi(\theta)d\theta$.
Q. 13
Statistics
Difficulty: hard
(1 Mark)
What is 'Gibbs Sampling' in multi-parameter Bayesian MCMC?
A
A special case of the Metropolis-Hastings algorithm where each scalar parameter $\theta_j$ is sequentially updated by sampling directly from its Full Conditional Distribution $\pi(\theta_j \mid \boldsymbol{\theta}_{-j}, \mathbf{x})$ with an acceptance probability of exactly 1.0
✓ Correct
B
Sampling with replacement from raw data
C
A method that works only on two variables
D
An optimization method for finding the global minimum
💡
Step-by-Step Explanation & Concept Rationale
Gibbs sampling updates one parameter at a time from its full univariate conditional distribution, accepting every step ($100\%$ acceptance rate) and avoiding manual tuning of proposal widths.
Q. 14
Statistics
Difficulty: hard
(1 Mark)
What does the Gelman-Rubin convergence diagnostic statistic ($\hat{R}$ / Potential Scale Reduction Factor) measure in MCMC?
A
The ratio of variance between multiple independent MCMC chains to variance within chains; $\hat{R} \approx 1.0$ (typically $< 1.05$) indicates convergence to the stationary posterior distribution
✓ Correct
B
The number of iterations per second
C
The correlation between prior and posterior means
D
The percentage of missing data in the chain
💡
Step-by-Step Explanation & Concept Rationale
The Gelman-Rubin $\hat{R}$ compares within-chain and between-chain variances across multiple chains initialized from dispersed starting points: values near $1.00$ indicate that all chains have converged and mixed thoroughly.
Q. 15
Statistics
Difficulty: medium
(1 Mark)
What is 'Effective Sample Size' (ESS) in MCMC posterior analysis?
A
The number of independent, uncorrelated samples that contain the same amount of information as the $N$ autocorrelated MCMC draws: $\text{ESS} = \frac{N}{1 + 2 \sum_{k=1}^\infty \rho_k}$
✓ Correct
B
The total number of iterations run by the computer
C
The sample size of the original survey
D
The number of parameters in the model
💡
Step-by-Step Explanation & Concept Rationale
Because successive MCMC states are autocorrelated, $N$ iterations contain less independent information than $N$ i.i.d. draws. The ESS quantifies the effective number of independent draws for computing posterior standard errors.
Q. 16
Statistics
Difficulty: easy
(1 Mark)
What is 'Covariance Stationarity' (Weak Stationarity) in Time Series Analysis?
A
A stochastic process where the Mean is constant ($E[Y_t] = \mu$), Variance is constant ($\text{Var}(Y_t) = \sigma^2$), and Autocovariance depends ONLY on the time lag $k$, not on actual time $t$: $\text{Cov}(Y_t, Y_{t-k}) = \gamma_k$
✓ Correct
B
A process where all observations are equal to zero
C
A process where the variance increases linearly over time
D
A deterministic sine wave
💡
Step-by-Step Explanation & Concept Rationale
Weak (second-order) stationarity requires constant mean, constant finite variance, and autocovariances $\gamma_k$ that depend solely on the lag separation $k$ rather than the absolute time index $t$.
Q. 17
Statistics
Difficulty: medium
(1 Mark)
What is a 'Random Walk' process $Y_t = Y_{t-1} + \epsilon_t$ (where $\epsilon_t \sim \text{WN}(0, \sigma^2)$)?
A
A non-stationary unit root process whose variance increases linearly with time ($\text{Var}(Y_t) = t \sigma^2$), exhibiting infinite memory of past shocks
✓ Correct
B
A stationary white noise process
C
A deterministic trend with constant variance
D
A seasonal process with period 12
💡
Step-by-Step Explanation & Concept Rationale
A random walk has a unit root ($|\phi| = 1$), non-constant variance $\text{Var}(Y_t) = t\sigma^2$, and persistent non-decaying shocks, making it non-stationary.
Q. 18
Statistics
Difficulty: medium
(1 Mark)
What is 'Spurious Regression' in time series econometrics (Granger & Newbold 1974)?
A
A misleading regression between two independent non-stationary unit root series ($I(1)$) that yields high $R^2$ and highly statistically significant t-ratios, even when the series have zero causal or economic relationship
✓ Correct
B
A regression with negative $R^2$
C
A regression with perfect collinearity
D
A calculation error caused by dividing by zero
💡
Step-by-Step Explanation & Concept Rationale
Regressing one random walk on another unrelated random walk produces spurious correlation: standard t-ratios diverge to $\pm \infty$ and Durbin-Watson statistic $d \to 0$, creating the illusion of a strong relationship.
Q. 19
Statistics
Difficulty: hard
(1 Mark)
What is 'Cointegration' (Robert Engle & Clive Granger 1987)?
A
A property where two or more non-stationary time series $I(1)$ share a common stochastic trend such that a linear combination of them is stationary ($I(0)$): $Y_t - \beta X_t = u_t \sim I(0)$
✓ Correct
B
Two time series having the same sample size
C
Two stationary series that are negatively correlated
D
Two time series with zero variance
💡
Step-by-Step Explanation & Concept Rationale
Cointegrated series drift together over time in a stable long-run equilibrium relationship ($u_t \sim I(0)$), preventing spurious regression and enabling Vector Error Correction Modeling (VECM).
Q. 20
Statistics
Difficulty: medium
(1 Mark)
What is the Augmented Dickey-Fuller (ADF) Test used for in time series modeling?
A
Testing for the presence of a Unit Root ($H_0: \gamma = 0$ / non-stationary) against the alternative of Stationarity ($H_1: \gamma < 0$)
✓ Correct
B
Testing for heteroscedasticity in OLS
C
Testing for seasonality in monthly data
D
Testing whether sample mean is zero
💡
Step-by-Step Explanation & Concept Rationale
The ADF test estimates $\Delta Y_t = \alpha + \beta t + \gamma Y_{t-1} + \sum_{i=1}^p \delta_i \Delta Y_{t-i} + \epsilon_t$. If the t-statistic is more negative than the non-standard Dickey-Fuller critical values, $H_0$ (unit root) is rejected.
Q. 21
Statistics
Difficulty: hard
(1 Mark)
What is the Null Hypothesis of the KPSS (Kwiatkowski-Phillips-Schmidt-Shin) Test, and how does it contrast with the ADF test?
A
The KPSS test tests the null hypothesis that the series is STATIONARY ($H_0: I(0)$), whereas the ADF test tests the null of a Unit Root ($H_0: I(1)$)
✓ Correct
B
KPSS tests for equality of two variances
C
KPSS tests for cointegration across 10 series
D
They share the exact same null hypothesis
💡
Step-by-Step Explanation & Concept Rationale
KPSS reverses the null hypothesis: $H_0$ is trend/level stationarity. Combining ADF and KPSS (confirmatory analysis) robustly verifies whether a series is genuinely stationary or unit-root non-stationary.
Q. 22
Statistics
Difficulty: easy
(1 Mark)
In an Autoregressive model of order 1 ($AR(1)$): $Y_t = \phi Y_{t-1} + \epsilon_t$, what is the condition for weak stationarity?
A
$|\phi| < 1$
✓ Correct
B
$\phi = 1$
C
$\phi > 1$
D
$\phi = 0$
💡
Step-by-Step Explanation & Concept Rationale
For an $AR(1)$ process to be stationary and causal, the autoregressive root must lie outside the unit circle in the complex plane, which translates to the real condition $|\phi| < 1$.
Q. 23
Statistics
Difficulty: medium
(1 Mark)
In the Box-Jenkins methodology for time series modeling, what are the theoretical shapes of the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) for a stationary $AR(p)$ process?
A
ACF decays exponentially / oscillates toward zero; PACF cuts off (drops to zero) abruptly after lag $p$
✓ Correct
B
ACF cuts off after lag $p$; PACF decays exponentially
C
Both ACF and PACF cut off abruptly after lag 1
D
Both ACF and PACF remain constant across all lags
💡
Step-by-Step Explanation & Concept Rationale
For an $AR(p)$ process: the ACF tails off (exponential decay or damped sinusoid), while the PACF cuts off sharply after lag $p$, which uniquely identifies the autoregressive order $p$.
Q. 24
Statistics
Difficulty: medium
(1 Mark)
What are the theoretical ACF and PACF patterns for a Moving Average process of order $q$ ($MA(q)$)?
A
ACF cuts off abruptly after lag $q$; PACF decays exponentially / oscillates toward zero
✓ Correct
B
ACF decays exponentially; PACF cuts off after lag $q$
C
Both ACF and PACF decay exponentially
D
Both ACF and PACF cut off at lag 1
💡
Step-by-Step Explanation & Concept Rationale
For an $MA(q)$ process: the ACF truncates to exactly zero for all lags $k > q$, while the PACF decays smoothly (tails off) toward zero.
Q. 25
Statistics
Difficulty: easy
(1 Mark)
In an $\text{ARIMA}(p, d, q)$ model, what does the parameter $d$ represent?
A
The order of Differencing applied to the non-stationary series to achieve stationarity ($\Delta^d Y_t = (1 - L)^d Y_t$)
✓ Correct
B
The number of autoregressive lags
C
The number of moving average lags
D
The seasonal period length (e.g., 12 for monthly)
💡
Step-by-Step Explanation & Concept Rationale
The parameter $d$ indicates the number of times the raw series is differenced to remove stochastic trends: $d=1$ means first difference $\Delta Y_t = Y_t - Y_{t-1}$; $d=2$ means second difference.
Study Stream Progress:
Showing 25 of 90 Questions (28%)
Jump to:
🎉
All 90 Solved Questions in Bayesian Inference, Time Series & Disaster Modeling Loaded!
You have studied the complete question pool with step-by-step rationales. Ready to test yourself?
Q. 1
Statistics
Difficulty: Easy
(1 Mark)
For a moderately skewed unimodal frequency distribution, what is Karl Pearson's empirical relationship between the Mean, Median, and Mode?
A
Mode = 3 Median - 2 Mean
✓ Correct
B
Mean = 3 Mode - 2 Median
C
Median = 3 Mean - 2 Mode
D
Mode = 2 Median - 3 Mean
💡
Step-by-Step Explanation & Concept Rationale
Karl Pearson observed that for moderately asymmetrical unimodal frequency curves, the distance between the mean and mode is approximately three times the distance between the mean and median: (Mean - Mode) ≈ 3(Mean - Median), which rearranges to Mode = 3 Median - 2 Mean.
Q. 2
Statistics
Difficulty: Easy
(1 Mark)
Which measure of central tendency is most severely distorted and influenced by extreme outlier values in a dataset?
A
Arithmetic Mean
✓ Correct
B
Median
C
Mode
D
Interquartile Range
💡
Step-by-Step Explanation & Concept Rationale
The arithmetic mean uses every numerical value in its calculation ($\sum x / n$), making it highly sensitive to extreme high or low outliers. The median and mode are resistant/robust measures of location.
Q. 3
Statistics
Difficulty: Medium
(1 Mark)
When calculating the average rate of speed, velocity, or price-per-unit ratios, which mathematical average is theoretically the most appropriate measure of central tendency?
A
Harmonic Mean
✓ Correct
B
Arithmetic Mean
C
Median
D
Mode
💡
Step-by-Step Explanation & Concept Rationale
The harmonic mean ($H = n / \sum (1/x_i)$) is the reciprocal of the arithmetic mean of reciprocals. It is mathematically the correct average for rates, ratios, speeds over equal distances, and prices per unit.
Q. 4
Statistics
Difficulty: Easy
(1 Mark)
For calculating the average annual growth rate of population, financial compound interest, or index numbers, which average is statistically optimal?
A
Geometric Mean
✓ Correct
B
Arithmetic Mean
C
Harmonic Mean
D
Mode
💡
Step-by-Step Explanation & Concept Rationale
The geometric mean ($G = (\prod x_i)^{1/n}$) is mathematically designed for averaging percentages, percentage changes, ratios, compounding financial interest, and multi-year demographic growth rates.
Q. 5
Statistics
Difficulty: Medium
(1 Mark)
For any set of positive, distinct real numbers ($x_i > 0$), what is the universal mathematical inequality relationship between the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM)?
A
AM > GM > HM
✓ Correct
B
HM > GM > AM
C
GM > AM > HM
D
AM = GM > HM
💡
Step-by-Step Explanation & Concept Rationale
The classical AM-GM-HM inequality states that for positive, non-identical numbers, the Arithmetic Mean is strictly greater than the Geometric Mean, which in turn is strictly greater than the Harmonic Mean: $AM > GM > HM$. They are equal only if all values are identical.
Q. 6
Statistics
Difficulty: Easy
(1 Mark)
The algebraic sum of the deviations of a set of observations from their arithmetic mean ($\sum (x_i - \bar{x})$) is ALWAYS equal to:
A
Zero (0)
✓ Correct
B
One (1)
C
The variance
D
The standard error
💡
Step-by-Step Explanation & Concept Rationale
A fundamental mathematical property of the arithmetic mean is that the sum of positive and negative deviations about the mean cancels out exactly: $\sum (x_i - \bar{x}) = \sum x_i - n\bar{x} = n\bar{x} - n\bar{x} = 0$.
Q. 7
Statistics
Difficulty: Medium
(1 Mark)
The sum of squared deviations of a set of values ($\sum (x_i - A)^2$) is minimized when $A$ is chosen as the:
A
Arithmetic Mean
✓ Correct
B
Median
C
Mode
D
Geometric Mean
💡
Step-by-Step Explanation & Concept Rationale
By differential calculus, $\sum (x_i - A)^2$ attains its global absolute minimum when $A = \bar{x}$ (the arithmetic mean). Conversely, the sum of absolute deviations $\sum |x_i - A|$ is minimized when $A$ is the median.
Q. 8
Statistics
Difficulty: Easy
(1 Mark)
In an ogive (cumulative frequency curve), the value on the horizontal axis corresponding to the intersection point of the 'less than' ogive and 'more than' ogive gives the:
A
Median
✓ Correct
B
Arithmetic Mean
C
Mode
D
Variance
💡
Step-by-Step Explanation & Concept Rationale
The intersection of the 'less than' cumulative frequency curve ($N/2$ from below) and the 'more than' cumulative frequency curve ($N/2$ from above) occurs precisely at the 50th percentile, which defines the Median.
Q. 9
Statistics
Difficulty: Easy
(1 Mark)
What relative measure of dispersion expresses the standard deviation as a percentage of the arithmetic mean to compare variability between datasets with different units of measurement?
A
Coefficient of Variation (CV)
✓ Correct
B
Variance
C
Standard Error
D
Range
💡
Step-by-Step Explanation & Concept Rationale
The Coefficient of Variation ($CV = (s / \bar{x}) \times 100\%$) is a dimensionless relative measure of dispersion introduced by Karl Pearson, allowing direct comparison of risk or variability across datasets with different units or disparate means.
Q. 10
Statistics
Difficulty: Easy
(1 Mark)
If every single observation in a dataset is multiplied by a constant $c = 5$, how does the standard deviation of the transformed dataset change?
A
It is multiplied by 5
✓ Correct
B
It is multiplied by 25
C
It remains unchanged
D
It increases by 5 units
💡
Step-by-Step Explanation & Concept Rationale
Multiplying data values by a constant $c$ scales the standard deviation by $|c|$ (so $s_{new} = 5 \times s_{old}$), while the variance scales by $c^2$ (so $s^2_{new} = 25 \times s^2_{old}$). Adding a constant changes neither standard deviation nor variance.
Q. 11
Statistics
Difficulty: Easy
(1 Mark)
If every observation in a dataset is increased by adding a constant $k = 10$, what happens to the standard deviation and the variance?
A
Both standard deviation and variance remain completely unchanged
✓ Correct
B
Standard deviation increases by 10
C
Variance increases by 100
D
Standard deviation is multiplied by 10
💡
Step-by-Step Explanation & Concept Rationale
Measures of dispersion (range, IQR, mean deviation, variance, standard deviation) are independent of change of origin. Adding or subtracting a constant shifts all points equally without altering their spread relative to each other.
Q. 12
Statistics
Difficulty: Easy
(1 Mark)
The Interquartile Range (IQR) is mathematically defined as the difference between which two percentiles?
A
Third Quartile ($Q_3$ / 75th percentile) and First Quartile ($Q_1$ / 25th percentile)
✓ Correct
B
90th percentile and 10th percentile
C
Maximum and Minimum value
D
Mean and Median
💡
Step-by-Step Explanation & Concept Rationale
The Interquartile Range ($IQR = Q_3 - Q_1$) spans the central 50% of ordered observations, serving as an outlier-resistant measure of statistical spread.
