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while confidence limits for the sum of the population parameters are given by S1 S2 zcsS1
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provided that the samples are independent. For example, confidence limits for the difference of two population means, in the case where the populations are infinite and have known standard deviations s1, s2, are given by # X1 # X2 zcsX2
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(10)
# # where X1, n1 and X2, n2 are the respective means and sizes of the two samples drawn from the populations. Similarly, confidence limits for the difference of two population proportions, where the populations are infinite, are given by P1 P2 zc P1(1 A n1 P1) P2(1 n2 P2) (11)
where P1 and P2 are the two sample proportions and n1 and n2 are the sizes of the two samples drawn from the populations.
Confidence Intervals for the Variance of a Normal Distribution
The fact that nS 2>s2 (n 1)S 2>s2 has a chi-square distribution with n 1 degrees of freedom enables us to 2 2 2 obtain confidence limits for s2 or s. For example, if x0.025 and x0.975 are the values of x for which 2.5% of the area lies in each tail of the distribution, then a 95% confidence interval is x2 0.025 nS 2 s2 x2 0.975 (12)
CHAPTER 6 Estimation Theory
or equivalently x2 0.025 (n 1)S 2 s2
x2 0.975
(13)
From these we see that s can be estimated to lie in the interval S !n x0.975 or equivalently S !n 1 x0.975
S!n x0.025
(14)
S !n 1 x0.025
(15)
with 95% confidence. Similarly, other confidence intervals can be found. It is in general desirable that the expected width of a confidence interval be as small as possible. For statistics with symmetric sampling distributions, such as the normal and t distributions, this is achieved by using tails of equal areas. However, for nonsymmetric distributions, such as the chi-square distribution, it may be desirable to adjust the areas in the tails so as to obtain the smallest interval. The process is illustrated in Problem 6.28.
Confidence Intervals for Variance Ratios
In 5, page 159, we saw that if two independent random samples of sizes m and n having variances S2, S2 1 2 are drawn from two normally distributed populations of variances s2, s2, respectively, then the random variable 1 2 ^ S 2 >s2 1 1 has an F distribution with m 1, n 1, degrees of freedom. For example, if we denote by F0.01 and F0.99 ^ 2 >s2 S2 2 the values of F for which 1% of the area lies in each tail of the F distribution, then with 98% confidence we have F0.01 S 2 >s2 1 1 S 2 >s2 2 2
F0.99
(16)
From this we can see that a 98% confidence interval for the variance ratio s2 >s2 of the two populations is given by 1 2
2 1 S1 ^ F0.99 S 2 2
s2 1 s2 2
2 1 S1 ^ F0.01 S 2 2
(17)
Note that F0.99 is read from one of the tables in Appendix F. The value F0.01 is the reciprocal of F0.99 with the degrees of freedom for numerator and denominator reversed, in accordance with Theorem 4-8, page 117. In a similar manner we could find a 90% confidence interval by use of the appropriate table in Appendix F. This would be given by
2 1 S1 ^ F0.95 S 2 2
s2 1 s2 2
2 1 S1 ^ F0.05 S 2 2
(18)
Maximum Likelihood Estimates
Although confidence limits are valuable for estimating a population parameter, it is still often convenient to have a single or point estimate. To obtain a best such estimate, we employ a technique known as the maximum likelihood method due to Fisher. To illustrate the method, we assume that the population has a density function that contains a population parameter, say, u, which is to be estimated by a certain statistic. Then the density function can be denoted by f (x, u). Assuming that there are n independent observations, X1, c, Xn, the joint density function for these observations is L f (x1, u) f (x2, u) c f (xn, u) (19)
which is called the likelihood. The maximum likelihood can then be obtained by taking the derivative of L with respect to u and setting it equal to zero. For this purpose it is convenient to first take logarithms and then take
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