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Special Tests of Significance for Small Samples
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In case samples are small (n 30), we can formulate tests of hypotheses and significance using other distributions besides the normal, such as Student s t, chi-square, and F. These involve exact sampling theory and so, of course, hold even when samples are large, in which case they reduce to those given above. The following are some examples. 1. MEANS. To test the hypothesis H0 that a normal population has mean, m, we use T # X S m !n 1 # X S
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# X m s> !n ^ for large n except that S !n>(n 1) S is used in place of s. The difference is that while Z is normally distributed, T has Student s t distribution. As n increases, these tend toward agreement. Tests of hypotheses similar to those for means on page 216, can be made using critical t values in place of critical z values. # where X is the mean of a sample of size n. This is analogous to using the standardized variable Z
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CHAPTER 7 Tests of Hypotheses and Significance
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2. DIFFERENCES OF MEANS. Suppose that two random samples of sizes n1 and n2 are drawn from normal (or approximately normal) populations whose standard deviations are equal, i.e., s1 s2. Suppose # # further that these two samples have means and standard deviations given by X1, X2 and S1, S2, respectively. To test the hypothesis H0 that the samples come from the same population (i.e., m1 m2 as well as s1 s2), we use the variable given by T # X1 1 s n A 1 # X2 1 n2 where s n1S 2 n2S 2 1 2 A n1 n2 2 (12)
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The distribution of T is Student s t distribution with n n1 n2 2 degrees of freedom. Use of (12) is made plausible on placing s1 s2 s in (12), page 157, and then using as an estimator of s2 the weighted average (n1 1) S 2 1 (n1 1)
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n1S 2 n2S 2 1 2 n1 n2 2
where S 2 and S 2 are the unbiased estimators of s2 and s2. This is the pooled variance obtained by combin1 2 1 2 ing the data. 3. VARIANCES. dom variables To test the hypothesis H0 that a normal population has variance s2, we consider the rannS2 s2 (n 1) S 2 s2
(13)
which (see pages 158 159) has the chi-square distribution with n 1 degrees of freedom. Then if a random sample of size n turns out to have variance s2, we would, on the basis of a two-tailed test, accept H0 (or at least not reject it) at the 0.05 level if x2 0.025 ns2 s2 x2 0.975 (14)
and reject it otherwise. A similar result can be obtained for the 0.01 or other level. To test the hypothesis H1 that the population variance is greater than s2, we would still use the null hypothesis H0 but would now employ a one-tailed test. Thus we would reject H0 at the 0.05 level (and thereby conclude that H1 is correct) if the particular sample variance s2 were such that ns2 s2 x2 0.95 (15)
and would accept H0 (or at least not reject it) otherwise. 4. RATIOS OF VARIANCES. In some problems we wish to decide whether two samples of sizes m and n, respectively, whose measured variances are s2 and s2, do or do not come from normal populations with the 1 2 same variance. In such cases, we use the statistic (see page 159). F S2 >s2 1 1 S 2 >s2 2 2
(16)
where s2, s2 are the variances of the two normal populations from which the samples are drawn. Suppose that 1 2 H0 denotes the null hypothesis that there is no difference between population variances, i.e., s2 s2. Then 1 2 under this hypothesis (16) becomes F
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