Q. 13
Statistics
Difficulty: Medium
(1 Mark)
In John Tukey's standard Box-and-Whisker plot, an observation is classified as an 'outlier' if it lies beyond:
A
$Q_1 - 1.5 \times IQR$ or $Q_3 + 1.5 \times IQR$
✓ Correct
B
$Mean \pm 1.0 \times SD$
C
$Q_1 - 0.5 \times IQR$ or $Q_3 + 0.5 \times IQR$
D
Beyond the 95th percentile only
💡
Step-by-Step Explanation & Concept Rationale
Tukey established the 'inner fences' of a boxplot at $[Q_1 - 1.5 \cdot IQR, Q_3 + 1.5 \cdot IQR]$. Data points falling outside these fences are plotted individually as suspected outliers.
Q. 14
Statistics
Difficulty: Easy
(1 Mark)
What is the value of the first central moment ($\mu_1$) about the arithmetic mean for any probability distribution?
A
Exactly 0
✓ Correct
B
Exactly 1
C
The variance
D
The standard deviation
💡
Step-by-Step Explanation & Concept Rationale
The $r$-th central moment is defined as $\mu_r = E[(X - \mu)^r]$. For $r = 1$, $\mu_1 = E[X - \mu] = E[X] - \mu = \mu - \mu = 0$ for every valid distribution.
Q. 15
Statistics
Difficulty: Easy
(1 Mark)
The second central moment about the mean ($\mu_2 = E[(X - \mu)^2]$) represents which fundamental statistical parameter?
A
Variance
✓ Correct
B
Standard error
C
Skewness
D
Kurtosis
💡
Step-by-Step Explanation & Concept Rationale
By definition, the second central moment $\mu_2$ is the variance ($\sigma^2$), measuring dispersion of observations around the population mean.
Q. 16
Statistics
Difficulty: Medium
(1 Mark)
Karl Pearson's moment coefficient of skewness ($\beta_1$) is calculated using the second and third central moments as:
A
$\beta_1 = \mu_3^2 / \mu_2^3$
✓ Correct
B
$\beta_1 = \mu_3 / \mu_2$
C
$\beta_1 = \mu_4 / \mu_2^2$
D
$\beta_1 = \mu_2^3 / \mu_3^2$
💡
Step-by-Step Explanation & Concept Rationale
Pearson defined $\beta_1 = \mu_3^2 / \mu_2^3$. The standardized skewness coefficient is $\gamma_1 = \sqrt{\beta_1} = \mu_3 / \sigma^3$. For a symmetric distribution, $\mu_3 = 0$ and $\beta_1 = 0$.
Q. 17
Statistics
Difficulty: Easy
(1 Mark)
A continuous frequency curve has a long tail stretching to the right side (positive values). What is the relative order of the three measures of central location?
A
Mean > Median > Mode
✓ Correct
B
Mode > Median > Mean
C
Median > Mean > Mode
D
Mean = Median = Mode
💡
Step-by-Step Explanation & Concept Rationale
In a positively (right) skewed distribution, extreme high values pull the mean farthest into the right tail, resulting in: $\text{Mean} > \text{Median} > \text{Mode}$. In a negatively (left) skewed distribution: $\text{Mode} > \text{Median} > \text{Mean}$.
Q. 18
Statistics
Difficulty: Easy
(1 Mark)
Karl Pearson's coefficient of kurtosis is defined as $\beta_2 = \mu_4 / \mu_2^2$. What is the value of $\beta_2$ for a perfectly normal (mesokurtic) distribution?
A
3.0 (or excess kurtosis $\gamma_2 = \beta_2 - 3 = 0$)
✓ Correct
B
0.0
C
1.0
D
0.5
💡
Step-by-Step Explanation & Concept Rationale
For any Gaussian (normal) distribution, the ratio of the fourth central moment to the square of the second central moment is exactly $\beta_2 = 3$. Excess kurtosis is defined as $\gamma_2 = \beta_2 - 3 = 0$.
Q. 19
Statistics
Difficulty: Easy
(1 Mark)
A distribution that is more peaked around its center than a normal distribution and has fatter, heavier tails (with $\beta_2 > 3$) is classified as:
A
Leptokurtic
✓ Correct
B
Platykurtic
C
Mesokurtic
D
Symmetrical
💡
Step-by-Step Explanation & Concept Rationale
Distributions with $\beta_2 > 3$ (excess kurtosis $> 0$) are termed 'leptokurtic' (sharp peak, heavy tails). Distributions with $\beta_2 < 3$ are 'platykurtic' (flatter peak, lighter tails), and $\beta_2 = 3$ is 'mesokurtic'.
Q. 20
Statistics
Difficulty: Easy
(1 Mark)
Two events $A$ and $B$ are said to be 'mutually exclusive' (disjoint) if and only if:
A
They cannot occur simultaneously ($P(A \cap B) = 0$)
✓ Correct
B
$P(A \cap B) = P(A) \times P(B)$
C
$P(A) + P(B) = 1$
D
$P(A|B) = P(A)$
💡
Step-by-Step Explanation & Concept Rationale
Mutually exclusive events have an empty intersection ($A \cap B = \emptyset$). Hence, the probability that both occur simultaneously is zero: $P(A \cap B) = 0$.
Q. 21
Statistics
Difficulty: Easy
(1 Mark)
Two events $A$ and $B$ are statistically 'independent' if and only if:
A
$P(A \cap B) = P(A) \times P(B)$
✓ Correct
B
$P(A \cap B) = 0$
C
$P(A \cup B) = P(A) + P(B)$
D
$P(A) = P(B)$
💡
Step-by-Step Explanation & Concept Rationale
By definition, events $A$ and $B$ are statistically independent when the occurrence of one does not affect the probability of the other, meaning $P(A|B) = P(A)$, which yields the multiplicative rule: $P(A \cap B) = P(A) \times P(B)$.
Q. 22
Statistics
Difficulty: Easy
(1 Mark)
If two fair six-sided dice are rolled simultaneously, what is the probability of obtaining a sum equal to 7?
A
6 / 36 (1 / 6)
✓ Correct
B
1 / 36
C
7 / 36
D
1 / 12
💡
Step-by-Step Explanation & Concept Rationale
There are $6 \times 6 = 36$ equally likely outcomes. The outcomes summing to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)—totaling 6 favorable outcomes. Thus, $P(\text{Sum} = 7) = 6/36 = 1/6$.
Q. 23
Statistics
Difficulty: Medium
(1 Mark)
Bayes' Theorem provides a mathematical framework for calculating:
A
Posterior probability given prior probabilities and conditional likelihoods
✓ Correct
B
The arithmetic mean of a sample
C
The exact range of a continuous variable
D
The degrees of freedom in an ANOVA table
💡
Step-by-Step Explanation & Concept Rationale
Bayes' Theorem ($P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum P(B|A_j)P(A_j)}$) updates the prior probability of an event $A_i$ upon observing new empirical evidence $B$ to calculate the revised posterior probability.
Q. 24
Statistics
Difficulty: Easy
(1 Mark)
For a discrete random variable $X$ with probability mass function $P(X = x)$, what two mathematical conditions must be satisfied?
A
$P(X = x) \ge 0$ for all $x$, and $\sum P(X = x) = 1$
✓ Correct
B
$P(X = x) \le 0$, and $\sum P(X = x) = 0$
C
$P(X = x) > 1$, and $\sum P(X = x) = 100$
D
$\sum P(X = x) = \infty$
💡
Step-by-Step Explanation & Concept Rationale
The axioms of probability mandate that every individual probability mass must be non-negative ($0 \le P(X=x) \le 1$), and the sum of probabilities across the entire sample space must equal exactly 1.
Q. 25
Statistics
Difficulty: Medium
(1 Mark)
For a continuous random variable $X$ with probability density function $f(x)$, what is the probability that $X$ takes on any single exact point value $c$ ($P(X = c)$)?
A
Exactly 0
✓ Correct
B
Exactly 1
C
$f(c)$
D
$0.5$
💡
Step-by-Step Explanation & Concept Rationale
For continuous random variables, probabilities correspond to areas under the density curve: $P(a \le X \le b) = \int_a^b f(x) dx$. Because the area over a single dimensionless point is zero, $P(X = c) = \int_c^c f(x) dx = 0$.
Study Stream Progress:
Showing 25 of 60 Questions (42%)
Jump to:
🎉
All 60 Solved Questions in Descriptive Statistics & Probability Loaded!
You have studied the complete question pool with step-by-step rationales. Ready to test yourself?
Q. 1
Statistics
Difficulty: Easy
(1 Mark)
In a Binomial experiment with $n$ independent trials and probability of success $p$ on each trial, what are the theoretical Mean and Variance?
A
Mean = $np$, Variance = $np(1-p)$
✓ Correct
B
Mean = $np$, Variance = $\sqrt{np}$
C
Mean = $p$, Variance = $p(1-p)$
D
Mean = $n/p$, Variance = $n(1-p)$
💡
Step-by-Step Explanation & Concept Rationale
For a Binomial distribution $B(n, p)$, the expected value is $E[X] = np$, and the variance is $Var(X) = npq = np(1-p)$. Note that the variance is strictly less than the mean since $0 < q < 1$.
Q. 2
Statistics
Difficulty: Easy
(1 Mark)
Which probability distribution has the unique property that its theoretical Mean is ALWAYS strictly equal to its Variance ($\mu = \sigma^2 = \lambda$)?
A
Poisson Distribution
✓ Correct
B
Binomial Distribution
C
Normal Distribution
D
Geometric Distribution
💡
Step-by-Step Explanation & Concept Rationale
In a Poisson distribution ($P(X=x) = e^{-\lambda}\lambda^x / x!$), the single parameter $\lambda$ represents both the expected value (mean) and the variance: $E[X] = Var(X) = \lambda$ (equidispersive property).
Q. 3
Statistics
Difficulty: Medium
(1 Mark)
Under what limiting conditions does the Binomial distribution converge asymptotically to a Poisson distribution?
A
When the number of trials $n \to \infty$, probability of success $p \to 0$, such that $np = \lambda$ remains constant
✓ Correct
B
When $n$ is small and $p = 0.5$
C
When $p \to 1$
D
When $n \to 0$
💡
Step-by-Step Explanation & Concept Rationale
The Poisson distribution arises as the limit of the Binomial distribution when trials $n$ become arbitrarily large and probability of success $p$ becomes very small (rare events), while the product $np = \lambda$ stays moderate and finite.
Q. 4
Statistics
Difficulty: Easy
(1 Mark)
What continuous probability distribution is characterized by a bell-shaped, symmetrical probability density curve defined entirely by its mean ($\mu$) and standard deviation ($\sigma$)?
A
Normal (Gaussian) Distribution
✓ Correct
B
Exponential Distribution
C
Chi-Square Distribution
D
F-Distribution
💡
Step-by-Step Explanation & Concept Rationale
The Normal distribution $N(\mu, \sigma^2)$ is continuous, symmetric about its mean, has points of inflection at $\mu \pm \sigma$, and its total area equals 1, serving as the cornerstone of classical statistical inference.
Q. 5
Statistics
Difficulty: Easy
(1 Mark)
According to the 'Empirical Rule' (68-95-99.7 Rule) for a Normal distribution, what percentage of observations lies within $\mu \pm 2\sigma$?
A
Approximately 95.45%
✓ Correct
B
Approximately 68.27%
C
Approximately 99.73%
D
Exactly 50.00%
💡
Step-by-Step Explanation & Concept Rationale
In any Normal distribution: ~68.27% of values fall within $\mu \pm 1\sigma$, ~95.45% fall within $\mu \pm 2\sigma$, and ~99.73% fall within $\mu \pm 3\sigma$.
Q. 6
Statistics
Difficulty: Easy
(1 Mark)
What are the numerical parameters (Mean and Variance) of the Standard Normal Distribution ($Z$)?
A
Mean $\mu = 0$, Variance $\sigma^2 = 1$
✓ Correct
B
Mean $\mu = 1$, Variance $\sigma^2 = 0$
C
Mean $\mu = 1$, Variance $\sigma^2 = 1$
D
Mean $\mu = 0$, Variance $\sigma^2 = 0$
💡
Step-by-Step Explanation & Concept Rationale
Standardizing a normal variable $X \sim N(\mu, \sigma^2)$ using the $Z$-transformation ($Z = (X - \mu)/\sigma$) converts it into the Standard Normal distribution $Z \sim N(0, 1)$ with mean 0 and variance 1.
Q. 7
Statistics
Difficulty: Medium
(1 Mark)
At what values on the horizontal axis do the two points of inflection occur on a standard Normal density curve $N(0, 1)$?
A
$Z = -1$ and $Z = +1$
✓ Correct
B
$Z = -2$ and $Z = +2$
C
$Z = -3$ and $Z = +3$
D
$Z = 0$
💡
Step-by-Step Explanation & Concept Rationale
The points of inflection (where curvature changes from concave downward to concave upward, $f''(x) = 0$) of a normal distribution occur at exactly one standard deviation from the mean: $x = \mu \pm \sigma$ (for standard normal, $Z = \pm 1$).
Q. 8
Statistics
Difficulty: Easy
(1 Mark)
The Central Limit Theorem (CLT) states that as the sample size $n$ becomes sufficiently large ($n \ge 30$), the sampling distribution of the sample mean ($\bar{X}$):
A
Approaches a Normal distribution, regardless of the underlying population's original distribution shape
✓ Correct
B
Approaches a Uniform distribution
C
Becomes skewed to the right
D
Collapses to zero variance
💡
Step-by-Step Explanation & Concept Rationale
The Central Limit Theorem is fundamental in statistics: the sampling distribution of the sample mean of independent random variables approaches a normal distribution with mean $\mu$ and variance $\sigma^2/n$ as $n \to \infty$, even if the parent population is highly skewed or non-normal.
Q. 9
Statistics
Difficulty: Easy
(1 Mark)
The standard deviation of the sampling distribution of a statistic (such as the sample mean $\bar{X}$) is formally termed the:
A
Standard Error
✓ Correct
B
Sampling Bias
C
Variance of population
D
Mean Deviation
💡
Step-by-Step Explanation & Concept Rationale
The standard deviation of an estimator's sampling distribution is called the Standard Error (e.g., $SE(\bar{x}) = \sigma / \sqrt{n}$ for the sample mean).
Q. 10
Statistics
Difficulty: Easy
(1 Mark)
If a simple random sample of size $n = 64$ is drawn from a large population with standard deviation $\sigma = 24$, what is the Standard Error of the sample mean ($SE(\bar{X})$)?
A
3.0
✓ Correct
B
0.375
C
8.0
D
192.0
💡
Step-by-Step Explanation & Concept Rationale
$SE(\bar{X}) = \sigma / \sqrt{n} = 24 / \sqrt{64} = 24 / 8 = 3.0$.
Q. 11
Statistics
Difficulty: Medium
(1 Mark)
When sampling without replacement from a finite population of size $N$, what correction factor must be applied to the Standard Error formula?
A
Finite Population Correction factor: $\sqrt{\frac{N - n}{N - 1}}$
✓ Correct
B
$N / (N - n)$
C
$\sqrt{N / n}$
D
$(N - 1) / N$
💡
Step-by-Step Explanation & Concept Rationale
When sample size $n$ exceeds 5% of a finite population $N$, the Finite Population Correction (FPC) factor $\sqrt{\frac{N - n}{N - 1}}$ reduces standard error to account for the reduction in sampling variability caused by not replacing drawn units.
Q. 12
Statistics
Difficulty: Easy
(1 Mark)
If $Z_1, Z_2, \dots, Z_k$ are $k$ independent Standard Normal random variables ($Z_i \sim N(0, 1)$), what is the probability distribution of the sum of their squares ($Y = \sum_{i=1}^k Z_i^2$)?
A
Chi-Square distribution with $k$ degrees of freedom ($\chi^2_k$)
✓ Correct
B
Student's $t$-distribution with $k$ degrees of freedom
C
F-distribution
D
Lognormal distribution
💡
Step-by-Step Explanation & Concept Rationale
By definition, the sum of the squares of $k$ independent standard normal variables follows a Chi-Square distribution with $k$ degrees of freedom: $\sum_{i=1}^k Z_i^2 \sim \chi^2_{(k)}$.
Q. 13
Statistics
Difficulty: Medium
(1 Mark)
What are the theoretical Mean and Variance of a Chi-Square distribution with $k$ degrees of freedom?
A
Mean = $k$, Variance = $2k$
✓ Correct
B
Mean = $k$, Variance = $k$
C
Mean = $2k$, Variance = $k$
D
Mean = 0, Variance = $k$
💡
Step-by-Step Explanation & Concept Rationale
For a Chi-Square distribution with $k$ degrees of freedom, the expected value is $E[\chi^2] = k$ and the variance is $Var(\chi^2) = 2k$.
Q. 14
Statistics
Difficulty: Easy
(1 Mark)
Student's $t$-distribution was developed in 1908 by William Sealy Gosset (under the pseudonym 'Student') for which primary practical purpose?
