# Step 1 if there is a difference we expect the mean of

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Step 1: If there is a difference, we expect the mean of the d values to be different from 0. This is expressed in symbolic form as μ d 6 = 0. Step 2: If the original claim is not true, we have μ d = 0. Step 3: The null hypothesis must contain equality, so we have H 0 : μ d = 0 H 1 : μ d 6 = 0 (original claim) 10

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Step 4: The significance level is α = 0 . 05. Step 5: Because we are testing a claim about the means of paired dependent data, we use the Student t distribution. Step 6: Before finding the value of the test statistic, we must first find the values of ¯ d and s d . When we evaluate the difference d for each subject, we find these differences d = right - left: -33, -74, -75, -42, -80, -36, -6, -10, -28, -86, 33, -61, -56, -41 ¯ d = d n = - 595 14 = - 42 . 5 s d = s n ( d 2 ) - ( d ) 2 n ( n - 1) = s 14(39 , 593) - ( - 595) 2 14(14 - 1) = 33 . 2 With these statistics and the assumption that μ d = 0, we can now find the value of the test statistic. t = ¯ d - μ d s d n = - 42 . 5 - 0 33 . 2 14 = - 4 . 790 . Fail to reject μ d = 0 Reject μ d = 0 Reject μ d = 0 μ d = 0 or t =0 t = 2.160 t = -2.160 Sample data: t = -4.790 The critical values of t = - 2 . 160 and t = 2 . 160 are found from Table A-3; use the column for 0.05 (two tails), and use the row with degrees of freedom of n - 1 = 13. The above figure shows the test statistic, critical values, and critical region. Step 7: Because the test statistic does fall in the critical region, we reject the null hypothesis of μ d = 0. Step 8: There is sufficient evidence to support the claim of a difference between the right- and left-hand reaction times. Because there does appear to be 11
such a difference, an engineer designing a fighter-jet cockpit should locate the ejection-seat activator so that it is readily accessible to the faster hand, which appears to be the right hand with seemingly lower reaction times. (We could require special training for left-handed pilots if a similar test of left-handed pilots if a similar test of left-handed pilots shows that their dominant hand is faster.) / £ ¡ ¢ Confidence Intervals The confidence interval estimate of the mean difference μ d is as follows: ¯ d - E < μ d < ¯ d + E where E = t α/ 2 s d n and degrees of freedom = n - 1. / £ ¡ ¢ EXAMPLE 6.8 Use the sample data from the preceding example to construct a 95% confi- dence interval estimate of μ d . SOLUTION Using the values of ¯ d = - 42 . 5, s d = 33 . 2, n = 14, and t α/ 2 = 2 . 160, we first find the value of the margin of error E . E = t α/ 2 s d n = 2 . 160 33 . 2 14 = 19 . 2 The confidence interval can now be found. ¯ d - E < μ d < ¯ d + E - 42 . 5 - 19 . 2 < μ d < - 42 . 5 + 19 . 2 - 61 . 7 < μ d < - 23 . 3 . ¤ § ¥ ƒ Crest and Dependent Samples In the late 1950s, Procter & Gamble introduced Crest toothpaste as the first such product with fluoride. To test the effectiveness of Crest in reducing cavities, researchers conducted experiments with several sets of twins. One of the twins in each set was given Crest with fluoride, while the other twin continued to use ordinary toothpaste without fluoride. It was believed that each pair of twins would have similar eating, brushing, and genetic characteristics.

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