Mathematical Statistics

Mathematical Statistics Practice 2

Advanced problems on hypothesis testing, ANOVA, regression, bootstrap, and multiple testing

8 Problems

Suggested: 2 hours

Instructions

• Try to solve each problem before viewing the solution
• Click "Show Solution" to reveal the answer and detailed explanation
• Focus on understanding the problem-solving methodology

1Chi-Square Goodness-of-Fit Test

Problem

A die is rolled 120 times with the following results:

Face	1	2	3	4	5	6
Observed	15	18	25	22	19	21

Test at α = 0.05 whether the die is fair.

Answer Summary

Compare observed and expected frequencies category by category, then judge the chi-square statistic against its degrees-of-freedom benchmark.

2Two-Sample t-Test

Problem

Two independent samples are drawn from normal populations with equal variances:

Sample 1: n₁ = 10, $\bar{x}_1 = 12.5$ , s₁² = 4.8
Sample 2: n₂ = 12, $\bar{x}_2 = 10.2$ , s₂² = 5.2

Test H₀: μ₁ = μ₂ vs H₁: μ₁ > μ₂ at α = 0.05.

Answer Summary

Build the standardized mean difference using the correct pooled or Welch standard error and interpret the test through the null hypothesis.

3Power of a Test

Problem

Let $X_1, \ldots, X_{25}$ be i.i.d. N(μ, 16). Consider testing H₀: μ = 10 vs H₁: μ > 10 at α = 0.05.

(1) Find the rejection region.

(2) Calculate the power when μ = 12.

(3) Find the sample size needed to achieve power 0.90 when μ = 12.

Answer Summary

Move from the rejection rule to the probability of rejection under an alternative so you can quantify sensitivity, not just significance.

4Analysis of Variance (ANOVA)

Problem

Three treatments are compared with the following sample data:

Treatment A: 12, 15, 14, 13 (n₁ = 4)
Treatment B: 18, 20, 19, 21 (n₂ = 4)
Treatment C: 16, 15, 17, 16 (n₃ = 4)

Perform a one-way ANOVA at α = 0.05 to test if the treatment means differ.

Answer Summary

Split total variation into between-group and within-group pieces, then compare the corresponding mean squares with an F statistic.

5Linear Regression Inference

Problem

A simple linear regression $Y = \beta_0 + \beta_1 X + \varepsilon$ is fitted to n = 15 observations with results:

$\hat{\beta}_0 = 2.5$ , $\hat{\beta}_1 = 1.8$
$\sum x_i = 75$ , $\sum x_i^2 = 450$
SSE = 28.5

(1) Test H₀: β₁ = 0 vs H₁: β₁ ≠ 0 at α = 0.05.

(2) Construct a 95% CI for β₁.

Answer Summary

Estimate the slope or fitted relationship first, then use standard-error formulas and t/F inference to judge uncertainty and signal strength.

6Bootstrap Confidence Intervals

Problem

Explain the bootstrap procedure for constructing a confidence interval for the population median.

Given a sample of size n = 20, describe:

(1) The bootstrap resampling process

(2) How to construct a 95% percentile bootstrap CI

(3) Advantages and assumptions of bootstrap

Answer Summary

Resample repeatedly, build the bootstrap distribution of the estimator, and read the interval from empirical quantiles or a bias-corrected rule.

7Non-parametric Tests: Wilcoxon Rank-Sum

Problem

Two independent samples are:

Sample A: 23, 25, 28, 31, 35
Sample B: 19, 22, 24, 26, 29, 32

Use the Wilcoxon rank-sum test to test H₀: the two populations have identical distributions vs H₁: they differ, at α = 0.05.

Answer Summary

Replace raw values with ranks, compare the rank totals across groups, and interpret the result as a distribution-free location comparison.

8Multiple Testing Correction

Problem

You conduct 10 independent hypothesis tests, each at significance level α = 0.05.

(1) What is the familywise error rate (probability of at least one false rejection) if all null hypotheses are true?

(2) Apply the Bonferroni correction. What should be the individual test level?

(3) If you observe p-values: 0.002, 0.018, 0.035, 0.048, 0.12, 0.23, 0.34, 0.45, 0.67, 0.89, which hypotheses would you reject using Bonferroni?

Answer Summary

Adjust either the rejection threshold or the p-values themselves so that family-wise error or false discovery is controlled across many tests.

Related Theory

Review Confidence Intervals and Interval Estimation

Revisit the theory that supports these worked practice problems.

Review Hypothesis Testing

Revisit the theory that supports these worked practice problems.

Review Nonparametric Hypothesis Testing

Revisit the theory that supports these worked practice problems.

Review Mathematical Statistics Fundamentals

Revisit the theory that supports these worked practice problems.