SOME THEORY AND PRACTICE OF STATISTICS by Howard G. Tucker Chapter 6. Statistical Inference Involving Discrete Distributions

6.1 A Basic Statistical Method in Hypothesis Testing. In chapter 4 we encountered our …rst problems in statistical inference. One dealt with the problem of sample size. The other was a problem of a statistical test of hypothesis. It dealt with the problem of whether two samples arose from the same distribution function or whether there was a shift and essentially a di¤erence in means between the two parent populations. What we did with that other problem was to assume that both parent distributions were the same, and, based on this assumption, we wondered whether the sample means could di¤er as much as they did without violating this assumption. If the separation or di¤erence observed was so large that, under this assumption of there being no di¤erence, the probability of the di¤erence being at least as large as that observed was “unbelievably small,”then we rejected the null hypothesis of there being no di¤erence in favor of the alternative that there was a di¤erence and in the direction indicated. We shall develop this notion further in this chapter but shall apply it only to discrete distributions. We shall develop this notion in two types of statistical problems: hypothesis testing and con…dence intervals. In the problem of hypothesis testing we consider a set of observable random variables X = (X1; ; Xn) whose joint distribution function depends upon some unknown constant (or scalar parameter) 2 . There might be a particular value of that we are interested in, call it 0. Based on our observations on the values of these random variables, our big problem will be if these data exhibit enough evidence to lead us to reject the idea that the true value of is 0. The traditional manner of stating this is to state that we are testing the null hypothesis that = 0 against the alternative that 6= 0. We shall only reject the null hypothesis if it is too unreasonable, based on our observed values of the random variables. Symbolically, statisticians write that they are testing the null hypothesis H0 : = 0 against the alternative H1 : 6= 0 or against the alternative H1 : > 0 or against the alternative H1 : < 0: An important thing to remember in all this is that we reject the null hypothesis in favor of an alternative only when there is overwhelming evidence to do so. A de…nition of “overwhelming evidence” will be introduced shortly.