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follows a t distribution with n 1 degrees of freedom. Then, since P( t 2,n 1 T t 2,n 1) 1 , we can substitute for T in the above expression and perform the necessary steps to isolate D )% 1 2 between the inequalities. This leads to the following 100(1 con dence interval on 1 2.
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De nition If d and sD are the sample mean and standard deviation of the difference of n random pairs of normally distributed measurements, a 100(1 )% con dence interval on the difference in means D is 1 2 d where t /2,n freedom.
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Table 10-3 Time in Seconds to Parallel Park Two Automobiles Automobile Subject 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1(x1j) 37.0 25.8 16.2 24.2 22.0 33.4 23.8 58.2 33.6 24.4 23.4 21.2 36.2 29.8 2(x2j) 17.8 20.2 16.8 41.4 21.4 38.4 16.8 32.2 27.8 23.2 29.6 20.6 32.2 53.8 Difference (dj) 19.2 5.6 0.6 17.2 0.6 5.0 7.0 26.0 5.8 1.2 6.2 0.6 4.0 24.0
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2 2 2 This con dence interval is also valid for the case where 2 1 2 , because sD estimates D V(X1 X2). Also, for large samples (say, n 30 pairs), the explicit assumption of normality is unnecessary because of the central limit theorem.
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EXAMPLE 10-10
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The journal Human Factors (1962, pp. 375-380) reports a study in which n 14 subjects were asked to parallel park two cars having very different wheel bases and turning radii. The time in seconds for each subject was recorded and is given in Table 10-3. From the column of observed differences we calculate d 1.21 and sD 12.68. The 90% con dence interval for D 1 2 is found from Equation 9-24 as follows: 1.771112.682 114 d t0.05,13 sD 1n
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Notice that the con dence interval on D includes zero. This implies that, at the 90% level of con dence, the data do not support the claim that the two cars have different mean parking times 1 and 2. That is, the value D 0 is not inconsistent with the observed data. 1 2 EXERCISES FOR SECTION 10-4
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10-33. Consider the shear strength experiment described in Example 10-9. Construct a 95% con dence interval on the difference in mean shear strength for the two methods. Is the result you obtained consistent with the ndings in Example 10-9 Explain why. 10-34. Reconsider the shear strength experiment described in Example 10-9. Do each of the individual shear strengths have to be normally distributed for the paired t-test to be appropriate, or is it only the difference in shear strengths that must be normal Use a normal probability plot to investigate the normality assumption. 10-35. Consider the parking data in Example 10-10. Use the paired t-test to investigate the claim that the two types of cars have different levels of dif culty to parallel park. Use 0.10. Compare your results with the con dence interval constructed in Example 10-10 and comment on why they are the same or different.
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10-36. Reconsider the parking data in Example 10-10. Investigate the assumption that the differences in parking times are normally distributed. 10-37. The manager of a eet of automobiles is testing two brands of radial tires and assigns one tire of each brand at random to the two rear wheels of eight cars and runs the cars until the tires wear out. The data (in kilometers) follow. Find a 99% con dence interval on the difference in mean life. Which brand would you prefer, based on this calculation Car 1 2 3 4 5 6 7 8 Brand 1 36,925 45,300 36,240 32,100 37,210 48,360 38,200 33,500 Brand 2 34,318 42,280 35,500 31,950 38,015 47,800 37,810 33,215
10-39. Fifteen adult males between the ages of 35 and 50 participated in a study to evaluate the effect of diet and exercise on blood cholesterol levels. The total cholesterol was measured in each subject initially and then three months after participating in an aerobic exercise program and switching to a low-fat diet. The data are shown in the accompanying table. Do the data support the claim that low-fat diet and aerobic exercise are of value in producing a mean reduction in blood cholesterol levels Use 0.05. Blood Cholesterol Level Subject 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Before 265 240 258 295 251 245 287 314 260 279 283 240 238 225 247 After 229 231 227 240 238 241 234 256 247 239 246 218 219 226 233
10-38. A computer scientist is investigating the usefulness of two different design languages in improving programming tasks. Twelve expert programmers, familiar with both languages, are asked to code a standard function in both languages, and the time (in minutes) is recorded. The data follow: Time Design Language 1 17 16 21 14 18 24 16 14 21 23 13 18 Design Language 2 18 14 19 11 23 21 10 13 19 24 15 20