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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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