SECTION 6.1 Confidence Intervals for the Mean (s Known) 309 55. When estimating the population mean, why not construct a 99% confidence interval every time? 56. When all other quantities remain the same, how does the indicated change affect the minimum sample size requirement? Explain. (a) Increase in the level of confidence (b) Increase in the error tolerance (c) Increase in the population standard deviation Extending Concepts Finite Population Correction Factor In Exercises 57 and 58, use the information below. In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n Ú 0.05N, however, the formula that determines the standard error of the mean sx needs to be adjusted, as shown below. sx = s1 nAN - n N - 1 Recall from the Section 5.4 exercises that the expression 21N- n2 1N- 12 is called a finite population correction factor. The margin of error is E = zc s1 nAN - n N - 1 . 57. Determine the finite population correction factor for each value of N and n. (a) N = 1000 and n = 500 (b) N = 1000 and n = 100 (c) N = 1000 and n = 75 (d) N = 1000 and n = 50 (e) N = 100 and n = 50 (f) N = 400 and n = 50 (g) N = 700 and n = 50 (h) N = 1200 and n = 50 What happens to the finite population correction factor as the sample size n decreases but the population size N remains the same? as the population size N increases but the sample size n remains the same? 58. Use the finite population correction factor to construct each confidence interval for the population mean. (a) c = 0.99, x = 8.6, s = 4.9, N = 200, n = 25 (b) c = 0.90, x = 10.9, s = 2.8, N = 500, n = 50 (c) c = 0.95, x = 40.3, s = 0.5, N = 300, n = 68 (d) c = 0.80, x = 56.7, s = 9.8, N = 400, n = 36 59. Sample Size The equation for determining the sample size n = a zc s E b 2 can be obtained by solving the equation for the margin of error E = zc s1 n for n. Show that this is true and justify each step.
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