1. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 If the P value is  : We reject H 0.

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 If the P value is >  : We do not reject H 0.

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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 z = Sample test statistic Area = P value

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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 z = Sample test statistic Area = P value

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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) z = Sample statistic Area = 1/2 of P value

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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) z = Sample statistic Area =1/2 of P value

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 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. H 0 :  = 1000 H 1 :  < 1000 Sample results: n = 36, mean = 999 hours, s = 3.4 hours, z = – 1.76 Compute the P value

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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. z = – 1.76 Area = P value

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Finding the P Value In a Left- Tailed Test P value = area to the left of the observed sample statistic z = – 1.76 Area = 0.0392

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Conclusion Since the P value is the smallest level of significance for which the sample data tells us to reject H 0, we reject H 0 for any   0.0392. For  < 0.0392, we fail to reject H 0.

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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. H 0 :  = 28 H 1 :   28 Sample results: n = 49, mean = 27.5 years, s = 2.3 years, z = –1.52

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 In a two-tailed test P value = sum of the areas in the two tails z = – 1.52 Area =1/2 of P value

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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 z = – 1.52 Area = 0.0643

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 Conclusion P = 0.1286 We reject H 0 for all   0.1286. We fail to reject H 0 for all  < 0.1286. z = – 1.52 Area = 0.0643

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 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. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Advantage of Knowing P Value We know all levels of significance for which the observed sample statistic tells us to reject H 0.

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Use of P Value If P value is , reject H 0. If P value is > , do not reject H 0.