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CHAPTER 7 Tests of Hypotheses and Significance
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7.26. A company manufactures rope whose breaking strengths have a mean of 300 lb and standard deviation 24 lb. It is believed that by a newly developed process the mean breaking strength can be increased, (a) Design a decision rule for rejecting the old process at a 0.01 level of significance if it is agreed to test 64 ropes, (b) Under the decision rule adopted in (a), what is the probability of accepting the old process when in fact the new process has increased the mean breaking strength to 310 lb Assume that the standard deviation is still 24 lb.
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(a) If m is the mean breaking strength, we wish to decide between the hypotheses H0: m H1: m 300 lb, and the new process is equivalent to the old one 300 lb, and the new process is better than the old one
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For a one-tailed test at a 0.01 level of significance, we have the following decision rule (refer to Fig. 7-8): (1) Reject H0 if the z score of the sample mean breaking strength is greater than 2.33. (2) Accept H0 otherwise. # m X # 300 # X Since Z , X 300 s> !n 24> !64 Therefore, the above decision rule becomes: # 2.33, X
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3z. Then if Z
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3(2.33)
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307.0 lb.
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(1) Reject H0 if the mean breaking strength of 64 ropes exceeds 307.0 lb. (2) Accept H0 otherwise.
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Fig. 7-9
(b) Consider the two hypotheses (H0: m 300 lb) and (H1: m 310 lb). The distributions of breaking strengths corresponding to these two hypotheses are represented respectively by the left and right normal distributions of Fig. 7-9. The probability of accepting the old process when the new mean breaking strength is actually 310 lb is represented by the region of area b in Fig. 7-9. To find this, note that 307.0 lb in standard units is (307.0 310) > 3 1.00; hence b area under right-hand normal curve to left of z 1.00 0.1587
CHAPTER 7 Tests of Hypotheses and Significance
This is the probability of accepting (H0: m probability of making a Type II error. 300 lb) when actually (H1: m
310 1b) is true, i.e., it is the
7.27. Construct (a) an OC curve, (b) a power curve for Problem 7.26, assuming that the standard deviation of breaking strengths remains at 24 lb.
By reasoning similar to that used in Problem 7.26(b), we can find b for the cases where the new process yields mean breaking strengths m equal to 305 lb, 315 lb, etc. For example, if m 305 lb, then 307.0 lb in standard units is (307.0 305)>3 0.67, and hence b area under right hand normal curve to left of z 0.67 0.7486
In this manner Table 7-4 is obtained. Table 7-4 m b 290 1.0000 295 1.0000 300 0.9900 305 0.7486 310 0.1587 315 0.0038 320 0.0000
(a) The OC curve is shown in Fig. 7-10(a). From this curve we see that the probability of keeping the old process if the new breaking strength is less than 300 lb is practically 1 (except for the level of significance of 0.01 when the new process gives a mean of 300 lb). It then drops rather sharply to zero so that there is practically no chance of keeping the old process when the mean breaking strength is greater than 315 lb. (b) The power curve shown in Fig. 7-10(b) is capable of exactly the same interpretation as that for the OC curve. In fact the two curves are essentially equivalent.
Fig. 7-10
7.28. To test the hypothesis that a coin is fair (i.e., p 0.5) by a number of tosses of the coin, we wish to impose the following restrictions: (A) the probability of rejecting the hypothesis when it is actually correct must be 0.05 at most; (B) the probability of accepting the hypothesis when actually p differs from 0.5 by 0.1 or more (i.e., p 0.6 or p 0.4) must be 0.05 at most. Determine the minimum sample size that is necessary and state the resulting decision rule.
Here we have placed limits on the risks of Type I and Type II errors. For example, the imposed restriction (A) requires that the probability of a Type I error is a 0.05 at most, while restriction (B) requires that the probability of a Type II error is b 0.05 at most. The situation is illustrated graphically in Fig. 7-11.
Fig. 7-11
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