A monograph written in 1902 states that the mean height of adult American males is 67.0 inches with a standard deviation of 3.5 inches. Wishing to see if these values have changed over the twentieth century the geneticist measured a random sample of 28 adult American males and found that X = 69.4 inches and s = 4.0 inches. Are these values significantly different from the values published in 1902?
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Step-by-step explanation:
EXAMPLE 6 (testing mean and variance in a single example) A geneticist interested in human populations has been studying growth patterns in American males since 1900. A monograph written in 1902 states that the mean height of adult American males is 67.0 inches with a standard deviation of 3.5 inches. Wishing to see if these values have changed over the twentieth century the geneticist measured a random sample of 28 adult American males and found that 4.69Xinches and s = 4.0 inches. Are these values significantly different from the values published in 1902? Solution There are two questions here –one about the mean and a second about the standard deviation or variance. Two questions require two sets of hypotheses and two test statistics. For the question about means, the hypotheses are inchesHinchesHa0.67:0.67:0With n = 28and 01.0This is a two-tailed test with the question and hypotheses (0Hand aH) formulated before the data were collected or analyzed: 16.376.04.2280.40.674.69nsXtUsing an alpha level of 0.01 for 271nv, we find the critical values to be 771.2(normal table). Since 3.16 > 2.771, we reject 0Hand conclude that the modern mean is significantly different from that reported in 1902 and, in fact, is higher than the reported value (because the value falls in the right-hand tail). For the question about variances, the hypotheses are inchesHinchesHa25.12:25.12:220Here n = 28. Then 3.3525.12161281222snThe question about variability is answered with a chi-square statistic. The 2value is expected to be close to 27 (n - I). if 0His true and significantly different from 27, if aHis true.