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Hypothesis Testing, Regression & ANOVA (Statistics) Solved Questions & Notes (2026) - Apex Rankers

Statistics & Data Science > Statistics > Hypothesis Testing, Regression & ANOVA

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Hypothesis Testing, Regression & ANOVA

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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.
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