A
Making inferences about the population mean when sample size is small ($n < 30$) and the population variance $\sigma^2$ is unknown
✓ Correct
B
Analyzing non-linear rocket trajectories
C
Testing differences between categorical ranks
D
Replacing the binomial distribution
💡
Step-by-Step Explanation & Concept Rationale
Gosset developed the $t$-distribution while working at the Guinness brewery to handle small samples ($n < 30$) where the true population standard deviation $\sigma$ is unknown and must be estimated by the sample standard deviation $s$.
Q. 15
Statistics
Difficulty: Easy
(1 Mark)
How does the curve of Student's $t$-distribution compare visually and mathematically to the Standard Normal ($Z$) curve?
A
It is symmetric and bell-shaped like $Z$, but has a lower peak and thicker, heavier tails, reflecting greater sampling variability
✓ Correct
B
It is skewed to the right
C
It is completely rectangular
D
It has a higher peak and thinner tails
💡
Step-by-Step Explanation & Concept Rationale
The $t$-distribution is symmetric about 0 but exhibits heavier tails and greater spread than the standard normal distribution because estimating $\sigma$ with $s$ introduces additional sampling uncertainty. As degrees of freedom $df \to \infty$, $t$ approaches $Z$.
Q. 16
Statistics
Difficulty: Hard
(1 Mark)
What is the ratio that defines Snedecor's $F$-distribution?
A
The ratio of two independent Chi-Square variables, each divided by its respective degrees of freedom: $F = (\chi^2_1 / d_1) / (\chi^2_2 / d_2)$
✓ Correct
B
The ratio of two normal variables
C
The product of two $t$-distributions
D
The ratio of mean to variance
💡
Step-by-Step Explanation & Concept Rationale
An $F$-distributed variable is defined as the ratio of two independent chi-square distributed random variables, each scaled by its degrees of freedom: $F = \frac{\chi^2_1 / d_1}{\chi^2_2 / d_2} \sim F_{(d_1, d_2)}$, used in ANOVA variance ratio tests.
Q. 17
Statistics
Difficulty: Medium
(1 Mark)
What continuous probability distribution possesses the unique 'Memoryless Property' ($P(X > s + t | X > s) = P(X > t)$), frequently used to model waiting times between independent Poisson events?
A
Exponential Distribution
✓ Correct
B
Normal Distribution
C
Gamma Distribution
D
Weibull Distribution
💡
Step-by-Step Explanation & Concept Rationale
The Exponential distribution is the only continuous distribution with the memoryless property: the probability that a component lasts an additional $t$ hours given that it has already survived $s$ hours is independent of $s$.
Q. 18
Statistics
Difficulty: Easy
(1 Mark)
What discrete probability distribution models the number of independent Bernoulli trials required to achieve the FIRST success?
A
Geometric Distribution
✓ Correct
B
Binomial Distribution
C
Poisson Distribution
D
Hypergeometric Distribution
💡
Step-by-Step Explanation & Concept Rationale
The Geometric distribution ($P(X = k) = (1-p)^{k-1} p$) models the trial number $k$ on which the very first success occurs in a sequence of independent Bernoulli trials.
Q. 19
Statistics
Difficulty: Medium
(1 Mark)
In finite sampling without replacement from a dichotomous population (e.g., drawing defective items from a batch without putting them back), which distribution governs the number of successes?
A
Hypergeometric Distribution
✓ Correct
B
Binomial Distribution
C
Poisson Distribution
D
Uniform Distribution
💡
Step-by-Step Explanation & Concept Rationale
When sampling without replacement, trials are dependent because probabilities change with each draw. The Hypergeometric distribution accurately computes the probability of $k$ successes from a finite population of size $N$ with $K$ successes.
Q. 20
Statistics
Difficulty: Hard
(1 Mark)
What probability distribution is widely utilized in survival analysis, industrial reliability engineering, and wind speed modeling due to its flexible shape parameter that can model increasing, constant, or decreasing hazard failure rates?
A
Weibull Distribution
✓ Correct
B
Binomial Distribution
C
Uniform Distribution
D
Cauchy Distribution
💡
Step-by-Step Explanation & Concept Rationale
The Weibull distribution is widely used in reliability analysis and engineering lifetime predictions because by adjusting its shape parameter $\beta$, it can model infant mortality ($\beta < 1$), constant random failures ($\beta = 1$, exponential), or wear-out aging failures ($\beta > 1$).
Q. 21
Statistics
Difficulty: Medium
(1 Mark)
In a continuous Uniform distribution $U(a, b)$, what is the formula for the variance?
A
$(b - a)^2 / 12$
✓ Correct
B
$(a + b) / 2$
C
$(b - a) / 2$
D
$(b - a)^2 / 2$
💡
Step-by-Step Explanation & Concept Rationale
For $X \sim U(a, b)$, the probability density is $f(x) = 1/(b-a)$. The expected value is $E[X] = (a+b)/2$, and the variance is $Var(X) = (b-a)^2 / 12$.
Q. 22
Statistics
Difficulty: Easy
(1 Mark)
What is the relationship between the standard normal distribution ($Z$) and Student's $t$-distribution when the sample size approaches infinity ($n \to \infty$)?
A
The $t$-distribution becomes mathematically identical to the standard normal distribution
✓ Correct
B
The $t$-distribution becomes uniform
C
The $t$-distribution variance becomes 0
D
The $t$-distribution becomes skewed
💡
Step-by-Step Explanation & Concept Rationale
As degrees of freedom approach infinity ($df \to \infty$), the sample variance $s^2$ converges in probability to the true population variance $\sigma^2$, and the Student's $t$-distribution curve converges exactly to the Standard Normal $N(0, 1)$ curve.
Q. 23
Statistics
Difficulty: Easy
(1 Mark)
In sampling theory, what sampling technique divides a heterogeneous population into non-overlapping homogeneous subpopulations (strata) and draws independent random samples from each stratum?
A
Stratified Random Sampling
✓ Correct
B
Cluster Sampling
C
Systematic Sampling
D
Convenience Sampling
💡
Step-by-Step Explanation & Concept Rationale
Stratified random sampling ensures proportional or optimal representation of distinct subgroups (e.g., urban vs rural, income brackets), minimizing sampling variance by exploiting homogeneity within strata.
Q. 24
Statistics
Difficulty: Medium
(1 Mark)
How does Cluster Sampling differ fundamentally from Stratified Random Sampling?
A
In cluster sampling, the population is divided into heterogeneous clusters, and a random sample of entire clusters is selected, whereas in stratified sampling, samples are drawn from every stratum
✓ Correct
B
Cluster sampling uses no random selection
C
Stratified sampling is used only for animals
D
Cluster sampling requires a complete census of all people
💡
Step-by-Step Explanation & Concept Rationale
In stratified sampling, every stratum is sampled (homogeneous within, heterogeneous between). In cluster sampling, clusters are mini-representations of the population (heterogeneous within, homogeneous between), and only selected whole clusters are surveyed to reduce travel and field costs.
Q. 25
Statistics
Difficulty: Easy
(1 Mark)
In Systematic Sampling, if a researcher selects every $k$-th element from an ordered sampling frame of size $N$ to obtain a sample of size $n$, what is the sampling interval $k$?
A
$k = N / n$
✓ Correct
B
$k = n / N$
C
$k = N \times n$
D
$k = \sqrt{N}$
💡
Step-by-Step Explanation & Concept Rationale
The sampling interval $k$ is calculated as $k = N / n$. A random integer between 1 and $k$ is selected as the random start, and every $k$-th subsequent item is sampled systematically.
Study Stream Progress:
Showing 25 of 50 Questions (50%)
Jump to:
🎉
All 50 Solved Questions in Distributions & Sampling Theory Loaded!
You have studied the complete question pool with step-by-step rationales. Ready to test yourself?
Q. 1
Statistics
Difficulty: easy
(1 Mark)
In statistical hypothesis testing, what is a 'Type I Error'?
A
Failing to reject the null hypothesis $H_0$ when it is false
B
Rejecting the null hypothesis $H_0$ when it is actually true
✓ Correct
C
Accepting the alternative hypothesis when sample size is too large
D
Making a calculation error during arithmetic computation
💡
Step-by-Step Explanation & Concept Rationale
A Type I error (probability $\alpha$, the significance level) occurs when a test rejects a true null hypothesis (a 'false positive').
Q. 2
Statistics
Difficulty: easy
(1 Mark)
What is a 'Type II Error' (denoted by $\beta$)?
A
Rejecting $H_0$ when $H_0$ is true
B
Failing to reject (accepting) the null hypothesis $H_0$ when it is actually false
✓ Correct
C
Choosing an incorrect statistical significance level
D
Using a one-tailed test instead of a two-tailed test
💡
Step-by-Step Explanation & Concept Rationale
A Type II error (probability $\beta$) occurs when the test fails to detect a genuine effect or difference, retaining a false null hypothesis (a 'false negative').
Q. 3
Statistics
Difficulty: easy
(1 Mark)
What is the 'Power of a Test' mathematically defined as?
A
$\alpha$
B
$1 - \beta$ (the probability of correctly rejecting a false null hypothesis)
✓ Correct
C
$1 - \alpha$
D
$\alpha / \beta$
💡
Step-by-Step Explanation & Concept Rationale
The Power of a test is $1 - \beta = P(\text{Reject } H_0 \mid H_0 \text{ is false})$, measuring the test's ability to detect a true effect of a given magnitude.
Q. 4
Statistics
Difficulty: medium
(1 Mark)
What is the definition of the 'p-value' in statistical testing?
A
The probability that the null hypothesis is true
B
The probability, assuming the null hypothesis is true, of observing a test statistic as extreme as, or more extreme than, the one actually calculated from the sample
✓ Correct
C
The probability that the alternative hypothesis is false
D
The probability of committing a Type II error
💡
Step-by-Step Explanation & Concept Rationale
The p-value is $P(T(\mathbf{X}) \ge t_{obs} \mid H_0)$. It is NOT the posterior probability that $H_0$ is true, but the probability of seeing such extreme data under the assumption that $H_0$ holds.
Q. 5
Statistics
Difficulty: easy
(1 Mark)
If the calculated p-value is less than the chosen significance level $\alpha$ ($p < \alpha$), what is the appropriate statistical decision?
A
Fail to reject the null hypothesis $H_0$
B
Reject the null hypothesis $H_0$ in favor of the alternative hypothesis $H_1$
✓ Correct
C
Accept both hypotheses simultaneously
D
Discard the sample data and restart the study
💡
Step-by-Step Explanation & Concept Rationale
When $p < \alpha$, the observed data is sufficiently improbable under $H_0$, leading to rejection of $H_0$ at the $\alpha$ significance level.
Q. 6
Statistics
Difficulty: medium
(1 Mark)
When comparing the means of two independent normal populations with UNKNOWN and UNEQUAL variances ($\sigma_1^2 \ne \sigma_2^2$), which test is appropriate?
A
Standard Student's two-sample pooled t-test
B
Welch's Satterthwaite t-test (Unequal Variance t-test)
✓ Correct
C
Paired t-test
D
Chi-Square test of goodness of fit
💡
Step-by-Step Explanation & Concept Rationale
Welch's t-test (the Behrens-Fisher problem solution) does not assume equal variances, adjusting the degrees of freedom using the Welch-Satterthwaite approximation.
Q. 7
Statistics
Difficulty: easy
(1 Mark)
What is the degree of freedom formula for the pooled two-sample Student's t-test with sample sizes $n_1$ and $n_2$ when equal variances ($\sigma_1^2 = \sigma_2^2$) are assumed?
A
$n_1 + n_2 - 1$
B
$n_1 + n_2 - 2$
✓ Correct
C
$n_1 + n_2$
D
$\min(n_1 - 1, n_2 - 1)$
💡
Step-by-Step Explanation & Concept Rationale
Pooling variance from two samples of sizes $n_1$ and $n_2$ uses $n_1 - 1$ degrees of freedom from the first and $n_2 - 1$ from the second, giving total $df = n_1 + n_2 - 2$.
Q. 8
Statistics
Difficulty: easy
(1 Mark)
When should a 'Paired Samples t-test' be used instead of an independent two-sample t-test?
A
When sample sizes $n_1$ and $n_2$ are unequal
B
When observations in the two groups are naturally paired or matched (e.g., pre-disaster vs. post-disaster measurements on the same geographical units)
✓ Correct
C
When the data follows a Poisson distribution
D
When the population variance is known exactly
💡
Step-by-Step Explanation & Concept Rationale
A paired t-test analyzes the differences $d_i = X_{1i} - X_{2i}$ within matched pairs, controlling for unit-level heterogeneity and increasing statistical power.
Q. 9
Statistics
Difficulty: easy
(1 Mark)
What is the degrees of freedom for a Paired t-test with $n$ matched pairs of observations?
A
$n - 1$
✓ Correct
B
$2n - 2$
C
$2n - 1$
D
$n$
💡
Step-by-Step Explanation & Concept Rationale
Because the paired test is mathematically equivalent to a one-sample t-test performed on the single sample of $n$ differences ($d_1, d_2, \dots, d_n$), the degrees of freedom is $n - 1$.
Q. 10
Statistics
Difficulty: easy
(1 Mark)
What is the test statistic for Pearson's Chi-Square Goodness of Fit test with $k$ categories, observed frequencies $O_i$, and expected frequencies $E_i$?
A
$\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}$
✓ Correct
B
$\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{O_i}$
C
$\chi^2 = \sum_{i=1}^k \frac{O_i - E_i}{\sqrt{E_i}}$
D
$\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)}{k}$
💡
Step-by-Step Explanation & Concept Rationale
Pearson's Chi-square statistic measures normalized squared deviations: $\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}$, which asymptotically follows a $\chi^2_{(k - 1 - p)}$ distribution under $H_0$ (where $p$ is the number of estimated parameters).
Q. 11
Statistics
Difficulty: easy
(1 Mark)
What is the degrees of freedom for Pearson's Chi-Square Test of Independence in an $r \times c$ contingency table?
A
$r \times c$
B
$(r - 1)(c - 1)$
✓ Correct
C
$r + c - 1$
D
$(r - 1) + (c - 1)$
💡
Step-by-Step Explanation & Concept Rationale
In an $r \times c$ contingency table with fixed marginal totals, there are $(r - 1)(c - 1)$ independent cell entries, giving degrees of freedom $df = (r - 1)(c - 1)$.
Q. 12
Statistics
Difficulty: medium
(1 Mark)
What is the rule of thumb regarding minimum expected frequencies in Pearson's Chi-Square test?
A
All expected frequencies $E_i$ should ideally be $\ge 5$ (or at least 80% of cells $\ge 5$ with none $< 1$)
✓ Correct
B
Observed frequencies must all exceed 100
C
Expected frequencies must be integers
D
No cell can have an observed count of zero
💡
Step-by-Step Explanation & Concept Rationale
Cochran's guideline states that the Chi-square continuous approximation to the discrete multinomial distribution becomes unreliable if expected frequencies fall below 5.
Q. 13
Statistics
Difficulty: medium
(1 Mark)
What is 'Yates' Correction for Continuity' in a $2 \times 2$ contingency table?
A
Subtracting 0.5 from the absolute difference $|O - E|$ before squaring: $\chi_{Yates}^2 = \sum \frac{(|O_i - E_i| - 0.5)^2}{E_i}$
✓ Correct
B
Multiplying the Chi-square statistic by 2
C
Dividing the p-value by the sample size
D
Adding 1 to every cell count
💡
Step-by-Step Explanation & Concept Rationale
Yates' continuity correction subtracts 0.5 from $|O - E|$ in $2 \times 2$ tables ($df = 1$) to adjust for approximating a discrete binomial/hypergeometric distribution with a continuous Chi-square distribution.
Q. 14
Statistics
Difficulty: easy
(1 Mark)
When cell counts in a $2 \times 2$ contingency table are very small ($E_i < 5$), which exact test should be used instead of the Chi-Square test?
A
Fisher's Exact Test
✓ Correct
B
Student's t-test
C
Welch's t-test
D
Kruskal-Wallis test
💡
Step-by-Step Explanation & Concept Rationale
Fisher's Exact Test calculates exact probabilities using the Hypergeometric distribution conditioning on the observed row and column margins, without relying on large-sample asymptotic approximations.
Q. 15
Statistics
Difficulty: easy
(1 Mark)
What hypothesis does the F-Test for equality of two variances test ($H_0: \sigma_1^2 = \sigma_2^2$ vs $H_1: \sigma_1^2 \ne \sigma_2^2$)?
A
Whether two sample means are equal
B
Whether two independent normal populations have identical variances, using $F = \frac{S_1^2}{S_2^2}$
✓ Correct
C
Whether regression residuals are normally distributed
D
Whether two categorical variables are independent
💡
Step-by-Step Explanation & Concept Rationale
The two-sample variance F-test evaluates $F = \frac{S_1^2}{S_2^2} \sim F(n_1-1, n_2-1)$ under $H_0: \sigma_1^2 = \sigma_2^2$, highly sensitive to departures from normality.
Q. 16
Statistics
Difficulty: hard
(1 Mark)
Why is the F-test for equality of variances considered 'Non-Robust' to departures from normality?
