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

This preview shows 10 out of 13 pages.

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
Image of page 10

Subscribe to view the full document.

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
Image of page 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.
Image of page 12

Subscribe to view the full document.

Image of page 13
You've reached the end of this preview.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern