Answer to Question #284063 in Civil and Environmental Engineering for Pope

Question

If only one city were tested,and the true mean for this city was equal to the national average of 487,what is the probability that null hypothesis would be rejected, leading to a true conclusion

Answer 1

The typical line of reasoning in a hypothesis test is as follows:

1. There is an initial research hypothesis of which the truth is unknown.
2. The first step is to state the relevant null and alternative hypotheses. This is important as mis-stating the hypotheses will muddy the rest of the process.
3. The second step is to consider the statistical assumptions being made about the sample in doing the test—for example, assumptions about the statistical independence or about the form of the distributions of the observations. This is important because invalid assumptions will mean that the results of the test are invalid.
4. Decide which test is appropriate, and state the relevant test statistic T
5. T
6. .
7. Derive the distribution of the test statistic under the null hypothesis from the assumptions.
8. Select a significance level (α
9. α
10. ), a probability threshold below which the null hypothesis will be rejected. Common values are 5% and 1%.
11. The distribution of the test statistic under the null hypothesis partitions the possible values of T
12. T
13.  into those for which the null hypothesis is rejected, the so called critical region, and those for which it is not. The probability of the critical region is α
14. α
15. .
16. Compute from the observations the observed value t
17. obs
18. t
19. obs
20.  of the test statistic T
21. T
22. .
23. Decide to either reject the null hypothesis in favor of the alternative or not reject it. The decision rule is to reject the null hypothesis H
24. 0
25. H
26. 0
27.  if the observed value t
28. obs
29. t
30. obs
31.  is in the critical region, and to accept or “fail to reject” the hypothesis otherwise.