A
Because it is computed using fractions
B
Because moderate departures from normality (skewness or heavy tails) severely distort the Type I error rate of the F-test; Levene's test or Brown-Forsythe test is preferred for non-normal data
✓ Correct
C
Because it requires equal sample sizes
D
Because the F-distribution only accepts integer degrees of freedom
💡
Step-by-Step Explanation & Concept Rationale
The variance ratio F-test is notoriously non-robust: non-normality leads to substantial inflation of false positive rates. Levene's test (based on deviations from the mean) or Brown-Forsythe (deviations from median) is robust.
Q. 17
Statistics
Difficulty: easy
(1 Mark)
What is the non-parametric equivalent of the independent two-sample Student's t-test?
A
Wilcoxon Signed-Rank Test
B
Mann-Whitney U Test (Wilcoxon Rank-Sum Test)
✓ Correct
C
Kruskal-Wallis H Test
D
Friedman Test
💡
Step-by-Step Explanation & Concept Rationale
The Mann-Whitney U test (Wilcoxon rank-sum test) tests whether two independent groups originate from the same continuous distribution by analyzing ranks rather than raw parametric values.
Q. 18
Statistics
Difficulty: easy
(1 Mark)
What is the non-parametric counterpart to the Paired Samples t-test?
A
Mann-Whitney U Test
B
Wilcoxon Signed-Rank Test
✓ Correct
C
Chi-Square Test of Homogeneity
D
One-Way ANOVA F-test
💡
Step-by-Step Explanation & Concept Rationale
The Wilcoxon Signed-Rank test evaluates paired or repeated measures data by ranking the absolute differences between pairs and assigning positive/negative signs to the ranks.
Q. 19
Statistics
Difficulty: easy
(1 Mark)
What is the non-parametric equivalent of One-Way Analysis of Variance (ANOVA) for comparing $k > 2$ independent groups?
A
Friedman Test
B
Kruskal-Wallis H Test
✓ Correct
C
Mann-Whitney U Test
D
Cochran's Q Test
💡
Step-by-Step Explanation & Concept Rationale
The Kruskal-Wallis H-test extends the Mann-Whitney U test to $k \ge 3$ independent groups, testing whether all group samples originate from identical population distributions using ranked data.
Q. 20
Statistics
Difficulty: medium
(1 Mark)
What is the non-parametric equivalent of Two-Way Repeated Measures ANOVA (or Randomized Complete Block Design)?
A
Kruskal-Wallis Test
B
Friedman Test
✓ Correct
C
Kolmogorov-Smirnov Test
D
Spearman's Rho
💡
Step-by-Step Explanation & Concept Rationale
The Friedman test is a non-parametric test for comparing $k \ge 3$ repeated measures or matched treatments across $n$ blocks (subjects/clusters), ranking treatments within each block.
Q. 21
Statistics
Difficulty: medium
(1 Mark)
What is the Kolmogorov-Smirnov (K-S) One-Sample Test used for?
A
Testing whether two population means are identical
B
Testing whether an empirical sample distribution $F_n(x)$ matches a specified theoretical continuous cumulative distribution $F_0(x)$ by computing the supremum distance $D = \sup_x |F_n(x) - F_0(x)|$
✓ Correct
C
Testing independence in $2 \times 2$ contingency tables
D
Testing for heteroscedasticity in linear regression
💡
Step-by-Step Explanation & Concept Rationale
The one-sample K-S test evaluates the maximum vertical distance $D$ between the empirical step function $F_n(x)$ and the continuous reference CDF $F_0(x)$.
Q. 22
Statistics
Difficulty: medium
(1 Mark)
What is the Kolmogorov-Smirnov Two-Sample Test used for?
A
Testing whether two independent samples are drawn from the same underlying continuous distribution
✓ Correct
B
Comparing two sample variances
C
Estimating the regression slope between two variables
D
Testing for serial correlation in time series
💡
Step-by-Step Explanation & Concept Rationale
The two-sample K-S test evaluates the maximum discrepancy $D = \sup_x |F_{1, n_1}(x) - F_{2, n_2}(x)|$ between the empirical CDFs of two independent samples, sensitive to differences in location, scale, and shape.
Q. 23
Statistics
Difficulty: medium
(1 Mark)
Which of the following is considered the most powerful omnibus test for Normality in moderate to small samples?
A
Chi-Square Goodness of Fit Test
B
Shapiro-Wilk Test ($W$-test)
✓ Correct
C
Runs Test
D
Sign Test
💡
Step-by-Step Explanation & Concept Rationale
The Shapiro-Wilk $W$-statistic evaluates the ratio of the squared best linear unbiased estimator of slope in a normal probability plot to the sample sum of squares, widely regarded as the most powerful test for normality.
Q. 24
Statistics
Difficulty: hard
(1 Mark)
What is the Anderson-Darling Test for Normality, and how does it differ from the Kolmogorov-Smirnov test?
A
It tests only discrete Poisson data
B
It is a distance test that applies higher weight to the tails of the distribution ($A^2 = -n - \sum \dots$), making it far more sensitive to tail deviations than the standard K-S test
✓ Correct
C
It uses ranks instead of continuous data
D
It only works for sample sizes $n < 10$
💡
Step-by-Step Explanation & Concept Rationale
Anderson-Darling is an empirical distribution function test that places greater weight on the tails, making it superior for detecting heavy-tailed or skewed departures from normality.
Q. 25
Statistics
Difficulty: medium
(1 Mark)
What does the Wald-Wolfowitz Runs Test evaluate?
A
The running speed of interviewers
B
The hypothesis of Randomness in a sequence of binary observations (e.g., above/below median, success/failure)
✓ Correct
C
The equality of two population variances
D
The linear relationship between two variables
💡
Step-by-Step Explanation & Concept Rationale
The Runs test analyzes the number of runs (consecutive sequences of identical symbols) to determine if a time sequence is random or exhibits clustering, trends, or cyclical patterns.
Study Stream Progress:
Showing 25 of 90 Questions (28%)
Jump to:
🎉
All 90 Solved Questions in Hypothesis Testing & Non-Parametric Methods Loaded!
You have studied the complete question pool with step-by-step rationales. Ready to test yourself?
Q. 1
Statistics
Difficulty: Easy
(1 Mark)
In hypothesis testing, what is a 'Type I Error' ($\alpha$)?
A
Rejecting a true null hypothesis (False Positive)
✓ Correct
B
Failing to reject a false null hypothesis (False Negative)
C
Accepting a true alternative hypothesis
D
Calculating an incorrect mean
💡
Step-by-Step Explanation & Concept Rationale
A Type I error occurs when the researcher rejects the null hypothesis ($H_0$) when $H_0$ is actually true in reality. The probability of committing a Type I error is the significance level $\alpha$.
Q. 2
Statistics
Difficulty: Easy
(1 Mark)
In hypothesis testing, what is a 'Type II Error' ($\beta$)?
A
Failing to reject (retaining) a false null hypothesis (False Negative)
✓ Correct
B
Rejecting a true null hypothesis
C
Rejecting an incorrect sample mean
D
Using an alpha level of 0.05
💡
Step-by-Step Explanation & Concept Rationale
A Type II error occurs when the test fails to reject $H_0$ when $H_0$ is false in reality (i.e., failing to detect a true effect). The probability of committing a Type II error is designated $\beta$.
Q. 3
Statistics
Difficulty: Easy
(1 Mark)
The 'Power of a Statistical Test' is defined mathematically as:
A
$1 - \beta$ (The probability of correctly rejecting a false null hypothesis)
✓ Correct
B
$1 - \alpha$
C
$\alpha + \beta$
D
$\alpha / \beta$
💡
Step-by-Step Explanation & Concept Rationale
Statistical power ($1 - \beta$) is the probability that a statistical test will correctly detect a true effect (rejecting $H_0$ when $H_0$ is indeed false). Standard desired power in research is typically 80% or 0.80.
Q. 4
Statistics
Difficulty: Medium
(1 Mark)
In statistical significance testing, what is the exact definition of a '$p$-value'?
A
The probability, under the assumption that the null hypothesis is true, of obtaining a test statistic as extreme as or more extreme than the observed value
✓ Correct
B
The probability that the null hypothesis is true
C
The probability that the alternative hypothesis is false
D
The probability of making a Type II error
💡
Step-by-Step Explanation & Concept Rationale
The $p$-value quantifies the compatibility of the sample data with the null hypothesis: it is the probability of observing a result at least as extreme as the actual observed result, assuming $H_0$ is true. If $p \le \alpha$, we reject $H_0$.
Q. 5
Statistics
Difficulty: Easy
(1 Mark)
Under what circumstance is a One-Sample $z$-test chosen over a One-Sample $t$-test to test a claim about a population mean $\mu$?
A
When the population standard deviation $\sigma$ is known (or sample size is very large by CLT)
✓ Correct
B
When sample size is small ($n < 30$) and $\sigma$ is unknown
C
When data is qualitative ordinal ranks
D
When comparing three population variances
💡
Step-by-Step Explanation & Concept Rationale
The $z$-test requires known population variance $\sigma^2$ (or very large samples where $s \approx \sigma$). When $\sigma$ is unknown and estimated by sample standard deviation $s$, the Student's $t$-test is mandated.
Q. 6
Statistics
Difficulty: Easy
(1 Mark)
What is the critical value of $Z$ for a two-tailed hypothesis test at the standard 5% level of significance ($\alpha = 0.05$)?
A
$\pm 1.96$
✓ Correct
B
$\pm 1.645$
C
$\pm 2.576$
D
$\pm 3.00$
💡
Step-by-Step Explanation & Concept Rationale
For a two-tailed test with $\alpha = 0.05$, each tail contains an area of $\alpha/2 = 0.025$. The standard normal critical values bounding the central 95% region are $Z = \pm 1.96$.
Q. 7
Statistics
Difficulty: Easy
(1 Mark)
When comparing the mean test scores of the SAME group of students before and after an educational intervention, which test is appropriate?
A
Paired-Samples $t$-test (Dependent $t$-test)
✓ Correct
B
Independent Two-Sample $t$-test
C
Chi-square test of independence
D
Kruskal-Wallis test
💡
Step-by-Step Explanation & Concept Rationale
A paired (dependent) $t$-test evaluates matched pairs or repeated measures on the same subjects, analyzing the differences $d_i = x_{after} - x_{before}$ with $n - 1$ degrees of freedom to remove between-subject variability.
Q. 8
Statistics
Difficulty: Easy
(1 Mark)
In a contingency table with $r$ rows and $c$ columns, what are the degrees of freedom for the Pearson Chi-Square test of independence ($\chi^2$)?
A
$(r - 1) \times (c - 1)$
✓ Correct
B
$r \times c$
C
$r + c - 1$
D
$(r - 1) + (c - 1)$
💡
Step-by-Step Explanation & Concept Rationale
In a two-way contingency table, row and column marginal totals impose constraints, leaving exactly $(r - 1)(c - 1)$ cell frequencies that can vary freely.
Q. 9
Statistics
Difficulty: Medium
(1 Mark)
In a Chi-Square goodness-of-fit test, what is the standard recommended minimum expected frequency ($E_{ij}$) in each cell to ensure test validity?
A
At least 5
✓ Correct
B
At least 1
C
At least 30
D
At least 100
💡
Step-by-Step Explanation & Concept Rationale
Under Cochran's rule for chi-square validity, no cell should have an expected frequency less than 1, and at least 80% of cells (or all cells in a $2 \times 2$ table) should have expected frequencies of at least 5. If violated, Yates' correction or Fisher's exact test is used.
Q. 10
Statistics
Difficulty: Medium
(1 Mark)
When cell counts in a $2 \times 2$ contingency table are very small (e.g., expected frequencies < 5), which non-parametric test calculates the EXACT hyper-geometric probability of the observed table?
A
Fisher's Exact Test
✓ Correct
B
Pearson's Chi-Square test
C
Mann-Whitney U test
D
Wald test
💡
Step-by-Step Explanation & Concept Rationale
Fisher's Exact Test computes exact hyper-geometric probabilities without asymptotic approximations, making it the gold standard for small-sample $2 \times 2$ contingency analyses.
Q. 11
Statistics
Difficulty: Easy
(1 Mark)
What is the primary objective of One-Way Analysis of Variance (One-Way ANOVA)?
A
Testing whether the population means of three or more independent groups are statistically equal
✓ Correct
B
Testing whether two variances are identical
C
Predicting a categorical variable from a continuous variable
D
Measuring non-linear correlation
💡
Step-by-Step Explanation & Concept Rationale
One-Way ANOVA tests the global null hypothesis $H_0: \mu_1 = \mu_2 = \dots = \mu_k$ against the alternative that at least one group mean differs, avoiding the inflation of Type I error rate incurred by multiple pairwise $t$-tests.
Q. 12
Statistics
Difficulty: Medium
(1 Mark)
Why is performing multiple pairwise two-sample $t$-tests to compare 5 treatment groups (requiring $\binom{5}{2} = 10$ individual $t$-tests at $\alpha = 0.05$) statistically flawed?
A
It inflates the overall experiment-wise (family-wise) Type I error rate ($1 - (1 - 0.05)^{10} \approx 40\%$) causing rampant false positives
✓ Correct
B
The degrees of freedom become negative
C
The sample variance becomes zero
D
$t$-tests cannot be calculated on computers
💡
Step-by-Step Explanation & Concept Rationale
Running multiple pairwise tests without adjustment leads to Family-Wise Error Rate inflation: $\alpha_{FWE} = 1 - (1 - \alpha)^c$. For 10 comparisons, the probability of at least one false positive escalates to ~40%. ANOVA maintains the overall alpha at 0.05.
Q. 13
Statistics
Difficulty: Easy
(1 Mark)
In an ANOVA table, how is the $F$-test statistic calculated?
A
Mean Square Between Groups divided by Mean Square Within Groups ($F = MSB / MSW$)
✓ Correct
B
$SSB / SSW$
C
$MSW / MSB$
D
Total Sum of Squares / Total df
💡
Step-by-Step Explanation & Concept Rationale
The $F$-statistic in ANOVA is the ratio of between-group variance estimate to within-group (error) variance estimate: $F = MSB / MSW$. Under $H_0$, both estimate error variance ($F \approx 1$); if treatment effects exist, $MSB > MSW$ ($F > 1$).
Q. 14
Statistics
Difficulty: Medium
(1 Mark)
When an omnibus ANOVA $F$-test rejects the null hypothesis ($p < 0.05$), which statistical post-hoc procedure is widely used to determine exactly which specific pairs of group means differ significantly while controlling the family-wise error rate?
A
Tukey's Honestly Significant Difference (HSD) test
✓ Correct
B
Paired $z$-test
C
Pearson correlation
D
Durbin-Watson test
💡
Step-by-Step Explanation & Concept Rationale
Tukey's HSD test performs all possible pairwise comparisons among group means using the studentized range statistic ($q$), holding the overall family-wise Type I error rate strictly at $\alpha$ (e.g., 0.05).
Q. 15
Statistics
Difficulty: Easy
(1 Mark)
Which non-parametric test is the distribution-free alternative to the independent Two-Sample $t$-test for comparing two independent groups when normality is violated?
A
Mann-Whitney U test (Wilcoxon Rank-Sum test)
✓ Correct
B
Kruskal-Wallis H test
C
McNemar's test
D
Cochran's Q test
💡
Step-by-Step Explanation & Concept Rationale
The Mann-Whitney U test (Wilcoxon rank-sum) evaluates whether the distributions of two independent populations differ by ranking all combined pooled observations, requiring no normality or equal variance assumptions.
Q. 16
Statistics
Difficulty: Easy
(1 Mark)
Which non-parametric test serves as the distribution-free counterpart to One-Way ANOVA for comparing three or more independent groups?
A
Kruskal-Wallis H test
✓ Correct
B
Friedman test
C
Wilcoxon signed-rank test
D
Spearman test
💡
Step-by-Step Explanation & Concept Rationale
The Kruskal-Wallis test is a rank-based non-parametric extension of the Mann-Whitney test used to compare three or more independent groups on an ordinal or continuous dependent variable without assuming normality.
Q. 17
Statistics
Difficulty: Easy
(1 Mark)
Which non-parametric test is the appropriate distribution-free alternative to the Paired $t$-test for evaluating matched-pair or before-and-after repeated measures?
A
Wilcoxon Signed-Rank Test
✓ Correct
B
Mann-Whitney U test
C
Chi-square test
D
Kruskal-Wallis test
💡
Step-by-Step Explanation & Concept Rationale
The Wilcoxon Signed-Rank test analyzes the signed differences between matched pairs, ranking the absolute differences to test whether the median difference between paired observations differs from zero.
