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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2:32 minutes

Problem 6.4.1

Textbook Question

Does a population have to be normally distributed to use the chi-square distribution?

Verified step by step guidance

Understand the purpose of the chi-square distribution: The chi-square distribution is commonly used in hypothesis testing, particularly for tests of independence, goodness-of-fit, and variance. It is important to know the assumptions underlying its use.

Recognize the key assumption: The chi-square distribution does not require the population to be normally distributed. Instead, it assumes that the data are drawn from a random sample and that the observations are independent of each other.

Focus on sample size: For the chi-square test to be valid, the sample size should be sufficiently large. Specifically, the expected frequencies in each category should generally be 5 or more to ensure the approximation to the chi-square distribution is accurate.

Clarify the role of normality: While normality is not a requirement for the chi-square distribution, the test statistic itself is derived from the sum of squared standardized differences, which follows a chi-square distribution under the null hypothesis.

Summarize the key point: The population does not need to be normally distributed to use the chi-square distribution, but the assumptions of random sampling, independence, and adequate sample size must be met for the test to be valid.

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Key Concepts

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

Chi-Square Distribution

The chi-square distribution is a statistical distribution commonly used in hypothesis testing, particularly for categorical data. It is defined by its degrees of freedom, which are determined by the number of categories or groups being analyzed. This distribution is crucial for tests such as the chi-square test of independence and the goodness-of-fit test.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. Many statistical methods assume that data follows a normal distribution, particularly in parametric tests. However, the chi-square distribution does not require the underlying population to be normally distributed, making it versatile for categorical data analysis.

Assumptions of Statistical Tests

Statistical tests often come with specific assumptions regarding the data, such as independence, sample size, and distribution shape. For the chi-square test, the primary assumptions include having a sufficiently large sample size and expected frequencies in each category. Understanding these assumptions helps determine the appropriateness of the test for the given data, regardless of the population's distribution.

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Related Practice

Textbook Question

In Exercises 1–10, use the following weights (g) of Hershey’s Kisses from Data Set 38 “Candies” in Appendix B.

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tTest The bag of Hershey’s Kisses includes 75 of the candies. According to the label, the total weight is 340 g, so the candies should have a mean weight of 340/75 = 4.5333 g. Use the sample of weights listed above to test the claim that they are from a population with a mean of 4.5333 g. Use a 0.05 significance level with a t test. What does the conclusion suggest about the claim of 340 g printed on the label?

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Textbook Question

In Exercises 1–10, use the following weights (g) of Hershey’s Kisses from Data Set 38 “Candies” in Appendix B.

Sign Test Repeat Exercise 3 using the sign test to test the claim that the sample of weights is from a population with a median of 4.5333 g.

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Textbook Question

In Exercises 1–10, use the following weights (g) of Hershey’s Kisses from Data Set 38 “Candies” in Appendix B.

Wilcoxon Signed-Ranks Test Repeat Exercise 3 using the Wilcoxon signed-ranks test to test the claim that the sample of weights is from a population with a median of 4.5333 g.

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Textbook Question

In Exercises 1–10, use a 0.05 significance level with the indicated test. If no particular test is specified, use the appropriate nonparametric test from this chapter.

World Series The last 114 baseball World Series ended with 66 wins by American League teams and 48 wins by National League teams. Use the sign test to test the claim that in each World Series, the American League team has a 0.5 probability of winning.

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Textbook Question

What happens to the shape of the chi-square distribution as the degrees of freedom increase?

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Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.98, n = 26

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Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.99, n = 30

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Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.80, n = 51

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