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7.22. In Problem 7.21 would your conclusions be changed if it turned out that there was a significant difference in the mean grades of the classes Explain your answer.
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Since the actual mean grades were not used at all in Problem 7.21, it makes no difference what they are. This is to be expected in view of the fact that we are not attempting to decide whether there is a difference in mean grades, but only whether there is a difference in variability of the grades.
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Operating characteristic curves 7.23. Referring to Problem 7.2, what is the probability of accepting the hypothesis that the coin is fair when the actual probability of heads is p 0.7
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The hypothesis H0 that the coin is fair, i.e., p 0.5, is accepted when the number of heads in 100 tosses lies between 39.5 and 60.5. The probability of rejecting H0 when it should be accepted (i.e., the probability of a Type I error) is represented by the total area a of the shaded region under the normal curve to the left in Fig. 7-6. As computed in Problem 7.2(a), this area a, which represents the level of significance of the test of H0, is equal to 0.0358.
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If the probability of heads is p 0.7, then the distribution of heads in 100 tosses is represented by the normal curve to the right in Fig. 7-6. The probability of accepting H0 when actually p 0.7 (i.e., the probability of a Type II error) is given by the cross-hatched area b. To compute this area, we observe that the distribution under the hypothesis p 0.7 has mean and standard deviation given by m np (100)(0.7) 70 s !npq 60.5 70 4.58 39.5 70 4.58 !(100)(0.7)(0.3) 2.07 6.66 4.58
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60.5 in standard units 39.5 in standard units
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Then b area under the standard normal curve between z 6.66 and z 2.07 0.0192. Therefore, with the given decision rule there is very little chance of accepting the hypothesis that the coin is fair when actually p 0.7. Note that in this problem we were given the decision rule from which we computed a and b. In practice two other possibilities may arise:
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CHAPTER 7 Tests of Hypotheses and Significance
(1) We decide on a (such as 0.05 or 0.01), arrive at a decision rule, and then compute b. (2) We decide on a and b and then arrive at a decision rule.
7.24. Work Problem 7.23 if (a) p
(a) If p
0.6, (b) p
0.8, (c) p
0.9, (d) p
0.6, the distribution of heads has mean and standard deviation given by m np (100)(0.6) 60 s !npq 60.5 60 4.90 39.5 60 4.90 !(100)(0.6)(0.4) 0.102 4.18 4.90
60.5 in standard units 39.5 in standard units
Then b area under the standard normal curve between z 4.18 and z 0.102 0.5405 Therefore, with the given decision rule there is a large chance of accepting the hypothesis that the coin is fair when actually p 0.6. (b) If p 0.8, then m np (100)(0.8) 80 and s !npq 60.5 4 39.5 4 80 80 !(100)(0.08)(0.2) 4.88 10.12 4.88 0.0000, very closely. 4.
60.5 in standard units 39.5 in standard units Then b
area under the standard curve between z
10.12 and z 0.9, b
(c) From comparison with (b) or by calculation, we see that if p (d) By symmetry, p 0.4 yields the same value of b as p
0 for all practical purposes. 0.5405. b) vs. p.
0.6, i.e., b
7.25. Represent the results of Problems 7.23 and 7.24 by constructing a graph of (a) b vs. p, (b) (1 Interpret the graphs obtained.
Table 7-3 shows the values of b corresponding to given values of p as obtained in Problems 7.23 and 7.24. Note that b represents the probability of accepting the hypothesis p 0.5 when actually p is a value other than 0.5. However, if it is actually true that p 0.5, we can interpret b as the probability of accepting p 0.5 when it should be accepted. This probability equals 1 0.0358 0.9642 and has been entered into Table 7-3.
Table 7-3 p b 0.1 0.0000 0.2 0.0000 0.3 0.0192 0.4 0.5405 0.5 0.9642 0.6 0.5405 0.7 0.0192 0.8 0.0000 0.9 0.0000
(a) The graph of b vs. p, shown in Fig. 7-7(a), is called the operating characteristic curve, or OC curve, of the decision rule or test of hypotheses. The distance from the maximum point of the OC curve to the line b 1 is equal to a 0.0358, the level of significance of the test. In general, the sharper the peak of the OC curve the better is the decision rule for rejecting hypotheses that are not valid. (b) The graph of (1 b) vs. p, shown in Fig. 7-7(b), is called the power curve of the decision rule or test of hypotheses. This curve is obtained simply by inverting the OC curve, so that actually both graphs are equivalent. The quantity 1 b is often called a power function since it indicates the ability or power of a test to reject hypotheses which are false, i.e., should be rejected. The quantity b is also called the operating characteristic function of a test.
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