Q. 18
Statistics
Difficulty: Easy
(1 Mark)
In simple linear regression analysis ($Y = \beta_0 + \beta_1 X + \epsilon$), the Ordinary Least Squares (OLS) method estimates parameters by minimizing the:
A
Sum of squared vertical residual errors ($\sum e_i^2 = \sum (y_i - \hat{y}_i)^2$)
✓ Correct
B
Sum of absolute errors
C
Maximum horizontal distance
D
Sum of dependent variables
💡
Step-by-Step Explanation & Concept Rationale
Ordinary Least Squares (OLS) finds the unique line that minimizes the sum of squared vertical distances (residuals $e_i = y_i - \hat{y}_i$) between the observed data points and the fitted regression line.
Q. 19
Statistics
Difficulty: Easy
(1 Mark)
In a multiple linear regression model, what serious econometric problem occurs when two or more independent predictor variables are highly linearly correlated with each other?
A
Multicollinearity
✓ Correct
B
Heteroscedasticity
C
Autocorrelation
D
Endogeneity
💡
Step-by-Step Explanation & Concept Rationale
Multicollinearity occurs when predictors share high mutual correlation. It inflates standard errors of estimated regression coefficients, rendering $t$-statistics insignificant and making individual predictor coefficients unstable and difficult to interpret.
Q. 20
Statistics
Difficulty: Medium
(1 Mark)
What diagnostic metric is commonly computed to detect harmful multicollinearity in multiple regression, where values exceeding 5 to 10 indicate severe collinearity?
A
Variance Inflation Factor (VIF)
✓ Correct
B
Durbin-Watson statistic
C
Cook's distance
D
Akaike Information Criterion (AIC)
💡
Step-by-Step Explanation & Concept Rationale
The Variance Inflation Factor ($VIF_j = 1 / (1 - R_j^2)$) quantifies how much the variance of an estimated regression coefficient is inflated due to collinearity with other predictors. A $VIF > 5$ or $10$ signals problematic multicollinearity.
Q. 21
Statistics
Difficulty: Medium
(1 Mark)
In linear regression modeling, what does the assumption of 'Homoscedasticity' require?
A
The variance of the error terms (residuals) must be constant across all levels of the independent variables ($Var(\epsilon_i) = \sigma^2$)
✓ Correct
B
The residuals must be positive numbers
C
The independent variables must have zero mean
D
The sample size must exceed 1,000
💡
Step-by-Step Explanation & Concept Rationale
Homoscedasticity requires that the residual variance around the regression line remains constant for all fitted values. The violation of this assumption (varying error variance, often widening like a funnel) is termed Heteroscedasticity.
Q. 22
Statistics
Difficulty: Medium
(1 Mark)
Which statistical test evaluates whether the residuals of a linear regression model exhibit first-order Autocorrelation (serial correlation) in time-series data?
A
Durbin-Watson test
✓ Correct
B
Breusch-Pagan test
C
Shapiro-Wilk test
D
Levene's test
💡
Step-by-Step Explanation & Concept Rationale
The Durbin-Watson test statistic ($d = \frac{\sum (e_t - e_{t-1})^2}{\sum e_t^2}$) tests for first-order serial correlation in residuals. A value near 2 indicates no autocorrelation; values approaching 0 indicate strong positive autocorrelation; values approaching 4 indicate negative autocorrelation.
Q. 23
Statistics
Difficulty: Medium
(1 Mark)
Which formal statistical test evaluates whether sample data conforms to a Normal distribution, recognized as one of the most powerful normality tests for small to moderate samples?
A
Shapiro-Wilk test
✓ Correct
B
Levene's test
C
Bartlett's test
D
Grubbs' test
💡
Step-by-Step Explanation & Concept Rationale
The Shapiro-Wilk test ($W$-statistic) evaluates the null hypothesis that sample data came from a normally distributed population. A significant $p < 0.05$ indicates departure from normality.
Q. 24
Statistics
Difficulty: Medium
(1 Mark)
Levene's Test and Bartlett's Test are used in statistical analysis to verify which critical assumption of ANOVA and two-sample $t$-tests?
A
Homogeneity of Variances (Homoscedasticity across groups)
✓ Correct
B
Independence of observations
C
Linearity of regression
D
Absence of outliers
💡
Step-by-Step Explanation & Concept Rationale
Levene's test (less sensitive to non-normality) and Bartlett's test test the null hypothesis that multiple group variances are equal: $\sigma_1^2 = \sigma_2^2 = \dots = \sigma_k^2$.
Q. 25
Statistics
Difficulty: Easy
(1 Mark)
What type of regression model is used when the dependent outcome variable is binary / dichotomous (e.g., Disease Present = 1, Disease Absent = 0)?
A
Binary Logistic Regression
✓ Correct
B
Simple Linear Regression
C
Polynomial Regression
D
Poisson Regression
💡
Step-by-Step Explanation & Concept Rationale
Logistic regression models the log-odds (logit) of a binary outcome as a linear combination of predictors: $\ln(p / (1-p)) = \beta_0 + \sum \beta_i X_i$, constraining predicted probabilities strictly between 0 and 1 via the sigmoid function.
Study Stream Progress:
Showing 25 of 36 Questions (69%)
Jump to:
🎉
All 36 Solved Questions in Hypothesis Testing, Regression & ANOVA Loaded!
You have studied the complete question pool with step-by-step rationales. Ready to test yourself?
Q. 1
Statistics
Difficulty: easy
(1 Mark)
In the classical simple linear regression model $Y_i = \beta_0 + \beta_1 X_i + u_i$, what does the Gauss-Markov Theorem state regarding the Ordinary Least Squares (OLS) estimator $\hat{\beta}_1$?
A
It is the maximum likelihood estimator for all non-linear equations
B
Under the classical linear regression assumptions, the OLS estimator is the Best Linear Unbiased Estimator (BLUE), having minimum variance among all linear unbiased estimators
✓ Correct
C
It guarantees that $R^2 = 1.0$ in large samples
D
It eliminates all measurement error in the dataset
💡
Step-by-Step Explanation & Concept Rationale
The Gauss-Markov theorem proves that under homoscedasticity, exogeneity, and no autocorrelation, OLS estimators $\hat{\boldsymbol{\beta}}$ achieve the lowest sampling variance among all possible linear unbiased estimators (BLUE).
Q. 2
Statistics
Difficulty: easy
(1 Mark)
In the matrix formulation of Multiple Linear Regression $\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{u}$, what is the OLS estimator $\hat{\boldsymbol{\beta}}$?
A
$\hat{\boldsymbol{\beta}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}$
✓ Correct
B
$\hat{\boldsymbol{\beta}} = \mathbf{X}^T \mathbf{y} (\mathbf{X}^T \mathbf{X})$
C
$\hat{\boldsymbol{\beta}} = (\mathbf{X} \mathbf{X}^T)^{-1} \mathbf{y}$
D
$\hat{\boldsymbol{\beta}} = \mathbf{X}^{-1} \mathbf{y}$
💡
Step-by-Step Explanation & Concept Rationale
Minimizing the sum of squared residuals $\mathbf{u}^T \mathbf{u} = (\mathbf{y} - \mathbf{X}\boldsymbol{\beta})^T (\mathbf{y} - \mathbf{X}\boldsymbol{\beta})$ with respect to $\boldsymbol{\beta}$ yields the normal equations $(\mathbf{X}^T \mathbf{X})\hat{\boldsymbol{\beta}} = \mathbf{X}^T \mathbf{y}$, giving $\hat{\boldsymbol{\beta}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}$.
Q. 3
Statistics
Difficulty: medium
(1 Mark)
What is the Variance-Covariance matrix of the OLS estimator $\hat{\boldsymbol{\beta}}$ under standard homoscedastic assumptions?
A
$\text{Var}(\hat{\boldsymbol{\beta}}) = \sigma^2 (\mathbf{X}^T \mathbf{X})^{-1}$
✓ Correct
B
$\text{Var}(\hat{\boldsymbol{\beta}}) = \sigma^2 (\mathbf{X}^T \mathbf{X})$
C
$\text{Var}(\hat{\boldsymbol{\beta}}) = \frac{\sigma^2}{n} \mathbf{I}$
D
$\text{Var}(\hat{\boldsymbol{\beta}}) = (\mathbf{X}^T \mathbf{X})^{-1}$
💡
Step-by-Step Explanation & Concept Rationale
Since $\hat{\boldsymbol{\beta}} = \boldsymbol{\beta} + (\mathbf{X}^T \mathbf{X})^{-1}\mathbf{X}^T\mathbf{u}$, $\text{Var}(\hat{\boldsymbol{\beta}}) = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T [\text{Var}(\mathbf{u})] \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1} = \sigma^2 (\mathbf{X}^T \mathbf{X})^{-1}$.
Q. 4
Statistics
Difficulty: easy
(1 Mark)
What is the Coefficient of Determination ($R^2$) in a linear regression model?
A
The ratio of the slope to the intercept
B
The proportion of the total variation in the dependent variable $Y$ that is explained by the regression model: $R^2 = 1 - \frac{SSE}{SST} = \frac{SSR}{SST}$
✓ Correct
C
The probability that the regression coefficients are non-zero
D
The correlation coefficient divided by degrees of freedom
💡
Step-by-Step Explanation & Concept Rationale
$R^2 = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}$ quantifies the percentage of variance in the response variable accounted for by the explanatory predictors in the model, bounded in $[0, 1]$.
Q. 5
Statistics
Difficulty: medium
(1 Mark)
Why is the 'Adjusted $R^2$' ($\bar{R}^2$) used instead of standard $R^2$ when comparing multiple regression models with different numbers of predictors?
A
Standard $R^2$ never decreases (and almost always increases) whenever a new predictor is added, whereas $\bar{R}^2 = 1 - \frac{SSE/(n-k-1)}{SST/(n-1)}$ penalizes the addition of irrelevant variables
✓ Correct
B
Standard $R^2$ can exceed 1.0 in multiple regression
C
Adjusted $R^2$ can only be computed using non-parametric statistics
D
Adjusted $R^2$ eliminates multicollinearity completely
💡
Step-by-Step Explanation & Concept Rationale
Standard $R^2$ mechanically inflates as more regressors are added, even if they are noise. Adjusted $R^2$ adjusts for degrees of freedom ($df$), increasing only if the new predictor improves the model more than expected by chance.
Q. 6
Statistics
Difficulty: easy
(1 Mark)
What is the relationship between the F-statistic for overall model significance in simple linear regression and the individual t-statistic for the slope coefficient?
A
$F = t^2$
✓ Correct
B
$F = \sqrt{t}$
C
$F = t / 2$
D
$F = 2t + 1$
💡
Step-by-Step Explanation & Concept Rationale
In simple linear regression ($k = 1$ predictor), testing $H_0: \beta_1 = 0$ via the two-sided t-test ($df = n-2$) is mathematically identical to the overall ANOVA F-test ($df_1 = 1, df_2 = n-2$), with $F = t^2$.
Q. 7
Statistics
Difficulty: medium
(1 Mark)
What is the Hat Matrix $\mathbf{H}$ in linear regression, defined as $\mathbf{H} = \mathbf{X}(\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T$?
A
A matrix that projects the observed response vector $\mathbf{y}$ onto the column space of $\mathbf{X}$ to yield predicted values: $\hat{\mathbf{y}} = \mathbf{H}\mathbf{y}$
✓ Correct
B
A matrix containing the names of the variables
C
A diagonal matrix of residual variances
D
An identity matrix of dimensions $n \times n$
💡
Step-by-Step Explanation & Concept Rationale
The Hat matrix (projection matrix) maps observed values to fitted values: $\hat{\mathbf{y}} = \mathbf{H}\mathbf{y}$. Its diagonal elements $h_{ii}$ measure the 'leverage' of each observation in predictor space.
Q. 8
Statistics
Difficulty: hard
(1 Mark)
What is the sum of the diagonal elements (trace) of the Hat Matrix $\mathbf{H} = \mathbf{X}(\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T$ with $p = k + 1$ parameters?
A
$\text{tr}(\mathbf{H}) = n$
B
$\text{tr}(\mathbf{H}) = p$ (the number of estimated regression parameters including intercept)
✓ Correct
C
$\text{tr}(\mathbf{H}) = n - p$
D
$\text{tr}(\mathbf{H}) = 1$
💡
Step-by-Step Explanation & Concept Rationale
Using cyclic property of trace: $\text{tr}(\mathbf{H}) = \text{tr}(\mathbf{X}(\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T) = \text{tr}((\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{X}) = \text{tr}(\mathbf{I}_p) = p$. The average leverage is $\bar{h} = p / n$.
Q. 9
Statistics
Difficulty: medium
(1 Mark)
An observation is commonly flagged as having 'High Leverage' in regression diagnostics if its diagonal Hat value $h_{ii}$ exceeds what threshold?
A
$h_{ii} > \frac{2p}{n}$ (or $h_{ii} > \frac{3p}{n}$)
✓ Correct
B
$h_{ii} > 1.0$
C
$h_{ii} < 0$
D
$h_{ii} > 0.5$
💡
Step-by-Step Explanation & Concept Rationale
Since the average leverage is $\bar{h} = p/n$, observations with $h_{ii} > 2p/n$ (or $3p/n$) are considered high leverage points that exert disproportionate potential influence on the fitted plane.
Q. 10
Statistics
Difficulty: medium
(1 Mark)
What does Cook's Distance ($D_i$) measure in regression diagnostics?
A
The physical distance between the interviewer and the respondent
B
The overall influence of the $i$-th observation on ALL fitted values when that observation is deleted: $D_i = \frac{\sum_{j=1}^n (\hat{Y}_j - \hat{Y}_{j(i)})^2}{p \cdot MSE}$
✓ Correct
C
The correlation between error terms and explanatory variables
D
The multicollinearity between two predictor variables
💡
Step-by-Step Explanation & Concept Rationale
Cook's Distance combines both the standardized residual (outlier in $Y$) and the leverage $h_{ii}$ (outlier in $X$) to measure how much all regression predictions shift when point $i$ is omitted. $D_i > 1$ or $D_i > 4/n$ indicates high influence.
Q. 11
Statistics
Difficulty: easy
(1 Mark)
What is 'Multicollinearity' in multiple regression?
A
The presence of high linear correlation among two or more independent explanatory variables ($X_j$)
✓ Correct
B
A non-linear relationship between the dependent variable and the independent variable
C
Unequal variance in the error terms across observations
D
Correlation between successive error terms over time
💡
Step-by-Step Explanation & Concept Rationale
Multicollinearity occurs when predictors are highly intercorrelated, making the $(\mathbf{X}^T\mathbf{X})$ matrix near-singular, which inflates standard errors of regression coefficients without biasing point estimates.
Q. 12
Statistics
Difficulty: easy
(1 Mark)
What is the Variance Inflation Factor (VIF) for predictor $X_j$ defined as?
A
$\text{VIF}_j = \frac{1}{1 - R_j^2}$, where $R_j^2$ is the coefficient of determination from regressing $X_j$ on all other predictors
✓ Correct
B
$\text{VIF}_j = 1 - R_j^2$
C
$\text{VIF}_j = \sqrt{R_j^2}$
D
$\text{VIF}_j = \frac{R_j^2}{n - k}$
💡
Step-by-Step Explanation & Concept Rationale
The VIF measures how much the variance of coefficient $\hat{\beta}_j$ is inflated due to multicollinearity: $\text{VIF}_j = \frac{1}{1 - R_j^2}$. A common rule of thumb considers $\text{VIF} > 5$ or $10$ as evidence of severe multicollinearity.
Q. 13
Statistics
Difficulty: easy
(1 Mark)
What is 'Tolerance' in regression diagnostics?
A
The patience of the statistician running the model
B
The reciprocal of the Variance Inflation Factor: $\text{Tolerance}_j = 1 - R_j^2 = \frac{1}{\text{VIF}_j}$
✓ Correct
C
The p-value of the slope coefficient
D
The margin of error in hypothesis testing
💡
Step-by-Step Explanation & Concept Rationale
Tolerance is $\text{Tol}_j = 1 - R_j^2$. A tolerance below $0.10$ or $0.20$ indicates severe collinearity with other explanatory variables.
Q. 14
Statistics
Difficulty: easy
(1 Mark)
What is 'Heteroscedasticity' in regression modeling?
A
A violation of the classical assumption where the variance of the error term is NOT constant across observations: $\text{Var}(u_i \mid X_i) = \sigma_i^2 \ne \sigma^2$
✓ Correct
B
Correlation between error terms across different time periods
C
A linear relationship among independent variables
D
A non-normal distribution of the dependent variable
💡
Step-by-Step Explanation & Concept Rationale
Heteroscedasticity means the error terms have non-constant variance (e.g., variance of income errors grows as income increases), rendering standard OLS standard errors invalid and test statistics biased.
Q. 15
Statistics
Difficulty: medium
(1 Mark)
What are the consequences of ignoring Heteroscedasticity when using Ordinary Least Squares (OLS)?
