1. P-values

2. Introduction
Using the rejection region method of testing, we calculate the test statistic’s value and the null hypothesis is rejected if that value lies in the rejection region.
An alternate approach is based on a probability called the p-value. P-values provide a measure of the strength of the evidence against the null hypothesis.

3. Definition
The p-value is the probability, calculated assuming that the null hypothesis is true, of obtaining a value of the test statistic at least as contradictory to as the value calculated from the available sample.

4. Example
A sample of 51 Panasonic AAA batteries gave a sample mean zinc mass of 2.06 g and a sample standard deviation of .141 g. Does this data provide compelling evidence for concluding that the population mean zinc mass exceeds 2.0?
Here we test using a z test (based on the CLT; n=51). The value of the test statistic is

5. Example (continued)
Large values of z are contradictory to (based on the form of ).
The p-value is
What should we do with this information?

6. Example (continued)
The smaller the p-value, the more unlikely it is that we would see such an extreme value if the null hypothesis is true.
Thus, the smaller the p-value, the more evidence there is against the null hypothesis and for the alternative hypothesis.
If we compare the p-value with a significance level , we would reject the null hypothesis if p-value
In the example, we would reject for

7. Proposition The p-value is the smallest significance level
at which the null hypothesis can be rejected.
Because of this, the p-value is alternatively referred to as the observed significance level for the data.

8. P-values for z tests
If and the observed value of the test statistic is z, then: Alternative Hypothesis P-value

9. P-values for t tests
The table that we used previously for t tests is not adequate for computing p-values. One should use Appendix Table A-8. If and the observed statistic is t, then: Alternative Hypothesis P-value

10. More on interpreting p-values
Since the test statistic is a random variable, p-values are random as well.
If the null hypothesis is false, we hope that the p-value is small so that the null is rejected, and if the null hypothesis is true, we hope for the p-value to exceed the selected significance level so that we do not reject it.

11. More on interpreting p-values (continued)
When is true, the probability distribution of p-values is uniform on the interval 0 to 1. The area under the curve to the left of is
When is false, the distribution of p-values is no longer uniform. There is a much greater probability of small p-values (where is rejected), that chance increasing as the true mean deviates from

12. More on interpreting p-values (continued)
The p-value technique is equivalent to the rejection region method if we compare the p-value to a specified (the same conclusion, (reject or do not reject, will be reached).
The p-value, however, gives more information (the observed significance), which allows each person to evaluate whether the evidence is strong enough.

13. More on interpreting p-values
If the test procedure indicates that the null hypothesis should be rejected, one should consider whether the deviation from (or
in general) is of any practical significance.
E.g., in the example given earlier, because of the small standard deviation of , 2.06 was considered to be a significant deviation from 2. In the context of the problem, is it?
This typically arises when n is very large, because then virtually any deviation from will give significant results (see Section 9.5).