SECTION 10.1 Goodness-of-Fit Test 535 15. Home Sizes An organization claims that the number of prospective home buyers who want their next house to be larger, smaller, or the same size as their current house is not uniformly distributed. To test this claim, you randomly select 800 prospective home buyers and ask them what size they want their next house to be. The table at the left shows the results. At a = 0.05, test the organization’s claim. (Adapted from Better Homes and Gardens) 16. Births by Day of the Week A doctor claims that the number of births by day of the week is uniformly distributed. To test this claim, you randomly select 700 births from a recent year and record the day of the week on which each takes place. The table shows the results. At a = 0.10, test the doctor’s claim. (Adapted from National Center for Health Statistics) Day Frequency, f Sunday 69 Monday 106 Tuesday 113 Wednesday 111 Thursday 112 Friday 111 Saturday 78 Extending Concepts Testing for Normality Using a chi-square goodness-of-fit test, you can decide, with some degree of certainty, whether a variable is normally distributed. In all chi-square tests for normality, the null and alternative hypotheses are as listed below. H0: The variable has a normal distribution. Ha: The variable does not have a normal distribution. To determine the expected frequencies when performing a chi-square test for normality, first estimate the mean and standard deviation of the frequency distribution. Then, use the mean and standard deviation to compute the z-score for each class boundary. Then, use the z-scores to calculate the area under the standard normal curve for each class. Multiplying the resulting class areas by the sample size yields the expected frequency for each class. In Exercises 17 and 18, (a) find the expected frequencies, (b) find the critical value and identify the rejection region, (c) find the chi-square test statistic, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. 17. Test Scores At a = 0.01, test the claim that the 200 test scores shown in the frequency distribution are normally distributed. Class boundaries 49.5–58.5 58.5–67.5 67.5–76.5 76.5–85.5 85.5–94.5 Frequency, f 19 61 82 34 4 18. Test Scores At a = 0.05, test the claim that the 400 test scores shown in the frequency distribution are normally distributed. Class boundaries 50.5–60.5 60.5–70.5 70.5–80.5 80.5–90.5 90.5–100.5 Frequency, f 46 88 151 97 18 Response Frequency, f Larger 285 Same size 224 Smaller 291 TABLE FOR EXERCISE 15
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