DSE Statistical Inference and Hypothesis Testing 2 — Questions and Answers
Question 1: A researcher sets α = 0.01 instead of α = 0.05. What is the direct consequence?
- Increased power to detect true effects
- Reduced probability of a Type I error (Correct answer)
- Reduced probability of a Type II error
- Wider confidence intervals at the same confidence level
Correct answer: Reduced probability of a Type I error
Lowering α makes the rejection region smaller, directly reducing the probability of falsely rejecting a true null hypothesis (Type I error).
Question 2: Which condition must hold for the Central Limit Theorem to justify using a z-test on non-normal data?
- Population variance must be known and small
- Sample size must be sufficiently large (typically n ≥ 30) (Correct answer)
- Data must come from a symmetric distribution
- The null hypothesis must be simple
Correct answer: Sample size must be sufficiently large (typically n ≥ 30)
The CLT states that sample means approach normality as n grows, making large samples the key requirement for z-tests on non-normal populations.
Question 3: A 95% confidence interval for a mean is (12.3, 17.7). What is the correct interpretation?
- There is a 95% chance the true mean lies between 12.3 and 17.7
- 95% of all data values fall between 12.3 and 17.7
- If we repeated this procedure many times, 95% of such intervals would contain the true mean (Correct answer)
- The null hypothesis mean of 15 is accepted at the 5% level
Correct answer: If we repeated this procedure many times, 95% of such intervals would contain the true mean
Confidence intervals are a long-run frequency statement about the procedure, not a probability statement about a fixed (unknown) parameter.
Question 4: The Bonferroni correction addresses which problem in multiple hypothesis testing?
- Inflated Type II error rate across tests
- Inflated familywise Type I error rate (Correct answer)
- Non-normality of test statistics
- Unequal sample sizes across groups
Correct answer: Inflated familywise Type I error rate
Running many tests at α = 0.05 each inflates the chance of at least one false positive; Bonferroni divides α by the number of tests to control the familywise error rate.
Question 5: A paired t-test is preferred over an independent samples t-test when:
- Sample sizes in the two groups are very different
- Each observation in one group is naturally matched to one in the other (Correct answer)
- The population variances are assumed equal
- Both samples are drawn from normal distributions
Correct answer: Each observation in one group is naturally matched to one in the other
Pairing (e.g., before/after measurements on the same subject) removes between-subject variability, increasing power via the paired design.
Question 6: Under the null hypothesis, a p-value of 0.03 means:
- The probability the null hypothesis is true is 3%
- There is a 3% chance the alternative hypothesis is false
- The observed result or one more extreme would occur 3% of the time by chance alone (Correct answer)
- The effect size is practically significant at the 3% level
Correct answer: The observed result or one more extreme would occur 3% of the time by chance alone
A p-value is the probability of observing data at least as extreme as the sample data, assuming H₀ is true.
Question 7: Which test statistic is appropriate for comparing the variances of two independent normal populations?
- Chi-squared statistic
- F-statistic (Correct answer)
- Z-statistic
- T-statistic
Correct answer: F-statistic
The F-test uses the ratio of two sample variances, which follows an F-distribution under the null hypothesis of equal population variances.
DSE Statistical Inference and Hypothesis Testing 2
A researcher sets α = 0.01 instead of α = 0.05.
What is the direct consequence?