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CHAPTER 6 Estimation Theory
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In Table 6-1 we give values of zc corresponding to various confidence levels used in practice. For confidence levels not presented in the table, the values of zc can be found from the normal curve area table in Appendix C. Table 6-1 Confidence Level zc 99.73% 3.00 99% 2.58 98% 2.33 96% 2.05 95.45% 2.00 95% 1.96 90% 1.645 80% 1.28 68.27% 1.00 50% 0.6745
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In cases where a statistic has a sampling distribution that is different from the normal distribution (such as chi square, t, or F), appropriate modifications to obtain confidence intervals have to be made.
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Confidence Intervals for Means
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# 1. LARGE SAMPLES (n 30). If the statistic S is the sample mean X, then 95% and 99% confidence # # limits for estimation of the population mean m are given by X 1.96sX and X 2.58sX, respectively. # More generally, the confidence limits are given by X zcsX where zc, which depends on the particular level of confidence desired, can be read from the above table. Using the values of sX obtained in 5, we see that the confidence limits for the population mean are given by # X zc s !n (1)
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in case sampling is from an infinite population or if sampling is with replacement from a finite population, and by # X zc N s !n A N n 1 (2)
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if sampling is without replacement from a population of finite size N. In general, the population standard deviation s is unknown, so that to obtain the above confidence limits, ^ we use the estimator S or S. 2. SMALL SAMPLES (n , 30) AND POPULATION NORMAL. In this case we use the t distribution to obtain confidence levels. For example, if t0.975 and t0.975 are the values of T for which 2.5% of the area lies in each tail of the t distribution, then a 95% confidence interval for T is given by (see page 159) t0.975 # (X m)!n S
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t0.975
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from which we see that m can be estimated to lie in the interval # X t0.975 S !n
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S !n
with 95% confidence. In general the confidence limits for population means are given by S !n
where the value tc can be read from Appendix D. A comparison of (5) with (1) shows that for small samples we replace zc by tc. For n 30, zc and tc are practically equal. It should be noted that an advantage of the small sampling theory (which can of course be ^ used for large samples as well, i.e., it is exact) is that S appears in (5) so that the sample standard deviation can be used instead of the population standard deviation (which is usually unknown) as in (1).
CHAPTER 6 Estimation Theory
Confidence Intervals for Proportions
Suppose that the statistic S is the proportion of successes in a sample of size n 30 drawn from a binomial population in which p is the proportion of successes (i.e., the probability of success). Then the confidence limits for p are given by P zcsP, where P denotes the proportion of successes in the sample of size n. Using the values of sP obtained in 5, we see that the confidence limits for the population proportion are given by P zc pq An P zc p(1 A n p) (6)
in case sampling is from an infinite population or if sampling is with replacement from a finite population. Similarly, the confidence limits are P zc pq N A n AN n 1 (7)
if sampling is without replacement from a population of finite size N. Note that these results are obtained from # (1) and (2) on replacing X by P and s by !pq. To compute the above confidence limits, we use the sample estimate P for p. A more exact method is given in Problem 6.27.
Confidence Intervals for Differences and Sums
If S1 and S2 are two sample statistics with approximately normal sampling distributions, confidence limits for the differences of the population parameters corresponding to S1 and S2 are given by S1 S2 zcsS1