MS-DS Master of Data science Master of Data science Statistical Inference Concepts 1 — Questions and Answers
Question 1: When performing a two-tailed hypothesis test at a significance level of α = 0.05, what is the critical region in each tail?
- 0.10
- 0.05
- 0.025 (Correct answer)
- 0.01
Correct answer: 0.025
In a two-tailed test, the significance level α is split equally between both tails. With α = 0.05, each tail contains 0.025 of the probability, so the critical region in each tail is 0.025.
Question 2: A researcher increases the sample size in a study while keeping all other factors constant. What is the most likely effect on the width of a 95% confidence interval?
- The interval becomes wider
- The interval becomes narrower (Correct answer)
- The interval remains unchanged
- The interval shifts to the right
Correct answer: The interval becomes narrower
Increasing sample size reduces the standard error (SE = σ/√n), which directly narrows the margin of error and thus produces a narrower confidence interval, reflecting greater precision.
Question 3: Which of the following best describes a p-value in the context of hypothesis testing?
- The probability that the null hypothesis is true
- The probability of observing results at least as extreme as the sample data, assuming the null hypothesis is true (Correct answer)
- The probability that the alternative hypothesis is true
- The probability of committing a Type I error in repeated experiments
Correct answer: The probability of observing results at least as extreme as the sample data, assuming the null hypothesis is true
A p-value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, calculated under the assumption that the null hypothesis is true. It does not give the probability that H₀ is true.
Question 4: What happens to the probability of a Type I error if a researcher lowers the significance level from 0.05 to 0.01?
- It increases
- It decreases (Correct answer)
- It remains the same
- It becomes equal to the probability of a Type II error
Correct answer: It decreases
The significance level α directly defines the probability of a Type I error (rejecting a true null hypothesis). Lowering α from 0.05 to 0.01 directly reduces the Type I error rate, though it simultaneously increases the risk of a Type II error.
Question 5: In a one-sample z-test, which assumption is required about the population variance?
- The population variance must be unknown and estimated from the sample
- The population variance must be known (Correct answer)
- The population variance must equal the sample variance
- The population variance must be less than 1
Correct answer: The population variance must be known
The one-sample z-test requires that the population variance (σ²) is known. When the population variance is unknown and must be estimated from the sample, a t-test is used instead.
Question 6: A 99% confidence interval is constructed instead of a 95% confidence interval from the same sample. How does this affect the interval?
- It becomes narrower and more precise
- It becomes wider but provides less certainty
- It becomes wider and provides greater certainty of capturing the true parameter (Correct answer)
- It becomes narrower and provides greater certainty
Correct answer: It becomes wider and provides greater certainty of capturing the true parameter
Increasing the confidence level from 95% to 99% requires a larger critical value (e.g., z = 2.576 vs 1.96), which widens the interval. The wider interval offers a higher probability (99%) of containing the true population parameter.
MS-DS Master of Data science Master of Data science Statistical Inference Concepts 1
When performing a two-tailed hypothesis test at a significance level of α = 0.05, what is the critical region in each tail?