CHAPTER 6 Estimation Theory

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Confidence limits for the difference in proportions of the two groups are given by P1 P2 zc P1Q1 A n1 P2Q2 n2

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1 P2. Here where subscripts 1 and 2 refer to teenagers and adults, respectively, and Q1 1 P1, Q2 P1 300>600 0.50 and P2 100>400 0.25 are, respectively, the proportion of teenagers and adults who liked the program. (a) 95% confidence limits: 0.50 0.25 1.96 !(0.50)(0.50)>600 (0.25)(0.75)>400 0.25 0.06.

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Therefore we can be 95% confident that the true difference in proportions lies between 0.19 and 0.31. (b) 99% confidence limits: 0.50 0.25 2.58 !(0.50)(0.50)>600 (0.25)(0.75)>400 0.25 0.08.

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Therefore, we can be 99% confident that the true difference in proportions lies between 0.17 and 0.33.

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6.18. The electromotive force (emf) of batteries produced by a company is normally distributed with mean 45.1 volts and standard deviation 0.04 volt. If four such batteries are connected in series, find (a) 95%, (b) 99%, (c) 99.73%, (d) 50% confidence limits for the total electromotive force.

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If E1, E2, E3, and E4 represent the emfs of the four batteries, we have mE1

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E2 E3 E4

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mE1 mE3

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mE2 mE4

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sE1 sE2 sE1

E2 E3 E4

2s2 1 E sE4

s2 2 E

s2 3 E

s2 4 E

Then, since mE1 mE1

45.1 volts and sE1 180.4 1.96(0.08) 2.58(0.08) 3(0.08) 0.6745(0.08) and 180.4 180.4 180.4

E2 E3 E4

0.04 volt, 0.08

E2 E3 E4

4(45.1)

24(0.04)2

(a) 95% confidence limits are 180.4 (b) 99% confidence limits are 180.4 (c) 99.73% confidence limits are 180.4 (d) 50% confidence limits are 180.4

0.16 volts. 0.21 volts. 0.24 volts. 0.054 volts.

The value 0.054 volts is called the probable error.

Confidence intervals for variances 6.19. The standard deviation of the lifetimes of a sample of 200 electric light bulbs was computed to be 100 hours. Find (a) 95%, (b) 99% confidence limits for the standard deviation of all such electric light bulbs.

In this case large sampling theory applies. Therefore (see Table 5-1, page 160) confidence limits for the population standard deviation s are given by S zcs> !2n, where zc indicates the level of confidence. We use the sample standard deviation to estimate s. (a) The 95% confidence limits are 100 1.96(100)> !400 100 9.8.

Therefore, we can be 95% confident that the population standard deviation will lie between 90.2 and 109.8 hours. (b) The 99% confidence limits are 100 2.58(100)> !400 100 12.9.

Therefore, we can be 99% confident that the population standard deviation will lie between 87.1 and 112.9 hours.

6.20. How large a sample of the light bulbs in Problem 6.19 must we take in order to be 99.73% confident that the true population standard deviation will not differ from the sample standard deviation by more than (a) 5%, (b) 10%

As in Problem 6.19, 99.73% confidence limits for s are S of s. Then the percentage error in the standard deviation is 3s> !2n s 3s> !2n s 3s> !2n, using s as an estimate

300 % !2n

CHAPTER 6 Estimation Theory

(a) If 300> !2n (b) If 300> !2n

5, then n 10, then n

1800. Therefore, the sample size should be 1800 or more. 450. Therefore, the sample size should be 450 or more.

6.21. The standard deviation of the heights of 16 male students chosen at random in a school of 1000 male students is 2.40 inches. Find (a) 95%, (b) 99% confidence limits of the standard deviation for all male students at the school. Assume that height is normally distributed.

(a) 95% confidence limits are given by S !n>x0.975 and S !n>x0.025. For n 16 1 15 degrees of freedom, x2 27.5 or x0.975 5.24 and x2 6.26 or 0.975 0.025 x0.025 2.50. Then the 95% confidence limits are 2.40 !16>5.24 and 2.40 !16>2.50, i.e., 1.83 and 3.84 inches. Therefore, we can be 95% confident that the population standard deviation lies between 1.83 and 3.84 inches. (b) 99% confidence limits are given by S !n>x0.995 and S !n>x0.005. For n 16 1 15 degrees of freedom, x2 32.8 or x0.995 5.73 and x2 4.60 or 0.995 0.005 x0.005 21.4. Then the 99% confidence limits are 2.40 !16>5.73 and 2.40 !16>2.14, i.e., 1.68 and 4.49 inches. Therefore, we can be 99% confident that the population standard deviation lies between 1.68 and 4.49 inches.