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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2:04 minutes

Problem 7.5.5

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?

Verified step by step guidance

Understand the purpose of the chi-square test: The chi-square test for variance or standard deviation is used to test hypotheses about the variability of a population, whereas z-tests and t-tests are used to test hypotheses about the population mean.

Identify the assumptions for the chi-square test: The chi-square test requires that the population being tested is normally distributed. This is a stricter assumption compared to z-tests and t-tests, which can sometimes be applied to approximately normal distributions or large sample sizes.

Recognize the difference in test statistics: The chi-square test statistic is calculated using the formula \( \chi^2 = \frac{(n-1)s^2}{\sigma^2} \), where \( n \) is the sample size, \( s^2 \) is the sample variance, and \( \sigma^2 \) is the hypothesized population variance. In contrast, z-tests and t-tests use formulas involving the sample mean and standard error.

Understand the degrees of freedom: For the chi-square test, the degrees of freedom are \( n-1 \), where \( n \) is the sample size. For t-tests, the degrees of freedom depend on the sample size and the type of test (e.g., one-sample or two-sample). Z-tests do not involve degrees of freedom as they are based on the standard normal distribution.

Note the type of hypothesis tested: The chi-square test is specifically designed to test hypotheses about variance or standard deviation, while z-tests and t-tests focus on hypotheses about the mean. This makes the chi-square test suitable for questions about variability rather than central tendency.

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Key Concepts

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

Chi-Square Test

The chi-square test is a statistical method used to determine if there is a significant difference between the expected and observed frequencies in categorical data. It is particularly useful for assessing variance or standard deviation in a population, requiring that the data be independent, randomly sampled, and that the sample size is sufficiently large to ensure the validity of the test.

Z-Test and T-Test

Z-tests and t-tests are statistical tests used to determine if there is a significant difference between the means of two groups. A z-test is appropriate when the population variance is known and the sample size is large (typically n > 30), while a t-test is used when the population variance is unknown and the sample size is smaller, relying on the t-distribution for more accurate results.

Assumptions of Tests

Each statistical test has specific assumptions that must be met for the results to be valid. For chi-square tests, the data should be categorical, while z-tests and t-tests assume that the data is continuous and normally distributed. Additionally, z-tests require known population variance, whereas t-tests do not, highlighting the differences in their application based on the nature of the data.

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Related Practice

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

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

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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

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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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.

Left-tailed test, n=7,α=0.01

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