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Chapter 11: The Chi-Square Distribution

Introduction to Chapter 11: The Chi-Square Distribution

This is a photo of a pile of grocery store receipts. The items and prices are blurred.
Figure 1. The chi-square distribution can be used to find relationships between two things, like grocery prices at different stores. (credit: Pete/flickr)

Have you ever wondered if lottery numbers were evenly distributed or if some numbers occurred with a greater frequency? How about if the types of movies people preferred were different across different age groups? What about if a coffee machine was dispensing approximately the same amount of coffee each time? You could answer these questions by conducting a hypothesis test.

You will now study a new distribution, one that is used to determine the answers to such questions. This distribution is called the chi-square distribution.

In this chapter, you will learn the three major applications of the chi-square distribution:

  1. the goodness-of-fit test, which determines if data fit a particular distribution, such as in the lottery example
  2. the test of independence, which determines if events are independent, such as in the movie example
  3. the test of a single variance, which tests variability, such as in the coffee example

"Traditional Statistics" versus Technology Note

The traditional method of calculating degrees of freedom, the test statistic, and p-value for hypothesis testing of the chi-square distribution using by-hand methods with formulas and statistical tables is covered in the main sections of the text.

An added section titled Using Technology is provided at the end of the chapter that provide instructions for calculating values to do hypothesis testing of the chi-square distribution by using the TI-83+ and TI-84 calculators and other technology tools.

Chi-Square Distribution - Solution Sheet

To do the homework problems for this chapter, use the following solution sheet.

  1. [latex]H_0: \underline{\hspace{2cm}}[/latex]
  2. [latex]H_a: \underline{\hspace{2cm}}[/latex]
  3. What are the degrees of freedom?
  4. State the distribution to use for the test.
  5. What is the test statistic?
  6. What is the p-value? In one to two complete sentences, explain what the p-value means for this problem.
  7. Use the previous information to sketch a picture of this situation on Figure a. Clearly label and scale the horizontal axis and shade the region(s) corresponding to the p-value.
    This is a right-skewed frequency curve with blank horizontal and vertical axes.
    Figure 2. Chi-square curve to sketch a picture of the p-value
  8. Indicate the correct decision (“reject” or “do not reject” the null hypothesis) and write appropriate conclusions, using complete sentences.
    • Alpha: [latex]\underline{\hspace{2cm}}[/latex]
    • Decision: [latex]\underline{\hspace{2cm}}[/latex]
    • Reason for decision: [latex]\underline{\hspace{2cm}}[/latex]
    • Conclusion: [latex]\underline{\hspace{2cm}}[/latex]

 

Collaborative Exercise

Section 11.1

Collaborative Exercise

Look in the sports section of a newspaper or on the Internet for some sports data (baseball averages, basketball scores, golf tournament scores, football odds, swimming times, and the like). Plot a histogram and a boxplot using your data. See if you can determine a probability distribution that your data fits. Have a discussion with the class about your choice.

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