13. Chi-Square Tests & Goodness of Fit

[DATA] Pedestrian Deaths A researcher wanted to determine whether pedestrian deaths were uniformly distributed over the days of the week. She randomly selected 300 pedestrian deaths, recorded the day of the week on which the death occurred, and obtained the following results (the data are based on information obtained from the Insurance Institute for Highway Safety). Test the belief that the day of the week on which a fatality happens involving a pedestrian occurs with equal frequency at the alpha = 0.05 level of significance.

In Section 10.2, we tested hypotheses regarding a population proportion using a z-test. However, we can also use the chi-square goodness-of-fit test to test hypotheses with k = 2 possible outcomes. In Problems 25 and 26, we test hypotheses with the use of both methods.

Living Alone? In 2000, 25.8% of Americans 15 years of age or older lived alone, according to the Census Bureau. A sociologist, who believes that this percentage is greater today, conducts a random sample of 400 Americans 15 years of age or older and finds that 164 are living alone.

b. Test the sociologist’s belief at the alpha=0.05 level of significance using the goodness-of-fit test.

A researcher wanted to determine if the distribution of educational attainment of Americans today is different from the distribution in 2000. The distribution of educational attainment in 2000 was as follows:

Source: Statistical Abstract of the United States.

The researcher randomly selects 500 Americans, learns their levels of education, and obtains the data shown in the table. Do the data suggest that the distribution of educational attainment has changed since 2000? Use the α = 0.1 level of significance.

[NOW WORK] Job Satisfaction Is there an association between one’s level of education and satisfaction with work? A random sample of 5244 employed individuals were asked to disclose their highest level of education and satisfaction with their work/job. The results are shown in the table below. The data are from the General Social Survey.

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a. Compute the expected values of each cell under the assumption of independence.

In Problems 5 and 6, determine the expected counts for each outcome.

[NOW WORK]

In Problems 5 and 6, determine the expected counts for each outcome.

[NOW WORK]

Benford’s Law

According to Benford’s law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below. In Exercises 21–24, test for goodness-of-fit with the distribution described by Benford’s law.

Detecting Fraud When working for the Brooklyn district attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 15, 0, 76, 479, 183, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed frequencies are substantially different from the frequencies expected with Benford’s law, the check amounts appear to result from fraud. Use a 0.01 significance level to test for goodness-of-fit with Benford’s law. Does it appear that the checks are the result of fraud?

Does It Matter Where I Sit? Does the location of your seat in a classroom play a role in attendance or grade? To answer this question, professors randomly assigned 400 students * in a general education physics course to one of four groups. Source: Perkins, Katherine K. and Wieman, Carl E, “The Surprising Impact of Seat Location on Student Performance” The Physics Teacher, Vol. 43, Jan. 2005.

The 100 students in group 1 sat 0 to 4 meters from the front of the class, the 100 students in group 2 sat 4 to 6.5 meters from the front, the 100 students in group 3 sat 6.5 to 9 meters from the front, and the 100 students in group 4 sat 9 to 12 meters from the front.

c. At the end of the semester, the proportion of students in the top 20% of the class was determined. Of the students in group 1, 25% were in the top 20%; of the students in group 2, 21% were in the top 20%; of the students in group 3, 15% were in the top 20%; of the students in group 4, 19% were in the top 20%. How many students would we expect to be in the top 20% of the class if seat location plays no role in grades? Is there a significant difference in the number of students in the top 20% of the class by group?

The following table contains the number of successes and failures for three categories of a variable.

Test whether the proportions are equal for each category at the level of significance.

The following table contains the number of successes and failures for three categories of a variable.

Test whether the proportions are equal for each category at the alpha=0.01 level of significance.

Celebrex Celebrex, a drug manufactured by Pfizer, Inc., is used to relieve symptoms associated with osteoarthritis and rheumatoid arthritis in adults. In clinical trials of the medication, some subjects reported dizziness as a side effect. The researchers wanted to discover whether the proportion of subjects taking Celebrex who reported dizziness as a side effect differed significantly from that for other treatment groups. The following data were collected.

a. Test whether the proportion of subjects within each treatment group who experienced dizziness are the same at the alpha=0.05 level of significance.

In Problems 5–8, find the critical values χ²₁₋ᵅ⁄₂ and χ²ᵅ⁄₂ for the given level of confidence and sample size.

6. 95% confidence, n = 25

In Problems 5–8, find the critical values χ²₁₋ᵅ⁄₂ and χ²ᵅ⁄₂ for the given level of confidence and sample size.

8. 99% confidence, n = 14

[DATA] Heights of Baseball PlayersData obtained from the National Center for Health Statistics show that men between the ages of 20 and 29 have a mean height of 69.3 inches, with a standard deviation of 2.9 inches. A baseball analyst wonders whether the standard deviation of heights of major-league baseball players is less than 2.9 inches. The heights (in inches) of 20 randomly selected players are shown in the table.

c. Test the notion at the α = 0.01 level of significance.

True or False: The shape of the chi-square distribution depends on the degrees of freedom.