A
OLS estimates of $\boldsymbol{\beta}$ become biased and inconsistent
B
OLS coefficient estimates remain unbiased and consistent, but standard errors are biased (usually underestimated), rendering t-tests, F-tests, and confidence intervals invalid, and OLS is no longer BLUE
✓ Correct
C
The $R^2$ becomes negative
D
The regression line passes through all points exactly
💡
Step-by-Step Explanation & Concept Rationale
Under heteroscedasticity, OLS point estimates remain unbiased ($E[\hat{\boldsymbol{\beta}}] = \boldsymbol{\beta}$), but standard formula for $\text{Var}(\hat{\boldsymbol{\beta}})$ is wrong, distorting p-values. Weighted Least Squares (WLS) or Huber-White robust SEs are required.
Q. 16
Statistics
Difficulty: medium
(1 Mark)
Which of the following statistical tests is specifically designed to detect Heteroscedasticity in regression models?
A
Durbin-Watson test
B
Breusch-Pagan Test / White's General Heteroscedasticity Test
✓ Correct
C
Dickey-Fuller test
D
Kruskal-Wallis test
💡
Step-by-Step Explanation & Concept Rationale
The Breusch-Pagan test regresses squared OLS residuals on the independent variables (or fitted values), while White's test includes squares and cross-products of regressors to test for general forms of heteroscedasticity.
Q. 17
Statistics
Difficulty: hard
(1 Mark)
What is White's Heteroscedasticity-Consistent Covariance Matrix Estimator (HCCME / 'Robust Standard Errors')?
A
$\widehat{\text{Var}}(\hat{\boldsymbol{\beta}}) = (\mathbf{X}^T \mathbf{X})^{-1} \left( \sum_{i=1}^n \hat{u}_i^2 \mathbf{x}_i \mathbf{x}_i^T \right) (\mathbf{X}^T \mathbf{X})^{-1}$ (the 'Sandwich' estimator)
✓ Correct
B
$\widehat{\text{Var}}(\hat{\boldsymbol{\beta}}) = \sigma^2 \mathbf{I}$
C
$\widehat{\text{Var}}(\hat{\boldsymbol{\beta}}) = \frac{1}{n} (\mathbf{X}^T \mathbf{X})$
D
$\widehat{\text{Var}}(\hat{\boldsymbol{\beta}}) = (\mathbf{X}^T \mathbf{X})^{-1}$
💡
Step-by-Step Explanation & Concept Rationale
Halbert White (1980) formulated the sandwich estimator $(X^TX)^{-1}(X^T \hat{\boldsymbol{\Omega}} X)(X^TX)^{-1}$, providing asymptotically valid standard errors and hypothesis tests even when the exact functional form of heteroscedasticity is unknown.
Q. 18
Statistics
Difficulty: easy
(1 Mark)
What is 'Autocorrelation' (Serial Correlation) in time series regression?
A
Correlation between error terms at different time points: $\text{Cov}(u_t, u_s) \ne 0$ for $t \ne s$
✓ Correct
B
Correlation between two independent variables
C
Correlation between the intercept and slope
D
The presence of multiple peaks in a histogram
💡
Step-by-Step Explanation & Concept Rationale
Autocorrelation occurs when regression disturbances in period $t$ are correlated with disturbances in earlier periods (e.g., $u_t = \rho u_{t-1} + \epsilon_t$), common in macro-economic and environmental time-series data.
Q. 19
Statistics
Difficulty: medium
(1 Mark)
What is the range and ideal null-value of the Durbin-Watson statistic ($d$) for detecting first-order autocorrelation in regression residuals?
A
Bounded between 0 and 4; $d \approx 2$ indicates no first-order autocorrelation ($\rho = 0$)
✓ Correct
B
Bounded between -1 and +1; $d = 0$ indicates no correlation
C
Bounded between 0 and $\infty$; $d = 100$ is ideal
D
Bounded between 0 and 1; $d = 0.5$ is ideal
💡
Step-by-Step Explanation & Concept Rationale
The Durbin-Watson statistic is $d = \frac{\sum_{t=2}^T (e_t - e_{t-1})^2}{\sum_{t=1}^T e_t^2} \approx 2(1 - \hat{\rho})$. $d \approx 2$ means no autocorrelation; $d \approx 0$ indicates strong positive autocorrelation; $d \approx 4$ indicates strong negative autocorrelation.
Q. 20
Statistics
Difficulty: hard
(1 Mark)
When is the Durbin-Watson test INVALID for testing serial correlation?
A
When the sample size $n > 100$
B
When the regression model contains a Lagged Dependent Variable among the regressors ($Y_{t-1}$), where the Durbin-h test or Breusch-Godfrey LM test must be used instead
✓ Correct
C
When the error terms are homoscedastic
D
When all variables are positive
💡
Step-by-Step Explanation & Concept Rationale
If a lagged dependent variable $Y_{t-1}$ is included as a predictor, the Durbin-Watson statistic is severely biased toward 2 (masking true serial correlation). The Breusch-Godfrey LM test is the general standard.
Q. 21
Statistics
Difficulty: hard
(1 Mark)
What are Newey-West Standard Errors (HAC - Heteroscedasticity and Autocorrelation Consistent)?
A
Standard errors that are robust to both unknown forms of heteroscedasticity AND arbitrary serial correlation up to a specified lag truncation length $L$
✓ Correct
B
Standard errors calculated using only the newest observations
C
Standard errors that are always equal to 1.0
D
Standard errors used exclusively in logistic regression
💡
Step-by-Step Explanation & Concept Rationale
Newey-West (1987) HAC standard errors use a Bartlett kernel weighting scheme to produce positive semi-definite covariance estimates robust to simultaneous heteroscedasticity and autocorrelation in time-series OLS.
Q. 22
Statistics
Difficulty: hard
(1 Mark)
What does the Ramsey RESET (Regression Equation Specification Error Test) test for?
A
General model misspecification, including omitted non-linear terms (e.g., squared or cubic regressors) and incorrect functional form, by testing the significance of powers of fitted values $\hat{Y}^2, \hat{Y}^3$
✓ Correct
B
Whether the computer reset properly during estimation
C
Equality of variances across subgroups
D
Multicollinearity among explanatory variables
💡
Step-by-Step Explanation & Concept Rationale
Ramsey's RESET tests for omitted non-linearities and functional form misspecification by adding powers of the fitted values $(\hat{y}_i^2, \hat{y}_i^3)$ to the auxiliary regression and testing their joint significance via F-test.
Q. 23
Statistics
Difficulty: easy
(1 Mark)
In Binary Logistic Regression, what is the 'Logit' Link Function defined as for probability $p = P(Y = 1 \mid X)$?
A
$\text{logit}(p) = \ln\left(\frac{p}{1 - p}\right) = \beta_0 + \beta_1 X$
✓ Correct
B
$\text{logit}(p) = \frac{p}{1 - p}$
C
$\text{logit}(p) = e^p$
D
$\text{logit}(p) = \Phi^{-1}(p)$
💡
Step-by-Step Explanation & Concept Rationale
The logit function is the natural logarithm of the odds: $\text{logit}(p) = \ln\left(\frac{p}{1-p}\right)$. It maps probability $p \in (0, 1)$ to $(-\infty, \infty)$, enabling linear parameterization.
Q. 24
Statistics
Difficulty: medium
(1 Mark)
In a Logistic Regression model $\ln\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 X$, how is the slope coefficient $\beta_1$ interpreted?
A
For a one-unit increase in $X$, the probability $p$ increases by $\beta_1$
B
For a one-unit increase in $X$, the Odds Ratio changes multiplicatively by a factor of $e^{\beta_1}$ (i.e., log-odds increases by $\beta_1$)
✓ Correct
C
For a one-unit increase in $X$, the variance increases by $\beta_1^2$
D
$\beta_1$ represents the correlation between $X$ and $Y$
💡
Step-by-Step Explanation & Concept Rationale
In logistic regression, exponentiating the slope gives the Odds Ratio: $\text{OR} = e^{\beta_1}$. A one-unit increase in $X$ multiplies the odds of the outcome by $e^{\beta_1}$.
Q. 25
Statistics
Difficulty: medium
(1 Mark)
What estimation method is standard for fitting Logistic Regression models?
A
Ordinary Least Squares (OLS)
B
Maximum Likelihood Estimation (MLE) using iterative numerical algorithms like Newton-Raphson or Iteratively Reweighted Least Squares (IRLS)
✓ Correct
C
Method of Moments
D
Sign Test
💡
Step-by-Step Explanation & Concept Rationale
Because the binary response $Y_i \sim \text{Bernoulli}(p_i)$ violates OLS homoscedasticity and normality, Logistic Regression parameters are estimated by maximizing the Bernoulli log-likelihood via Newton-Raphson / IRLS.
Study Stream Progress:
Showing 25 of 90 Questions (28%)
Jump to:
🎉
All 90 Solved Questions in Linear & Multiple Regression, Correlation & ANOVA Loaded!
You have studied the complete question pool with step-by-step rationales. Ready to test yourself?
Q. 1
Statistics
Difficulty: medium
(1 Mark)
If a random variable $X$ follows a Poisson distribution with parameter $\lambda = 4$, what is the coefficient of skewness $\gamma_1 = \frac{\mu_3}{\sigma^3}$?
A
0.25
B
0.50
✓ Correct
C
1.00
D
2.00
💡
Step-by-Step Explanation & Concept Rationale
For a Poisson distribution with parameter $\lambda$, the variance $\sigma^2 = \lambda$ and the third central moment $\mu_3 = \lambda$. Therefore, skewness $\gamma_1 = \frac{\lambda}{\lambda^{3/2}} = \frac{1}{\sqrt{\lambda}} = \frac{1}{\sqrt{4}} = 0.5$.
Q. 2
Statistics
Difficulty: easy
(1 Mark)
For a Binomial distribution $B(n, p)$, under what condition does the distribution become perfectly symmetrical?
A
When $n \to \infty$ for any $p$
B
When $p = 0.5$
✓ Correct
C
When $n p = 1$
D
When $p = 0.25$
💡
Step-by-Step Explanation & Concept Rationale
The Binomial distribution is perfectly symmetrical (skewness $\gamma_1 = \frac{q-p}{\sqrt{npq}} = 0$) if and only if $p = q = 0.5$, regardless of the sample size $n$.
Q. 3
Statistics
Difficulty: medium
(1 Mark)
What is the Moment Generating Function (MGF) $M_X(t)$ of an Exponential random variable $X \sim \text{Exp}(\lambda)$ with probability density function $f(x) = \lambda e^{-\lambda x}$ for $x > 0$ and $t < \lambda$?
A
$\frac{\lambda}{\lambda - t}$
✓ Correct
B
$\frac{\lambda}{\lambda + t}$
C
$e^{\lambda t}$
D
$\frac{1}{1 - \lambda t^2}$
💡
Step-by-Step Explanation & Concept Rationale
The MGF is $M_X(t) = \int_0^\infty e^{tx} \lambda e^{-\lambda x} dx = \lambda \int_0^\infty e^{-(\lambda - t)x} dx = \frac{\lambda}{\lambda - t}$ for $t < \lambda$.
Q. 4
Statistics
Difficulty: easy
(1 Mark)
If $X$ and $Y$ are two independent standard normal random variables ($X, Y \sim N(0, 1)$), what is the probability distribution of $Z = X^2 + Y^2$?
A
Standard Normal distribution $N(0, 2)$
B
Chi-Square distribution with 2 degrees of freedom $\chi^2_{(2)}$ (or Exponential with mean 2)
✓ Correct
C
Student's t-distribution with 2 degrees of freedom
D
F-distribution with $(1, 1)$ degrees of freedom
💡
Step-by-Step Explanation & Concept Rationale
The sum of squares of $k$ independent standard normal variables follows a Chi-square distribution with $k$ degrees of freedom. For $k=2$, $\chi^2_{(2)}$ is identical to an Exponential distribution with parameter $\lambda = 1/2$ (mean 2).
Q. 5
Statistics
Difficulty: easy
(1 Mark)
Which probability distribution possesses the unique 'Memoryless Property' among continuous random variables?
A
Normal distribution
B
Exponential distribution
✓ Correct
C
Uniform distribution
D
Weibull distribution with shape parameter $k = 2$
💡
Step-by-Step Explanation & Concept Rationale
The Exponential distribution satisfies $P(X > s + t \mid X > s) = P(X > t)$, meaning the past duration has no effect on future probability; it is the only continuous distribution with this property.
Q. 6
Statistics
Difficulty: easy
(1 Mark)
Which discrete probability distribution possesses the memoryless property ($P(X > s + t \mid X > s) = P(X > t)$)?
A
Binomial distribution
B
Geometric distribution
✓ Correct
C
Poisson distribution
D
Hypergeometric distribution
💡
Step-by-Step Explanation & Concept Rationale
The Geometric distribution is the sole discrete distribution that is memoryless, modeling the number of Bernoulli trials until the first success.
Q. 7
Statistics
Difficulty: medium
(1 Mark)
If a random variable $X$ has a standard Cauchy distribution ($f(x) = \frac{1}{\pi(1+x^2)}$), what is its mathematical expectation $E[X]$?
A
0
B
Undefined (does not exist)
✓ Correct
C
1
D
$\pi$
💡
Step-by-Step Explanation & Concept Rationale
The integral $\int_{-\infty}^\infty \frac{x}{\pi(1+x^2)} dx$ is not absolutely convergent (diverges to $\infty - \infty$). Hence, the mean and all higher moments of the Cauchy distribution do not exist.
Q. 8
Statistics
Difficulty: easy
(1 Mark)
If $X_1, X_2, \dots, X_n$ are independent random variables from a Poisson distribution with parameters $\lambda_1, \lambda_2, \dots, \lambda_n$, what is the distribution of their sum $S_n = \sum_{i=1}^n X_i$?
A
Normal distribution with mean $\sum \lambda_i$
B
Poisson distribution with parameter $\lambda = \sum_{i=1}^n \lambda_i$
✓ Correct
C
Binomial distribution with $p = \sum \lambda_i / n$
D
Negative Binomial distribution
💡
Step-by-Step Explanation & Concept Rationale
By the additive (reproductive) property of the Poisson distribution, the sum of independent Poisson variables is itself Poisson distributed with parameter equal to the sum of their individual means.
Q. 9
Statistics
Difficulty: easy
(1 Mark)
What is the variance of a Continuous Uniform distribution $U(a, b)$ on the interval $[a, b]$?
A
$\frac{(b - a)^2}{12}$
✓ Correct
B
$\frac{(b - a)^2}{6}$
C
$\frac{b - a}{2}$
D
$\frac{(b + a)^2}{12}$
💡
Step-by-Step Explanation & Concept Rationale
For a uniform distribution $U(a, b)$, the mean is $\mu = \frac{a+b}{2}$ and the variance is $\sigma^2 = \frac{(b-a)^2}{12}$.
Q. 10
Statistics
Difficulty: medium
(1 Mark)
If $X \sim N(\mu, \sigma^2)$, what are the values of the third and fourth central moments $\mu_3$ and $\mu_4$?
A
$\mu_3 = 0$ and $\mu_4 = 3\sigma^4$
✓ Correct
B
$\mu_3 = \sigma^3$ and $\mu_4 = \sigma^4$
C
$\mu_3 = 0$ and $\mu_4 = 0$
D
$\mu_3 = 3\sigma^3$ and $\mu_4 = 3\sigma^4$
💡
Step-by-Step Explanation & Concept Rationale
Due to symmetry around $\mu$, all odd central moments of the normal distribution are zero ($\mu_3 = 0$). The fourth central moment is $\mu_4 = 3\sigma^4$, giving a kurtosis coefficient $\beta_2 = \mu_4 / \sigma^4 = 3$ (mesokurtic).
Q. 11
Statistics
Difficulty: medium
(1 Mark)
What is the relationship between the Hypergeometric distribution and the Binomial distribution?
A
The Binomial distribution models sampling without replacement, whereas Hypergeometric models sampling with replacement
B
The Hypergeometric distribution converges to the Binomial distribution as the population size $N \to \infty$ with a fixed sampling fraction $\frac{n}{N} \to 0$
✓ Correct
C
The Hypergeometric distribution always has a higher variance than the Binomial distribution
D
They are mathematically identical for all population sizes
💡
Step-by-Step Explanation & Concept Rationale
The Hypergeometric distribution models sampling without replacement from a finite population. As population size $N \to \infty$, sampling without replacement becomes effectively equivalent to sampling with replacement (Binomial).
Q. 12
Statistics
Difficulty: medium
(1 Mark)
In a Negative Binomial distribution $NB(r, p)$, what does the random variable $X$ represent?
A
The number of trials required to achieve exactly $r$ successes
B
The number of failures before the $r$-th success is observed
✓ Correct
C
The number of successes in $r$ independent trials
D
The total time elapsed between $r$ independent Poisson events
💡
Step-by-Step Explanation & Concept Rationale
The Negative Binomial distribution models the number of failures $k$ (or total trials $n$) preceding the $r$-th success in independent Bernoulli trials with success probability $p$.
