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.

Bicycle Deaths A researcher wanted to determine whether bicycle deaths were uniformly distributed over the days of the week. She randomly selected 200 deaths that involved a bicycle, 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). Is there reason to believe that bicycle fatalities occur with equal frequency with respect to day of the week at the alpha = 0.05 level of significance?

[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.

Roulette Wheel A pit boss suspects that a roulette wheel is out of balance. A roulette wheel has 18 black slots, 18 red slots, and 2 green slots. The pit boss spins the wheel 500 times and records the following frequencies:

Is the wheel out of balance? Use the α = 0.05 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.

[Image]

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?