9-2 Two Means: Independent Samples 457 Confidence Interval Estimate of M1 − M2: Independent Samples The confidence interval estimate of the difference m1 - m2 is 1x1 - x22 - E 6 1m1 - m22 6 1x1 - x22 + E where E = ta>2Bs2 1 n1 + s2 2 n2 and the number of degrees of freedom df is as described for hypothesis tests. (In this book, we use df = smaller of n1 - 1 and n2 - 1.) Equivalent Methods The P-value method of hypothesis testing, the critical value method of hypothesis testing, and confidence intervals all use the same distribution and standard error, so they are all equivalent in the sense that they result in the same conclusions. P-Value Method Are People Getting Taller? EXAMPLE 1 Listed below are heights (mm) of randomly selected U.S. Army male personnel measured in 1988 (from Data Set 2 “ANSUR I 1988”) and different heights (mm) of randomly selected U.S. Army male personnel measured in 2012 (from Data Set 3 “ANSUR II 2012”). Use a 0.05 significance level to test the claim that the mean height of the 1988 population is less than the mean height of the 2012 population. ANSUR I 1988 1698 1727 1734 1684 1667 1680 1785 1885 1841 1702 1738 1732 ANSUR II 2012 1810 1850 1777 1811 1780 1733 1814 1861 1709 1740 1694 1766 1748 1794 1780 SOLUTION REQUIREMENT CHECK (1) The values of the two population standard deviations are not known and we are not making an assumption that they are equal. (2) The two samples are independent because the measurements are from different people, and they are not matched or paired in any way. (3) The samples are simple random samples. (4) Both samples are small (30 or fewer), so we need to determine whether both samples come from populations having normal distributions. Normal quantile plots of the two samples suggest that the samples are from populations having distributions that are not far from normal. The requirements are all satisfied. Using the P-value method summarized in Figure 8-1 on page 376, we can test the claim as follows. Step1: The claim that “the mean height of the 1988 population is less than the mean height of the 2012 population” can be expressed as m1 6 m2. Step2: If the original claim is false, then m1 Ú m2. Step3: The alternative hypothesis is the expression not containing equality, and the null hypothesis is an expression of equality, so we have H0: m1 = m2 H1: m1 6 m2 We now proceed with the assumption that m1 = m2, or m1 - m2 = 0. continued
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