Q. 13
Statistics
Difficulty: medium
(1 Mark)
If a continuous random variable $X$ has a Gamma distribution $\text{Gamma}(\alpha, \beta)$ with PDF $f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}$ for $x > 0$, what are its mean and variance?
A
Mean $= \frac{\alpha}{\beta}$, Variance $= \frac{\alpha}{\beta^2}$
✓ Correct
B
Mean $= \alpha \beta$, Variance $= \alpha \beta^2$
C
Mean $= \frac{\alpha}{\beta^2}$, Variance $= \frac{\alpha}{\beta}$
D
Mean $= \frac{1}{\alpha \beta}$, Variance $= \frac{1}{\alpha \beta^2}$
💡
Step-by-Step Explanation & Concept Rationale
For the rate parameterization of the Gamma distribution $\text{Gamma}(\alpha, \beta)$, $E[X] = \frac{\alpha}{\beta}$ and $\text{Var}(X) = \frac{\alpha}{\beta^2}$.
Q. 14
Statistics
Difficulty: easy
(1 Mark)
What distribution results from the ratio of two independent Chi-square random variables, each divided by their respective degrees of freedom: $F = \frac{U / d_1}{V / d_2}$?
A
Student's t-distribution
B
Snedecor's F-distribution with $(d_1, d_2)$ degrees of freedom
✓ Correct
C
Beta distribution of the first kind
D
Gamma distribution with parameters $(d_1, d_2)$
💡
Step-by-Step Explanation & Concept Rationale
By definition, Snedecor's F-statistic is the ratio of two independent Chi-square variables scaled by their degrees of freedom: $F(d_1, d_2) = \frac{\chi^2_{(d_1)}/d_1}{\chi^2_{(d_2)}/d_2}$.
Q. 15
Statistics
Difficulty: easy
(1 Mark)
If $Z \sim N(0, 1)$ is a standard normal variable and $V \sim \chi^2_{(k)}$ is an independent Chi-square variable with $k$ degrees of freedom, what is the distribution of $T = \frac{Z}{\sqrt{V / k}}$?
A
Standard Normal distribution
B
Student's t-distribution with $k$ degrees of freedom ($t_{(k)}$)
✓ Correct
C
F-distribution with $(1, k)$ degrees of freedom
D
Cauchy distribution with parameter $k$
💡
Step-by-Step Explanation & Concept Rationale
Gosset defined Student's t-distribution as the ratio of a standard normal deviate to the square root of an independent Chi-square deviate divided by its degrees of freedom.
Q. 16
Statistics
Difficulty: medium
(1 Mark)
What is the relationship between the Student's t-distribution with $k$ degrees of freedom and the F-distribution?
A
$T_{(k)}^2 \sim F(1, k)$
✓ Correct
B
$T_{(k)} \sim F(k, 1)$
C
$T_{(k)}^2 \sim F(k, k)$
D
$\sqrt{T_{(k)}} \sim F(1, k)$
💡
Step-by-Step Explanation & Concept Rationale
Squaring a t-distributed variable with $k$ degrees of freedom yields $T^2 = \frac{Z^2}{V/k} = \frac{\chi^2_{(1)}/1}{\chi^2_{(k)}/k} \sim F(1, k)$.
Q. 17
Statistics
Difficulty: easy
(1 Mark)
As the degrees of freedom $k \to \infty$, what does Student's t-distribution converge to?
A
Standard Uniform distribution $U(-1, 1)$
B
Standard Normal distribution $N(0, 1)$
✓ Correct
C
Chi-square distribution $\chi^2_{(1)}$
D
Cauchy distribution
💡
Step-by-Step Explanation & Concept Rationale
As degrees of freedom approach infinity, the heavier tails of the t-distribution diminish, and the density function converges asymptotically to the standard normal distribution $N(0, 1)$.
Q. 18
Statistics
Difficulty: medium
(1 Mark)
What is the value of the integral $\int_{-\infty}^{\infty} e^{-x^2} dx$ (the Gaussian Gaussian integral)?
A
$\pi$
B
$\sqrt{\pi}$
✓ Correct
C
$\frac{\sqrt{\pi}}{2}$
D
$2\pi$
💡
Step-by-Step Explanation & Concept Rationale
The standard Gaussian integral $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$. (This is derived using double integration in polar coordinates).
Q. 19
Statistics
Difficulty: medium
(1 Mark)
For a Weibull distribution with PDF $f(x) = \frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k}$ ($x \ge 0$), what happens when the shape parameter $k = 1$?
A
It becomes a Standard Normal distribution
B
It reduces exactly to the Exponential distribution with rate $\frac{1}{\lambda}$
✓ Correct
C
It becomes a Rayleigh distribution
D
It becomes a Lognormal distribution
💡
Step-by-Step Explanation & Concept Rationale
When shape parameter $k = 1$, the Weibull PDF simplifies to $f(x) = \frac{1}{\lambda} e^{-x/\lambda}$, which is the Exponential distribution with scale parameter $\lambda$ (rate $\beta = 1/\lambda$).
Q. 20
Statistics
Difficulty: medium
(1 Mark)
What is the Rayleigh distribution in terms of the Weibull distribution?
A
Weibull distribution with shape parameter $k = 0.5$
B
Weibull distribution with shape parameter $k = 2$
✓ Correct
C
Weibull distribution with shape parameter $k = 3.6$
D
Weibull distribution with shape parameter $k \to \infty$
💡
Step-by-Step Explanation & Concept Rationale
The Rayleigh distribution is a special case of the Weibull distribution when the shape parameter $k = 2$. It models wind speed distributions and wave heights.
Q. 21
Statistics
Difficulty: easy
(1 Mark)
If $X \sim \text{Lognormal}(\mu, \sigma^2)$, what is the distribution of $Y = \ln(X)$?
A
Exponential distribution with mean $\mu$
B
Normal distribution $N(\mu, \sigma^2)$
✓ Correct
C
Standard Uniform distribution
D
Gamma distribution
💡
Step-by-Step Explanation & Concept Rationale
By definition, a positive random variable $X$ is lognormally distributed if its natural logarithm $\ln(X)$ is normally distributed with mean $\mu$ and variance $\sigma^2$.
Q. 22
Statistics
Difficulty: medium
(1 Mark)
What is the mean of a Lognormal random variable $X \sim \text{Lognormal}(\mu, \sigma^2)$?
A
$e^{\mu}$
B
$e^{\mu + \frac{\sigma^2}{2}}$
✓ Correct
C
$\mu + \frac{\sigma^2}{2}$
D
$e^{\mu + \sigma}$
💡
Step-by-Step Explanation & Concept Rationale
The expectation of a lognormal variable is $E[X] = e^{\mu + \frac{\sigma^2}{2}}$, which is always strictly greater than the geometric mean $e^\mu$ for $\sigma > 0$.
Q. 23
Statistics
Difficulty: medium
(1 Mark)
What is Chebyshev's Inequality for any random variable $X$ with finite mean $\mu$ and variance $\sigma^2$, for any $k > 1$?
A
$P(|X - \mu| \ge k\sigma) \le \frac{1}{k^2}$
✓ Correct
B
$P(|X - \mu| \ge k\sigma) \ge 1 - \frac{1}{k^2}$
C
$P(|X - \mu| < k\sigma) \le \frac{1}{k^2}$
D
$P(|X - \mu| \ge k\sigma) \le \frac{\sigma^2}{k}$
💡
Step-by-Step Explanation & Concept Rationale
Chebyshev's inequality guarantees that for any distribution, the probability of observing a value beyond $k$ standard deviations from the mean is at most $\frac{1}{k^2}$, i.e., $P(|X - \mu| \ge k\sigma) \le \frac{1}{k^2}$.
Q. 24
Statistics
Difficulty: easy
(1 Mark)
According to Markov's Inequality, if $X$ is a non-negative random variable ($X \ge 0$) with expectation $E[X]$, what is the upper bound on $P(X \ge a)$ for any $a > 0$?
A
$P(X \ge a) \le \frac{E[X]}{a}$
✓ Correct
B
$P(X \ge a) \le \frac{E[X^2]}{a^2}$
C
$P(X \ge a) \ge 1 - \frac{E[X]}{a}$
D
$P(X \ge a) \le \frac{a}{E[X]}$
💡
Step-by-Step Explanation & Concept Rationale
Markov's inequality states that for any non-negative random variable $X$ and constant $a > 0$, $P(X \ge a) \le \frac{E[X]}{a}$.
Q. 25
Statistics
Difficulty: medium
(1 Mark)
What does the Weak Law of Large Numbers (WLLN) state regarding the sample mean $\bar{X}_n$ of $n$ independent and identically distributed (i.i.d.) random variables with finite mean $\mu$?
A
$\bar{X}_n$ converges in distribution to a standard normal variable
B
$\bar{X}_n$ converges in probability to the population mean $\mu$ as $n \to \infty$ (i.e., $\lim_{n \to \infty} P(|\bar{X}_n - \mu| > \epsilon) = 0$)
✓ Correct
C
The variance of the population decreases to zero as $n \to \infty$
D
$\bar{X}_n$ equals $\mu$ with probability 1 for any sample size $n$
💡
Step-by-Step Explanation & Concept Rationale
The Weak Law of Large Numbers (Khinchin's theorem) guarantees convergence in probability: as sample size grows arbitrarily large, the sample mean converges in probability to the expected value $\mu$.
Study Stream Progress:
Showing 25 of 90 Questions (28%)
Jump to:
🎉
All 90 Solved Questions in Probability Distributions & Random Variables Loaded!
You have studied the complete question pool with step-by-step rationales. Ready to test yourself?
Q. 1
Statistics
Difficulty: Hard
(1 Mark)
How do changes in disease prevalence affect Positive Predictive Value (PPV) and Negative Predictive Value (NPV)?
A
Higher prevalence increases PPV and decreases NPV, while test Sensitivity and Specificity remain unchanged
✓ Correct
B
Higher prevalence increases both Sensitivity and PPV
C
Prevalence has no mathematical impact on predictive values
D
Higher prevalence decreases PPV and increases NPV
💡
Step-by-Step Explanation & Concept Rationale
Unlike sensitivity and specificity (intrinsic test properties), predictive values depend heavily on disease prevalence. As prevalence rises, True Positives increase relative to False Positives, increasing PPV, while NPV decreases.
Q. 2
Statistics
Difficulty: Medium
(1 Mark)
What distribution describes the survival time of medical patients when the instantaneous hazard rate (failure rate) is constant over time?
A
Exponential Distribution
✓ Correct
B
Normal Distribution
C
Weibull Distribution with shape parameter > 1
D
Beta Distribution
💡
Step-by-Step Explanation & Concept Rationale
The exponential distribution is the only continuous distribution with a constant failure/hazard rate (memoryless property: h(t) = lambda). When hazard changes with time, the Weibull distribution is used.
Q. 3
Statistics
Difficulty: Hard
(1 Mark)
In survival analysis and medical statistics, what is Cox Proportional Hazards Regression model used for?
A
Semi-parametric regression modeling time-to-event outcomes with multiple covariates without assuming a baseline hazard distribution
✓ Correct
B
Classifying images into medical diagnosis categories
C
Predicting mean blood pressure using ordinary least squares
D
Modeling count data with excessive zeros
💡
Step-by-Step Explanation & Concept Rationale
The Cox proportional hazards model is semi-parametric: it assumes covariate effects multiply the baseline hazard by exp(beta * X), but leaves the baseline hazard function h_0(t) completely unspecified.
Q. 4
Statistics
Difficulty: Medium
(1 Mark)
In randomized clinical trials, what does 'Intention-to-Treat' (ITT) analysis dictate?
A
All participants are analyzed according to the group they were originally randomly assigned, regardless of compliance or dropouts
✓ Correct
B
Only participants who completed 100% of prescribed medication doses are analyzed
C
Patients who switched treatment groups are re-assigned to the new group
D
Non-compliant patients are permanently deleted from sample calculations
💡
Step-by-Step Explanation & Concept Rationale
Intention-to-treat (ITT) analysis preserves the benefits of randomization, prevents attrition bias, and mirrors real-world effectiveness by analyzing every subject in their originally assigned trial arm.
Study Stream Progress:
Showing 4 of 4 Questions (100%)
🎉
All 4 Solved Questions in Probability Distributions & Sampling Loaded!
You have studied the complete question pool with step-by-step rationales. Ready to test yourself?
Q. 1
Statistics
Difficulty: easy
(1 Mark)
In Simple Random Sampling Without Replacement (SRSWOR) of size $n$ from a finite population of size $N$, what is the variance of the sample mean $\bar{y}$?
A
$\text{Var}(\bar{y}) = \frac{S^2}{n}$
B
$\text{Var}(\bar{y}) = \left(\frac{N - n}{N}\right) \frac{S^2}{n}$
✓ Correct
C
$\text{Var}(\bar{y}) = \left(\frac{N - 1}{N}\right) \frac{S^2}{n}$
D
$\text{Var}(\bar{y}) = \frac{S^2}{Nn}$
💡
Step-by-Step Explanation & Concept Rationale
In SRSWOR, the variance of the sample mean includes the Finite Population Correction (FPC) factor: $\text{Var}(\bar{y}) = \left(1 - \frac{n}{N}\right) \frac{S^2}{n} = \left(\frac{N - n}{N}\right) \frac{S^2}{n}$, where $S^2 = \frac{1}{N-1}\sum (Y_i - \bar{Y})^2$.
Q. 2
Statistics
Difficulty: easy
(1 Mark)
What is the Finite Population Correction (FPC) factor defined as in survey sampling?
A
$1 - \frac{n}{N}$ (or $\frac{N - n}{N}$)
✓ Correct
B
$\frac{n}{N}$
C
$\frac{N}{n}$
D
$1 + \frac{n}{N}$
💡
Step-by-Step Explanation & Concept Rationale
The FPC factor is $1 - f = 1 - \frac{n}{N} = \frac{N-n}{N}$, where $f = n/N$ is the sampling fraction. When $f < 0.05$ (less than 5% of the population), FPC is often omitted as it approaches 1.
Q. 3
Statistics
Difficulty: medium
(1 Mark)
Under Neyman's Optimum Allocation in Stratified Random Sampling, how is the sample size $n_h$ in stratum $h$ allocated when sampling costs per unit are equal?
A
$n_h = n \frac{N_h}{N}$
B
$n_h = n \frac{N_h S_h}{\sum N_h S_h}$
✓ Correct
C
$n_h = n \frac{S_h}{\sum S_h}$
D
$n_h = \frac{n}{L}$
💡
Step-by-Step Explanation & Concept Rationale
Neyman allocation determines that the stratum sample size $n_h$ is directly proportional to both the stratum size $N_h$ and the stratum standard deviation $S_h$: $n_h = n \frac{N_h S_h}{\sum_{i=1}^L N_i S_i}$, minimizing the variance of the stratified estimator for a fixed total sample size.
Q. 4
Statistics
Difficulty: medium
(1 Mark)
When is Stratified Random Sampling most effective at reducing the variance of the overall population mean estimator compared to Simple Random Sampling?
A
When strata are heterogeneous within and homogeneous between
B
When strata are homogeneous within (low internal variance $S_h^2$) and heterogeneous between (large differences among stratum means)
✓ Correct
C
When all strata have identical sample sizes
D
When the population size $N$ is less than 50
💡
Step-by-Step Explanation & Concept Rationale
Stratification maximizes precision when elements within each stratum are as homogeneous as possible (minimizing within-stratum variance), and means between different strata are widely separated.
Q. 5
Statistics
Difficulty: easy
(1 Mark)
What is the fundamental condition for an estimator $T_n$ to be 'Consistent' for parameter $\theta$?
A
$E[T_n] = \theta$ for all sample sizes $n$
B
$T_n$ converges in probability to $\theta$ as $n \to \infty$ (i.e., $\text{plim}_{n \to \infty} T_n = \theta$)
✓ Correct
C
$\text{Var}(T_n) = 0$ for all $n$
D
$T_n$ follows a normal distribution for any $n$
💡
Step-by-Step Explanation & Concept Rationale
Consistency is an asymptotic property stating that as sample size $n$ grows infinitely large, the probability of the estimator deviating from the true parameter by any $\epsilon > 0$ approaches zero: $\lim_{n \to \infty} P(|T_n - \theta| > \epsilon) = 0$.
Q. 6
Statistics
Difficulty: medium
(1 Mark)
If an estimator $T_n$ is asymptotically unbiased ($\\lim_{n \to \infty} E[T_n] = \theta$) and its variance vanishes as $n \to \infty$ ($\\lim_{n \to \infty} \text{Var}(T_n) = 0$), does this guarantee consistency?
A
No, consistency requires higher moments to exist
B
Yes, by Chebyshev's inequality, a vanishing mean squared error (MSE) is a sufficient condition for consistency
✓ Correct
C
Only if the population is normally distributed
D
No, it only guarantees unbiasedness
💡
Step-by-Step Explanation & Concept Rationale
Since $\text{MSE}(T_n) = \text{Var}(T_n) + [\text{Bias}(T_n)]^2$, if both variance and bias vanish as $n \to \infty$, $\text{MSE} \to 0$, which by Chebyshev's inequality ensures convergence in probability (consistency).
