The P Value

1. The P Value • The P value is the smallest level of significance for which the observed sample statistic tells us to reject the null hypothesis. • The P value is also called the probability of chance or the attained level of significance.

2. If the P value is : We reject H0.

3. If the P value is >: We do not reject H0.

4. In a right-tailed test • P value = area to the right of the observed sample statistic • P value = probability that the mean computed from any random sample of size n will be greater than the observed sample statistic

5. In a right-tailed test • P value = Probability that the mean computed from any random sample of size n will be > observed sample statistic Area = P value z = Sample test statistic

6. In a left-tailed test • P value = area to the left of the observed sample statistic • P value = P(any sample statistic < observed sample statistic Area = P value z = Sample test statistic

7. In a two-tailed test • P value = sum of the areas in the two tails • If the observed sample statistic falls in the right half of a symmetric curve, P value = 2P(sample statistic > observed sample statistic) Area = 1/2 of P value z = Sample statistic

8. In a two-tailed test • P value = sum of the areas in the two tails • If the observed sample statistic falls in the left half of a symmetric curve, P value = 2P(sample statistic < observed sample statistic) Area =1/2 of P value z = Sample statistic

9. Compute the P value The manufacturer of light bulbs claims that they will burn for 1000 hours. ...The bulbs will be returned ... if my sample indicates that they will burn less than 1000 hours. H0:  = 1000 H1:  < 1000 Sample results: n = 36, mean = 999 hours, s = 3.4 hours, z = – 1.76

10. In a left-tailed test • P value = area to the left of the observed sample statistic • Use Table 5 in Appendix II to find the area. Area = P value z = – 1.76

11. Finding the P Value In a Left-Tailed Test • P value = area to the left of the observed sample statistic Area = 0.0392 z = – 1.76

12. Conclusion • Since the P value is the smallest level of significance for which the sample data tells us to reject H0, we reject H0 for any   0.0392. • For  < 0.0392, we fail to reject H0.

13. Hypothesis Test Example Your college claims that the mean age of its students is 28 years. You wish to check the validity of this statement. H0:  = 28 H1:   28 Sample results: n = 49, mean = 27.5 years, s = 2.3 years, z = –1.52

14. In a two-tailed test • P value = sum of the areas in the two tails Area =1/2 of P value z = – 1.52

15. Finding the P Value In a Two-Tailed Test • P value = sum of the areas in the two tails • P value = 2(0.0643) • P = 0.1286 Area = 0.0643 z = – 1.52

16. Conclusion • P = 0.1286 • We reject H0 for all   0.1286. • We fail to reject H0 for all  < 0.1286. Area = 0.0643 z = – 1.52

17. Caution • Establish the level of significance  before doing the hypothesis test. • The level of significance  should reflect your willingness to risk a Type I error and may be affected by the accuracy and reliability of your measurement instruments.

18. Advantage of Knowing P Value We know all levels of significance for which the observed sample statistic tells us to reject H0.

19. Use of P Value • If P value is , reject H0. • If P value is >, do not reject H0.