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

Previous problemNext problem

1:56 minutes

Problem 6.4.2

Textbook Question

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

Verified step by step guidance

Understand the chi-square distribution: The chi-square distribution is a continuous probability distribution that is commonly used in hypothesis testing and confidence interval estimation for variance. It is defined by its degrees of freedom (df).

Recall the relationship between degrees of freedom and the shape of the chi-square distribution: The degrees of freedom (df) determine the shape of the chi-square distribution. As df increases, the distribution changes its characteristics.

Note the behavior for small degrees of freedom: When the degrees of freedom are small (e.g., df = 1 or 2), the chi-square distribution is highly skewed to the right, meaning it has a long tail on the positive side.

Describe the effect of increasing degrees of freedom: As the degrees of freedom increase, the chi-square distribution becomes less skewed and starts to resemble a normal distribution. This is because the central limit theorem implies that the sum of a large number of independent squared standard normal variables approaches a normal distribution.

Summarize the trend: In summary, as the degrees of freedom increase, the chi-square distribution becomes more symmetric and approaches the shape of a normal distribution, with the peak shifting slightly to the right and the spread increasing.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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 continuous probability distribution that arises in statistics, particularly in hypothesis testing and confidence interval estimation for variance. It is defined by its degrees of freedom, which are typically related to the number of independent standard normal variables being squared and summed. The distribution is positively skewed, especially with low degrees of freedom.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation. In the context of the chi-square distribution, the degrees of freedom are often determined by the number of categories minus one. As the degrees of freedom increase, the distribution becomes less skewed and approaches a normal distribution.

Shape of the Distribution

The shape of a distribution describes how the values are spread out across different ranges. For the chi-square distribution, as the degrees of freedom increase, the distribution becomes more symmetric and approaches a bell-shaped curve. This transition indicates that with more data, the variability in the sample estimates becomes more stable, leading to a distribution that resembles the normal distribution.

Watch next

Master Step 1: Write Hypotheses with a bite sized video explanation from Patrick

Start learning

Related Videos

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.

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.

views

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.

views

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.

views

Textbook Question

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

views

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

views

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

views

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

views

Textbook Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.99, n = 10

views

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap