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2 b 1

5.3. Assume that the heights of 3000 male students at a university are normally distributed with mean 68.0 inches and standard deviation 3.0 inches. If 80 samples consisting of 25 students each are obtained, what would be the mean and standard deviation of the resulting sample of means if sampling were done (a) with replacement, (b) without replacement

Fig. 14-44 14.34 For the coupled circuit shown in Fig. 14-45, nd the components of the current I2 resulting from each source V1 and V2 . Ans: 0:77 112:68 A; 1:72 86:058 A

The numbers of samples of size 25 that could be obtained theoretically from a group of 3000 students with and without replacement are (3000)25 and 3000C25, which are much larger than 80. Hence, we do not get a true sampling distribution of means but only an experimental sampling distribution. Nevertheless, since the number of samples is large, there should be close agreement between the two sampling distributions. Hence, the mean and standard deviation of the 80 sample means would be close to those of the theoretical distribution. Therefore, we have (a) mX m 68.0 inches and sX s !n 3 !25 n 1 0.6 inches

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Fig. 14-45 14.35 Determine the coupling coe cient k in the circuit shown in Fig. 14-46, if the power in the 10- resistor is 32 W. Ans: 0:791 In (14a), replace a, X11 , X22 , and XM by their expressions in terms of X1 , X2 , and k, thereby obtaining (14b).

25 1

See Fig. 9-11.

which is only very slightly less than 0.6 inches and can for all practical purposes be considered the same as in sampling with replacement. Thus we would expect the experimental sampling distribution of means to be approximately normally distributed with mean 68.0 inches and standard deviation 0.6 inches.

CHAP. 14]

5.4. In how many samples of Problem 5.3 would you expect to find the mean (a) between 66.8 and 68.3 inches, (b) less than 66.4 inches

Fig. 14-46 14.37 For the coupled circuit shown in Fig. 14-47, nd the input impedance at terminals ab. Ans: 3 j36:3

# The mean X of a sample in standard units is here given by Z (a) 66.8 in standard units 68.3 in standard units (66.8 (68.3 68.0)>0.6 68.0)>0.6 2.0 0.5 # X mX sX # X 68.0 . 0.6

Proportion of samples with means between 66.8 and 68.3 inches (area under normal curve between z (area between z (area between z 0.4772 0.1915 2 and z 0 and z 0.6687 (80) (0.6687) or 53 (Fig. 5-2). 0) 0.5) 2.0 and z 0.5)

Fig. 14-47 14.38 Find the input impedance at terminals ab of the coupled circuit shown in Fig. 14-48. Ans: 1 j1:5

Fig. 5-2

axyz a x x <= x x x 0 axyz > 0 ayz x x y y z a <= z z a x x a a a a <= a y y a 1 z z <= <= <= x x y y z b z z z a x b y y 1 axyz 0 y y <= y y y 1 z z z z <= z 1 xyz b

Fig. 14-48 14.39 Find the input impedance at terminals ab of the coupled circuit shown in Fig. 14-49. Ans: 6:22 j4:65

(66.4

Proportion of samples with means less than 66.4 inches (area under normal curve to left of z (area to left of z (area between z 0.5 0.4962 0) 2.67 and z 0) 0.0038 0.304 or zero (Fig. 5-3). 2.67)

Fig. 14-49

(80)(0.0038)

Obtain Thevenin and Norton equivalent circuits at terminals ab of the coupled circuit shown in Fig. 14-50. Ans: V 0 7:07 458 V; I 0 1:04 27:98 A; Z 0 6:80 72:98

Fig. 5-3

5.5. Five hundred ball bearings have a mean weight of 5.02 oz and a standard deviation of 0.30 oz. Find the probability that a random sample of 100 ball bearings chosen from this group will have a combined weight, (a) between 496 and 500 oz, (b) more than 510 oz.

Fig. 14-50

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