Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 26m
11. Correlation1h 6m
12. Regression1h 35m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 0m

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

Statistics

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

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1:55 minutes

Problem 7.5.3

Textbook Question

How do the critical values for a two-tailed test change as alpha decreases?

Verified step by step guidance

Understand the concept of alpha (α): Alpha represents the significance level of a hypothesis test, which is the probability of rejecting the null hypothesis when it is actually true. Common values for alpha are 0.05, 0.01, etc.

Recognize the relationship between alpha and critical values: In a two-tailed test, the critical values define the boundaries of the rejection region. As alpha decreases, the rejection region becomes smaller, meaning the critical values move further away from the center of the distribution.

Recall the distribution used: For most hypothesis tests, the critical values are determined using a standard normal (Z) distribution or a t-distribution, depending on the sample size and whether the population standard deviation is known.

Determine the new critical values: To find the critical values for a smaller alpha, calculate the z-scores or t-scores that correspond to the new alpha level. For a two-tailed test, divide alpha by 2 to account for both tails, and then find the values that leave the remaining probability in the center of the distribution.

Interpret the change: As alpha decreases, the critical values become more extreme (larger in magnitude for positive and negative values), indicating that it becomes harder to reject the null hypothesis because the rejection region is smaller.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Critical Values

Critical values are the threshold points that define the boundaries for rejecting the null hypothesis in hypothesis testing. In a two-tailed test, critical values are determined based on the significance level (alpha) and the distribution of the test statistic. They mark the points beyond which the test statistic is considered extreme enough to reject the null hypothesis.

Alpha Level (α)

The alpha level, often denoted as α, represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Common alpha levels are 0.05, 0.01, and 0.10. As alpha decreases, the criteria for rejecting the null hypothesis become stricter, leading to a higher threshold for critical values.

Two-Tailed Test

A two-tailed test is a statistical test that evaluates whether a sample mean is significantly different from a population mean in either direction (greater or less). This type of test is used when the alternative hypothesis does not specify a direction of the effect. As alpha decreases, the critical values move further from the mean, reflecting the need for stronger evidence to reject the null hypothesis.

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Textbook Question

In Exercises 11 and 12, find the P-value for the hypothesis test with the standardized test statistic z. Decide whether to reject H0 for the level of significance α.

Two-tailed test, z = 2.57, α = 0.10

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Textbook Question

In Exercises 13–16, find the critical value(s) and rejection region(s) for the type of z-test with level of significance . Include a graph with your answer.

Left-tailed test, α=0.02

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Textbook Question

In Exercises 13–16, find the critical value(s) and rejection region(s) for the type of z-test with level of significance . Include a graph with your answer.

Right-tailed test, α=0.025

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Textbook Question

In Exercises 13–16, find the critical value(s) and rejection region(s) for the type of z-test with level of significance . Include a graph with your answer.

Two-tailed test, α=0.03

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Textbook Question

Describe the difference between calculating the standardized test statistic, Z^2, for a chi-square test for variance and a chi-square test for standard deviation.

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Textbook Question

How do the requirements for a chi-square test for a variance or standard deviation differ from a z-test or a t-test for a mean?

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Textbook Question

Explain how to test a population variance or a population standard deviation.

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Textbook Question

In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.

Right-tailed test, n=10,α=0.10

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