Hypothesis test of a single population mean

Suppose our interest is in testing whether the population mean was equal to a particular hypothesized value, pg. A hypothesis testing process typically starts with a statement of the null and alternate hypotheses. The null hypothesis can be stated in the following manner  

If data are found to contradict the null hypothesis, it will be rejected in favor of the alternate hypothesis  

The alternate hypothesis is two sided in the sense that values clearly less than p would be consistent with it as would values that were clearly greater than Pg. Rejection of the null hypothesis because Pg p (p is much greater than the hypothesized value pg) may lead to one decision (for example, continue with the development of the new drug with a larger study) whereas rejection of the null hypothesis because Pg p (p is much less than the hypothesized value Pg) may lead to a completely different decision (for example, to stop development of the new drug because it has no effect on SBP or actually increases SBP). What is important is that, a priori, either outcome is possible. 

The next step of the hypothesis testing process is to identify a numeric criterion by which the plausibility of the null hypothesis is tested. This numeric criterion is called the test statistic, and we use it to decide if the value that resulted from the study contradicts the null hypothesis or not. The test statistic to be used in this case is  

If the null hypothesis is true - that is, the population mean is the hypothesized value, Pg - the value of the test statistic will be close to 0. The further the test statistic value is from 0 (either negative or positive) the less plausible is the hypothesized value, pg - that is, the null hypothesis should be rejected in favor of the alternate. 

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