The question is: Could the sample with XX = 67, the test statistic, have come from the population with the mean value of 69? Intuitively, we may not reject the null hypothesis if XX is "sufficiently close" to x"; otherwise we may reject it in favor of the alternative hypothesis. But how do we decide that XX is "sufficiently close" to x? We can adopt two approaches, (1) confidence interval and (2) test of significance, both leading to identical conclusions in any specific application.
Since Xi — N(x, a2), we know that the test statistic XX is distributed as
Since we know the probability distribution of Xi, why not establish, say, a 100(1 — a) confidence interval for x based on XX and see whether this confidence interval includes x = x? If it does, we may not reject the null hypothesis; if it does not, we may reject the null hypothesis. Thus, if a = 0.05, we will have a 95% confidence interval and if this confidence interval includes x , we may not reject the null hypothesis—95 out of 100 intervals thus established are likely to include x~.
The actual mechanics are as follows: since XX — N(x, a2/n), it follows that
a/ n that is, a standard normal variable. Then from the normal distribution table we know that
which, on rearrangement, gives a
This is a 95% confidence interval for Once this interval has been established, the test of the null hypothesis is simple. All that we have to do is to see whether x = fa lies in this interval. If it does, we may not reject the null hypothesis; if it does not, we may reject it.
Turning to our example, we have already established a 95% confidence interval for x, which is
This interval obviously does not include x = 69. Therefore, we can reject the null hypothesis that the true x is 69 with a 95% confidence coefficient. Geometrically, the situation is as depicted in Figure A.12.
In the language of hypothesis testing, the confidence interval that we have established is called the acceptance region and the area(s) outside the acceptance region is (are) called the critical region(s), or region(s) of rejection of the null hypothesis. The lower and upper limits of the acceptance region (which demarcate it from the rejection regions) are called the critical values. In this language of hypothesis testing, if the hypothesized value falls inside the acceptance region, one may not reject the null hypothesis; otherwise one may reject it.
It is important to note that in deciding to reject or not reject H0, we are likely to commit two types of errors: (1) we may reject H0 when it is, in fact,
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