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Regression Analysis: Strength versus Length The regression equation is Strength 5.11 2.90 Length Predictor Constant Length Coef 5.115 2.9027 SE Coef 1.146 0.1170 T 4.46 24.80 P 0.000 0.000 R-Sq(adj) 96.2%

R-Sq 96.4% R-Sq(pred) 95.54%

F 615.08

... .

P 0.000

Note that r 2 (0.9818)2 0.9640 (which is reported in the Minitab output), or that approximately 96.40% of the variability in pull strength is explained by the linear relationship to wire length. Now suppose that we wish to test the hypothesis H0: H1: 0 0

4 5 1 5

0.9818123 11 0.9640

This statistic is also reported in the Minitab output as a test of H0: 1 0. Because t0.025,23 2.069, we reject H0 and conclude that the correlation coef cient 0. Finally, we may construct an approximate 95% con dence interval on from Equation 10-57. Since arctanh r arctanh 0.9818 2.3452, Equation 11-50 becomes tanh a2.3452 which reduces to 0.9585 0.9921 1.96 b 122 tanh a2.3452 1.96 b 122

1<:3 = 9

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11-55. The nal test and exam averages for 20 randomly selected students taking a course in engineering statistics and a course in operations research follow. Assume that the nal averages are jointly normally distributed. (a) Find the regression line relating the statistics nal average to the OR nal average. (b) Test for signi cance of regression using 0.05. Statistics OR Statistics OR Statistics OR Statistics 86 80 94 95 84 85 75 75 81 83 80 71 72 71 69 75 86 81 62 65 76 75 81 71 76 90 93 84 90 92 65 72 83 81 97 (c) (d) (e) (f) Systolic BP 130 133 150 128 151 146 150 140 148 125 133 135 150 Systolic BP 153 128 132 149 158 150 163 156 124 170 165 160 159

114/143

(c) (d) (e) (f)

OR 70 73 72 80 98 Estimate the correlation coef cient. Test the hypothesis that 0, using 0.05. Test the hypothesis that 0.5, using 0.05. Construct a 95% con dence interval for the correlation coef cient. 11-56. The weight and systolic blood pressure of 26 randomly selected males in the age group 25 to 30 are shown in the following table. Assume that weight and blood pressure are jointly normally distributed. (a) Find a regression line relating systolic blood pressure to weight. (b) Test for signi cance of regression using 0.05.

... ...

Estimate the correlation coef cient. Test the hypothesis that 0, using 0.05. Test the hypothesis that 0.6, using 0.05. Construct a 95% con dence interval for the correlation coef cient. 11-57. Consider the NFL data introduced in Exercise 11-4. (a) Estimate the correlation coef cient between the number of games won and the yards rushing by the opponents. (b) Test the hypothesis H0: 0 versus H1: 0 using 0.05. What is the P-value for this test (c) Construct a 95% con dence interval for . (d) Test the hypothesis H0: 0.7 versus H1: 0.7 using 0.05. Find the P-value for this test.

11-58. Show that the t-statistic in Equation 11-46 for testing H0: 0 is identical to the t-statistic for testing H0: 1 0. 11-59. A random sample of 50 observations was made on the diameter of spot welds and the corresponding weld shear strength. (a) Given that r 0.62, test the hypothesis that 0, using 0.01. What is the P-value for this test (b) Find a 99% con dence interval for . (c) Based on the con dence interval in part (b), can you conclude that 0.5 at the 0.01 level of signi cance 11-60. Suppose that a random sample of 10,000 (X, Y ) pairs yielded a sample correlation coef cient of r 0.02. (a) What is the conclusion that you would reach if you tested H0: 0 using 0.05 What is the P-value for this test (b) Comment on the practical signi cance versus the statistical signi cance of your answer. 11-61. The following data gave X the water content of snow on April 1 and Y the yield from April to July (in inches) on the Snake River watershed in Wyoming for 1919 to 1935. (The data were taken from an article in Research Notes, Vol. 61, 1950, Paci c Northwest Forest Range Experiment Station, Oregon)

assuming that 9 V K3 > 6 V when the sums are taken mod 256. Carrying this one step further, we have i = 4 and

32 30.012 0.0052 4

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