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Notice the general relationship between sample size, desired length of the con dence interval 2E, con dence level 100(1 ), and standard deviation : As the desired length of the interval 2E decreases, the required sample size n increases for a xed value of and speci ed con dence.
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8-2 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN
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As increases, the required sample size n increases for a xed desired length 2E and speci ed con dence. As the level of con dence increases, the required sample size n increases for xed desired length 2E and standard deviation .
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8-2.3
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One-Sided Con dence Bounds
The con dence interval in Equation 8-7 gives both a lower con dence bound and an upper con dence bound for . Thus it provides a two-sided CI. It is also possible to obtain one-sided con dence bounds for by setting either l or u and replacing z 2 by z . De nition A 100(1 )% upper-con dence bound for u and a 100(1 1n x z 1n is (8-10)
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(8-9)
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)% lower-con dence bound for x z l
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8-2.4
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General Method to Derive a Con dence Interval
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It is easy to give a general method for nding a con dence interval for an unknown parameter . Let X1, X2, p , Xn be a random sample of n observations. Suppose we can nd a statistic g(X1, X2, p , Xn; ) with the following properties: In the case considered in this section, the parameter . The random variable g(X1, X2, p , 2 1 1n2 and satis es both conditions above; it depends on the sample and Xn; ) 1X on , and it has a standard normal distribution since is known. Now one must nd constants CL and CU so that P3CL g 1X1, X2, p , Xn; 2 CU 4 1 (8-11) 1. g(X1, X2, p , Xn; ) depends on both the sample and . 2. The probability distribution of g(X1, X2, p , Xn; ) does not depend on or any other unknown parameter.
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z 2 and Because of property 2, CL and CU do not depend on . In our example, CL CU z 2. Finally, you must manipulate the inequalities in the probability statement so that This gives L(X1, X2, p , Xn) and U(X1, X2, p , Xn) as the lower and upper con dence limits de ning the 100(1 )% con dence interval for . The quantity g(X1, X2, p , Xn; ) is often called a pivotal quantity because we pivot on this quantity in Equation 8-11 to pro2 1 1n2 duce Equation 8-12. In our example, we manipulated the pivotal quantity 1X to obtain L 1X1, X2, p , Xn 2 X z 2 1n and U 1X1, X2, p , Xn 2 X z 2 1n. P3L1X1, X2, p , Xn 2 U1X1, X2, p , Xn 2 4 1 (8-12)
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254 8-2.5
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CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE
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A Large-Sample Con dence Interval for
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We have assumed that the population distribution is normal with unknown mean and known standard deviation . We now present a large-sample CI and that does not require these assumptions. Let X1, X2, p , Xn be a random sample from a population with unknown mean and variance 2 . Now if the sample size n is large, the central limit theorem implies that X has approximately a normal distribution with mean and variance 2 n. Therefore Z 1X 2 1 1n2 has approximately a standard normal distribution. This ratio could be used as a pivotal quantity and manipulated as in Section 8-2.1 to produce an approximate CI for . However, the standard deviation is unknown. It turns out that when n is large, replacing by the sample standard deviation S has little effect on the distribution of Z. This leads to the following useful result.
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