530 CHAPTER 10 Chi-Square Tests and the F -Distribution Ages Previous age distribution Survey results 0 – 9 16% 76 10 – 19 20% 84 20 – 29 8% 30 30 – 39 14% 60 40 – 49 15% 54 50 – 59 12% 40 60 – 69 10% 42 70+ 5% 14 TRY IT YOURSELF 2 A sociologist claims that the age distribution for the residents of a city is different from the distribution 10 years ago. The distribution of ages 10 years ago is shown in the table at the left. You randomly select 400 residents and record the age of each. The survey results are shown in the table. At a = 0.05, perform a chi-square goodness-of-fit test to test whether the distribution has changed. Answer: Page A42 The chi-square goodness-of-fit test is often used to determine whether a distribution is uniform. For such tests, the expected frequencies of the categories are equal. When testing a uniform distribution, you can find the expected frequency of each category by dividing the sample size by the number of categories. For instance, suppose a company believes that the number of sales made by its sales force is uniform throughout a five-day workweek. If the sample consists of 1000 sales, then the expected value of the sales for each day will be 1000 5 = 200. Performing a Chi-Square Goodness-of-Fit Test A researcher claims that the number of different-colored candies in bags of dark chocolate M&M’s® is uniformly distributed. To test this claim, you randomly select a bag that contains 500 dark chocolate M&M’s® and determine the frequency of each color. The results are shown in the table below. At a = 0.10, test the researcher’s claim. (Adapted from Mars, Incorporated) Color Frequency, f Brown 80 Yellow 95 Red 88 Blue 83 Orange 76 Green 78 SOLUTION The claim is that the distribution is uniform, so the expected frequencies of the colors are equal. To find each expected frequency, divide the sample size by the number of colors. So, for each color, E = 500 6 ≈ 83.333. Because each expected frequency is at least 5 and the M&M’s® were randomly selected, you can use the chi-square goodness-of-fit test to test the expected distribution. Here are the null and alternative hypotheses. H0: The expected distribution of the different-colored candies in bags of dark chocolate M&M’s® is uniform. (Claim) Ha: The distribution of the different-colored candies in bags of dark chocolate M&M’s® is not uniform. Because there are 6 categories, the chi-square distribution has d.f. = k - 1 = 6 - 1 = 5 degrees of freedom. Using d.f. = 5 and a = 0.10, the critical value is x 2 0 = 9.236. The rejection region is x 2 7 9.236. To find the chi-square test statistic using a table, use the observed and expected frequencies, as shown on the next page. EXAMPLE 3
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