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Hypothesis Testing & Non-Parametric Methods (Statistics) Solved Questions & Notes (2026) - Apex Rankers

Statistics & Data Science > Statistics > Hypothesis Testing & Non-Parametric Methods

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Hypothesis Testing & Non-Parametric Methods

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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.
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