Q. 7
Statistics
Difficulty: medium
(1 Mark)
What does the Neyman-Pearson Factorization Theorem state regarding a statistic $T = T(\mathbf{X})$ and parameter $\theta$?
A
$T$ is unbiased if and only if $E[T] = \theta$
B
$T$ is a sufficient statistic for $\theta$ if and only if the joint likelihood can be factored as $L(\mathbf{x}; \theta) = g(T(\mathbf{x}), \theta) \cdot h(\mathbf{x})$, where $h(\mathbf{x})$ does not depend on $\theta$
✓ Correct
C
$T$ is efficient if its variance achieves the Cramér-Rao lower bound
D
$T$ is consistent if sample size exceeds 30
💡
Step-by-Step Explanation & Concept Rationale
The Factorization Criterion provides the definitive mathematical test for sufficiency: the likelihood factors into a function $g(T,\theta)$ depending on the data only through $T$, and a factor $h(\mathbf{x})$ completely free of the parameter $\theta$.
Q. 8
Statistics
Difficulty: easy
(1 Mark)
What is the sufficient statistic for parameter $\lambda$ in a random sample $X_1, X_2, \dots, X_n$ from a Poisson distribution $\text{Poisson}(\lambda)$?
A
Sample maximum $X_{(n)}$
B
Sample sum $\sum_{i=1}^n X_i$ (or sample mean $\bar{X}$)
✓ Correct
C
Sample median
D
Sample variance $S^2$
💡
Step-by-Step Explanation & Concept Rationale
The joint likelihood is $L = \frac{e^{-n\lambda} \lambda^{\sum x_i}}{\prod x_i!} = \left[e^{-n\lambda} \lambda^{\sum x_i}\right] \left[\frac{1}{\prod x_i!}\right]$. By the Factorization Theorem, $T = \sum_{i=1}^n X_i$ is jointly sufficient for $\lambda$.
Q. 9
Statistics
Difficulty: medium
(1 Mark)
What is the minimal sufficient statistic for $(\mu, \sigma^2)$ in a random sample from a Normal distribution $N(\mu, \sigma^2)$?
A
$(\bar{X}, S^2)$ or equivalently $\left(\sum_{i=1}^n X_i, \sum_{i=1}^n X_i^2\right)$
✓ Correct
B
Only the sample mean $\bar{X}$
C
Only the sample variance $S^2$
D
The sample range and median
💡
Step-by-Step Explanation & Concept Rationale
Factoring the joint normal likelihood reveals that the pair of statistics $\left(\sum X_i, \sum X_i^2\right)$ (or equivalently sample mean $\bar{X}$ and sample variance $S^2$) captures all information about both unknown parameters $(\mu, \sigma^2)$.
Q. 10
Statistics
Difficulty: hard
(1 Mark)
What does the Rao-Blackwell Theorem state regarding an unbiased estimator $T$ and a sufficient statistic $S$?
A
The estimator $T$ should be discarded in favor of the sample median
B
The conditional expectation $\delta = E[T \mid S]$ is an unbiased estimator of $\theta$ whose variance is strictly less than or equal to the variance of $T$ ($\text{Var}(\delta) \le \text{Var}(T)$)
✓ Correct
C
A sufficient statistic can never be unbiased
D
Conditioning on a sufficient statistic doubles the mean squared error
💡
Step-by-Step Explanation & Concept Rationale
Rao-Blackwellization improves any crude unbiased estimator $T$ by conditioning it on a sufficient statistic $S$: the resulting estimator $\delta = E[T \mid S]$ is independent of $\theta$, unbiased, and has variance $\le \text{Var}(T)$.
Q. 11
Statistics
Difficulty: hard
(1 Mark)
What does the Lehmann-Scheffé Theorem establish when an estimator is a function of a 'Complete and Sufficient' statistic?
A
The estimator is asymptotically normal
B
If $S$ is a complete sufficient statistic and $g(S)$ is an unbiased estimator of $\theta$, then $g(S)$ is the unique Uniformly Minimum Variance Unbiased Estimator (UMVUE) of $\theta$
✓ Correct
C
The estimator achieves the Cramér-Rao lower bound in all finite samples
D
The estimator has maximum likelihood properties
💡
Step-by-Step Explanation & Concept Rationale
Lehmann-Scheffé proves that conditioning any unbiased estimator on a complete sufficient statistic yields the unique, optimal UMVUE (no other unbiased estimator can achieve a smaller variance anywhere in parameter space).
Q. 12
Statistics
Difficulty: medium
(1 Mark)
What is Fisher Information $I(\theta)$ for a single observation from a probability distribution $f(x; \theta)$ defined as under standard regularity conditions?
A
$I(\theta) = E\left[\left(\frac{\partial \ln f(X; \theta)}{\partial \theta}\right)^2\right] = -E\left[\frac{\partial^2 \ln f(X; \theta)}{\partial \theta^2}\right]$
✓ Correct
B
$I(\theta) = \text{Var}(X) \cdot \theta$
C
$I(\theta) = \frac{1}{E[X^2]}$
D
$I(\theta) = \int f(x; \theta) dx$
💡
Step-by-Step Explanation & Concept Rationale
Fisher Information measures the amount of information an observable random variable carries about an unknown parameter, defined as the variance of the score function (or the negative expected second derivative of the log-likelihood).
Q. 13
Statistics
Difficulty: medium
(1 Mark)
What is the Cramér-Rao Lower Bound (CRLB) for the variance of any unbiased estimator $T$ of a scalar parameter $\theta$ from a sample of $n$ i.i.d. observations?
A
$\text{Var}(T) \ge \frac{1}{n I(\theta)}$
✓ Correct
B
$\text{Var}(T) \le \frac{1}{n I(\theta)}$
C
$\text{Var}(T) \ge \frac{I(\theta)}{n}$
D
$\text{Var}(T) = \frac{\sigma^2}{n}$
💡
Step-by-Step Explanation & Concept Rationale
The Cramér-Rao inequality establishes an absolute lower theoretical bound for the variance of any unbiased estimator: $\text{Var}(T) \ge \frac{1}{I_n(\theta)} = \frac{1}{n I(\theta)}$. An unbiased estimator attaining this bound is called 'Efficient'.
Q. 14
Statistics
Difficulty: medium
(1 Mark)
What is the Fisher Information $I(\lambda)$ for a single observation from a Poisson distribution $P(X=x) = \frac{e^{-\lambda}\lambda^x}{x!}$?
A
$\lambda$
B
$\frac{1}{\lambda}$
✓ Correct
C
$\lambda^2$
D
$\frac{1}{\lambda^2}$
💡
Step-by-Step Explanation & Concept Rationale
$\ln f(x; \lambda) = -\lambda + x \ln\lambda - \ln(x!)$. The second derivative is $\frac{\partial^2 \ln f}{\partial \lambda^2} = -\frac{x}{\lambda^2}$. Taking the negative expectation: $-E\left[-\frac{X}{\lambda^2}\right] = \frac{E[X]}{\lambda^2} = \frac{\lambda}{\lambda^2} = \frac{1}{\lambda}$.
Q. 15
Statistics
Difficulty: medium
(1 Mark)
For a random sample of size $n$ from $N(\mu, \sigma^2)$ with known $\sigma^2$, does the sample mean $\bar{X}$ attain the Cramér-Rao Lower Bound for estimating $\mu$?
A
No, its variance is higher than the CRLB
B
Yes, $\text{Var}(\bar{X}) = \frac{\sigma^2}{n}$, which is exactly equal to the CRLB $\frac{1}{n I(\mu)}$
✓ Correct
C
Only when $n \ge 100$
D
The CRLB cannot be computed for normal distributions
💡
Step-by-Step Explanation & Concept Rationale
For $N(\mu, \sigma^2)$, $I(\mu) = \frac{1}{\sigma^2}$. The CRLB is $\frac{1}{n(1/\sigma^2)} = \frac{\sigma^2}{n}$. Since $\text{Var}(\bar{X}) = \frac{\sigma^2}{n}$, $\bar{X}$ attains the bound with 100% efficiency.
Q. 16
Statistics
Difficulty: easy
(1 Mark)
What is the Invariance Property of Maximum Likelihood Estimators (MLE)?
A
The MLE never changes when new data is added
B
If $\hat{\theta}$ is the MLE of parameter $\theta$, then for any function $g(\theta)$, the MLE of $g(\theta)$ is $g(\hat{\theta})$
✓ Correct
C
The variance of the MLE is invariant to sample size
D
The MLE is invariant under change of coordinate origin only
💡
Step-by-Step Explanation & Concept Rationale
The Zehna Invariance Property states that if $\hat{\theta}$ is the MLE of $\theta$, then $g(\hat{\theta})$ is automatically the MLE of $g(\theta)$ (e.g., if $\hat{\sigma}$ is the MLE of $\sigma$, then $\hat{\sigma}^2$ is the MLE of $\sigma^2$).
Q. 17
Statistics
Difficulty: medium
(1 Mark)
What is the Maximum Likelihood Estimator (MLE) of the population variance $\sigma^2$ for a random sample from $N(\mu, \sigma^2)$ when both $\mu$ and $\sigma^2$ are unknown?
A
$\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})^2$
✓ Correct
B
$S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$
C
$\hat{\sigma}^2 = \frac{1}{n+1} \sum_{i=1}^n (X_i - \bar{X})^2$
D
$\hat{\sigma}^2 = \sqrt{\frac{1}{n} \sum X_i^2}$
💡
Step-by-Step Explanation & Concept Rationale
Maximizing the normal log-likelihood yields the divisor $n$: $\hat{\sigma}_{MLE}^2 = \frac{1}{n} \sum (X_i - \bar{X})^2$. Note that this MLE is biased in finite samples ($E[\hat{\sigma}^2] = \frac{n-1}{n}\sigma^2$), but asymptotically unbiased and consistent.
Q. 18
Statistics
Difficulty: easy
(1 Mark)
What is the expectation of the sample variance $S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$ for any population with finite variance $\sigma^2$?
A
$E[S^2] = \frac{n-1}{n} \sigma^2$
B
$E[S^2] = \sigma^2$ (it is strictly unbiased)
✓ Correct
C
$E[S^2] = \sigma^2 + \mu^2$
D
$E[S^2] = \frac{\sigma^2}{n}$
💡
Step-by-Step Explanation & Concept Rationale
Bessel's correction using $n-1$ in the denominator corrects the downward bias caused by estimating the population mean with the sample mean, ensuring $E[S^2] = \sigma^2$ for all sample sizes.
Q. 19
Statistics
Difficulty: hard
(1 Mark)
Under standard regularity conditions, what asymptotic distribution does the Maximum Likelihood Estimator $\hat{\theta}_n$ follow as $n \to \infty$?
A
Uniform distribution
B
Asymptotically normal: $\sqrt{n}(\hat{\theta}_n - \theta) \xrightarrow{d} N\left(0, \frac{1}{I(\theta)}\right)$
✓ Correct
C
Chi-square distribution with 1 degree of freedom
D
Student's t-distribution with $n-1$ degrees of freedom
💡
Step-by-Step Explanation & Concept Rationale
Asymptotic normality and asymptotic efficiency are hallmark properties of MLEs: as $n \to \infty$, $\hat{\theta}$ converges in distribution to a normal variable with mean $\theta$ and variance attaining the CRLB $\frac{1}{n I(\theta)}$.
Q. 20
Statistics
Difficulty: hard
(1 Mark)
In a sample of $n$ observations from a Uniform distribution $U(0, \theta)$, what is the Maximum Likelihood Estimator of $\theta$?
A
$2\bar{X}$
B
Sample maximum $X_{(n)} = \max(X_1, X_2, \dots, X_n)$
✓ Correct
C
Sample minimum $X_{(1)}$
D
$\frac{X_{(1)} + X_{(n)}}{2}$
💡
Step-by-Step Explanation & Concept Rationale
The likelihood $L(\theta) = \frac{1}{\theta^n}$ for $\theta \ge X_{(n)}$ is a monotonically decreasing function of $\theta$. It is maximized at the smallest allowable value of $\theta$, which is the sample maximum $X_{(n)}$. (Note: Regularity conditions do not hold here as support depends on $\theta$).
Q. 21
Statistics
Difficulty: easy
(1 Mark)
For $X_1, X_2, \dots, X_n \sim U(0, \theta)$, what is the Method of Moments estimator $\tilde{\theta}_{MOM}$?
A
$\tilde{\theta}_{MOM} = 2\bar{X}$
✓ Correct
B
$\tilde{\theta}_{MOM} = X_{(n)}$
C
$\tilde{\theta}_{MOM} = \bar{X} + S$
D
$\tilde{\theta}_{MOM} = \frac{\bar{X}}{2}$
💡
Step-by-Step Explanation & Concept Rationale
The population mean for $U(0, \theta)$ is $E[X] = \frac{\theta}{2}$. Equating sample mean to theoretical mean: $\bar{X} = \frac{\tilde{\theta}}{2} \implies \tilde{\theta}_{MOM} = 2\bar{X}$.
Q. 22
Statistics
Difficulty: hard
(1 Mark)
For $X \sim U(0, \theta)$, how does the Method of Moments estimator $\tilde{\theta} = 2\bar{X}$ compare to the adjusted MLE $\hat{\theta}^* = \frac{n+1}{n} X_{(n)}$?
A
The MOM estimator has smaller variance than the MLE
B
The adjusted MLE is the UMVUE and has variance $O(1/n^2)$, whereas MOM has much larger variance $O(1/n)$
✓ Correct
C
Both estimators are identical in all samples
D
The MOM estimator is sufficient, while the MLE is not
💡
Step-by-Step Explanation & Concept Rationale
Because the order statistic $X_{(n)}$ converges at rate $n$ (variance $\propto 1/n^2$), the adjusted MLE $\frac{n+1}{n}X_{(n)}$ is drastically superior to the MOM estimator $2\bar{X}$ (variance $\propto 1/n$).
Q. 23
Statistics
Difficulty: easy
(1 Mark)
What is the Method of Moments (MOM) estimation principle formulated by Karl Pearson?
A
Maximizing the log-likelihood function using calculus
B
Equating the first $k$ sample moments $m_r = \frac{1}{n}\sum X_i^r$ to the corresponding theoretical population moments $\mu_r'(\theta_1, \dots, \theta_k)$ and solving the system of equations
✓ Correct
C
Minimizing the sum of squared errors between observations
D
Conditioning on a minimal sufficient statistic
💡
Step-by-Step Explanation & Concept Rationale
Method of Moments estimates parameters by matching theoretical moments $E[X^k]$ with empirical sample moments $\frac{1}{n}\sum X_i^k$ and solving the resulting equations for the parameters.
Q. 24
Statistics
Difficulty: medium
(1 Mark)
What is a 'Pivotal Quantity' in interval estimation?
A
A statistic whose value is always equal to 0
B
A function of the sample data and the parameter of interest whose probability distribution is completely known and does NOT depend on any unknown parameters
✓ Correct
C
The ratio of the sample mean to the sample median
D
A parameter that determines the shape of the prior distribution
💡
Step-by-Step Explanation & Concept Rationale
A pivotal quantity $Q(\mathbf{X}, \theta)$ has a known distribution free of $\theta$ (e.g., $Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0, 1)$ or $T = \frac{\bar{X} - \mu}{S/\sqrt{n}} \sim t_{(n-1)}$), allowing exact algebraic construction of confidence intervals.
Q. 25
Statistics
Difficulty: easy
(1 Mark)
What is the exact $(1 - \alpha)$ confidence interval for the population mean $\mu$ of a normal distribution with UNKNOWN variance $\sigma^2$, based on a sample of size $n$?
A
$\bar{X} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
B
$\bar{X} \pm t_{\alpha/2, n-1} \frac{S}{\sqrt{n}}$
✓ Correct
C
$\bar{X} \pm \chi^2_{\alpha/2, n-1} \frac{S}{n}$
D
$\bar{X} \pm F_{\alpha/2} S^2$
💡
Step-by-Step Explanation & Concept Rationale
When population variance $\sigma^2$ is unknown and estimated by sample standard deviation $S$, the pivotal quantity follows Student's t-distribution with $n-1$ degrees of freedom, yielding $\bar{X} \pm t_{\alpha/2, n-1} \frac{S}{\sqrt{n}}$.
Study Stream Progress:
Showing 25 of 90 Questions (28%)
Jump to:
🎉
All 90 Solved Questions in Sampling Theory, Estimation & Statistical Inference Loaded!
You have studied the complete question pool with step-by-step rationales. Ready to test yourself?
Ready to Test Your Retention & Speed?
Now that you have reviewed the study questions and rationales, test yourself in our interactive 1-by-1 practice engine or take the full official timed mock exam.