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CHAPTER 7 Tests of Hypotheses and Significance
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(b) If the level of significance is 0.01, the z1 value in Fig. 7-5 is (1) Reject H0 if Z is less than 2.33.
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2.33. Hence we adopt the decision rule:
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(2) Accept H0 (or withhold any decision) otherwise. Since, as in Problem 7.7(a), the z score is 2.50, which is less than 2.33, we reject H0 at a 0.01 level of significance. Note that this decision is not the same as that reached in Problem 7.7(b) using a two-tailed test. It follows that decisions concerning a given hypothesis H0 based on one-tailed and two-tailed tests are not always in agreement. This is, of course, to be expected since we are testing H0 against a different alternative in each case. (c) The P value of the test is P(Z 1570) 0.0062, which is the probability that a mean lifetime of less than 1570 hours would occur by chance if H0 were true.
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7.9. The breaking strengths of cables produced by a manufacturer have mean 1800 lb and standard deviation 100 lb. By a new technique in the manufacturing process it is claimed that the breaking strength can be increased. To test this claim, a sample of 50 cables is tested, and it is found that the mean breaking strength is 1850 lb. (a) Can we support the claim at a 0.01 level of significance (b) What is the P value of the test
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(a) We have to decide between the two hypotheses H0: m H1: u 1800 lb, and there is really no change in breaking strength 1800 lb, and there is a change in breaking strength
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A one-tailed test should be used here (see Fig. 7-4). At a 0.01 level of significance the decision rule is: (1) If the z score observed is greater than 2.33, the results are significant at the 0.01 level and H0 is rejected. (2) Otherwise H0 is accepted (or the decision is withheld). Under the hypothesis that H0 is true, we find Z # X m s> !n 1850 1800 100> !50 3.55
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which is greater than 2.33. Hence we conclude that the results are highly significant and the claim should be supported. (b) The P value of the test is P(Z 3.55) 0.0002, which is the probability that a mean breaking strength of 1850 lb or more would occur by chance if H0 were true.
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Tests involving differences of means and proportions 7.10. An examination was given to two classes consisting of 40 and 50 students, respectively. In the first class the mean grade was 74 with a standard deviation of 8, while in the second class the mean grade was 78 with a standard deviation of 7. Is there a significant difference between the performance of the two classes at a level of significance of (a) 0.05, (b) 0.01 (c) What is the P value of the test
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Suppose the two classes come from two populations having the respective means m1 and m2. Then we have to decide between the hypotheses H0: m1 m2, and the difference is merely due to chance
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H1: m1 2 m2, and there is a significant difference between classes Under the hypothesis H0, both classes come from the same population. The mean and standard deviation of the difference in means are given by mX1
s2 1 A n1
s2 2 n2
82 40 A
72 50
where we have used the sample standard deviations as estimates of s1 and s2.
CHAPTER 7 Tests of Hypotheses and Significance
# # X1 X2 sX1 X2
Then
74 78 1.606
(a) For a two-tailed test, the results are significant at a 0.05 level if Z lies outside the range 1.96 to 1.96. Hence we conclude that at a 0.05 level there is a significant difference in performance of the two classes and that the second class is probably better. (b) For a two-tailed test the results are significant at a 0.01 level if Z lies outside the range 2.58 and 2.58. Hence we conclude that at a 0.01 level there is no significant difference between the classes. Since the results are significant at the 0.05 level but not at the 0.01 level, we conclude that the results are probably significant, according to the terminology used at the end of Problem 7.5. (c) The P value of the two-tailed test is P(Z 2.49) P(Z 2.49) that the observed statistics would occur in the same population. 0.0128, which is the probability
7.11. The mean height of 50 male students who showed above-average participation in college athletics was 68.2 inches with a standard deviation of 2.5 inches, while 50 male students who showed no interest in such participation had a mean height of 67.5 inches with a standard deviation of 2.8 inches. (a) Test the hypothesis that male students who participate in college athletics are taller than other male students. (b) What is the P value of the test
(a) We must decide between the hypotheses H0: m1 H1: m1 m2, and there is no difference between the mean heights m2, and mean height of first group is greater than that of second group
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