Hypothesis Testing & Non-Parametric Methods
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Hypothesis Testing & Non-Parametric Methods
Official curriculum roadmap, subject/topic distribution, negative marking rules, pacing guidelines, and solved sample questions.
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90 MCQs
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Hypothesis Testing & Non-Parametric Methods
90 MCQs
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90 MCQs across fundamental, intermediate, and advanced concept tiers.
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45 to 60 seconds per MCQ. Flag complex problems and preserve 10 minutes for final revision.
⚖️ Scoring & Negative Marking
+1 mark per correct answer. In competitive tests with negative marking, -0.25 applies for incorrect guesses.
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To maximize your score on Hypothesis Testing & Non-Parametric Methods, candidates are advised to follow a structured three-pass approach. In the First Pass, solve all direct recall and formula-based questions within 30 seconds each to secure foundational marks. In the Second Pass, tackle multi-step analytical and quantitative reasoning problems. In the Third Pass, review marked questions and verify calculations.
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📝 Pre-Rendered Solved Sample Questions & Detailed Solutions
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Sample Question 1
Hypothesis Testing & Non-Parametric Methods
easy • Statistics
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
C
Accepting the alternative hypothesis when sample size is too large
D
Making a calculation error during arithmetic computation
✓ Correct Answer: B - Rejecting the null hypothesis $H_0$ when it is actually true
📖 Step-by-Step Solution & Conceptual Rationale:
A Type I error (probability $\alpha$, the significance level) occurs when a test rejects a true null hypothesis (a 'false positive').
Sample Question 2
Hypothesis Testing & Non-Parametric Methods
easy • Statistics
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
C
Choosing an incorrect statistical significance level
D
Using a one-tailed test instead of a two-tailed test
✓ Correct Answer: B - Failing to reject (accepting) the null hypothesis $H_0$ when it is actually false
📖 Step-by-Step Solution & Conceptual 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').
Sample Question 3
Hypothesis Testing & Non-Parametric Methods
easy • Statistics
What is the 'Power of a Test' mathematically defined as?
A
$\alpha$
B
$1 - \beta$ (the probability of correctly rejecting a false null hypothesis)
C
$1 - \alpha$
D
$\alpha / \beta$
✓ Correct Answer: B - $1 - \beta$ (the probability of correctly rejecting a false null hypothesis)
📖 Step-by-Step Solution & Conceptual 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.
Sample Question 4
Hypothesis Testing & Non-Parametric Methods
medium • Statistics
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
C
The probability that the alternative hypothesis is false
D
The probability of committing a Type II error
✓ Correct Answer: 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
📖 Step-by-Step Solution & Conceptual 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.
Sample Question 5
Hypothesis Testing & Non-Parametric Methods
easy • Statistics
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 $H_1$
C
Accept both hypotheses simultaneously
D
Discard the sample data and restart the study
✓ Correct Answer: B - Reject the null hypothesis $H_0$ in favor of the alternative hypothesis $H_1$
📖 Step-by-Step Solution & Conceptual Rationale:
When $p < \alpha$, the observed data is sufficiently improbable under $H_0$, leading to rejection of $H_0$ at the $\alpha$ significance level.
Sample Question 6
Hypothesis Testing & Non-Parametric Methods
medium • Statistics
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)
C
Paired t-test
D
Chi-Square test of goodness of fit
✓ Correct Answer: B - Welch's Satterthwaite t-test (Unequal Variance t-test)
📖 Step-by-Step Solution & Conceptual 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.
Sample Question 7
Hypothesis Testing & Non-Parametric Methods
easy • Statistics
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
$n_1 + n_2 - 2$
C
$n_1 + n_2$
D
$\min(n_1 - 1, n_2 - 1)$
✓ Correct Answer: B - $n_1 + n_2 - 2$
📖 Step-by-Step Solution & Conceptual 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 $df = n_1 + n_2 - 2$.
Sample Question 8
Hypothesis Testing & Non-Parametric Methods
easy • Statistics
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)
C
When the data follows a Poisson distribution
D
When the population variance is known exactly
✓ Correct Answer: 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)
📖 Step-by-Step Solution & Conceptual 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.
Sample Question 9
Hypothesis Testing & Non-Parametric Methods
easy • Statistics
What is the degrees of freedom for a Paired t-test with $n$ matched pairs of observations?
A
$n - 1$
B
$2n - 2$
C
$2n - 1$
D
$n$
✓ Correct Answer: A - $n - 1$
📖 Step-by-Step Solution & Conceptual 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 $n - 1$.
Sample Question 10
Hypothesis Testing & Non-Parametric Methods
easy • Statistics
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
$\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}$
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
$\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)}{k}$
✓ Correct Answer: A - $\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}$
📖 Step-by-Step Solution & Conceptual 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).
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