1. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Procedure for Hypothesis Testing 1. Establish the null hypothesis, H 0. 2.Establish the alternate hypothesis: H 1. 3.Use the level of significance and the alternate hypothesis to determine the critical region. 4.Find the critical values that form the boundaries of the critical region(s). 5.Use the sample evidence to draw a conclusion regarding whether or not to reject the null hypothesis.

2. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Null Hypothesis Claim about  or historical value of  H 0 :  = k

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 H 0 :  = k If you believe  is less than the value stated in H 0, use a left-tailed test. H 1 :  < k

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 H 0 :  = k If you believe  is more than the value stated in H 0, use a right-tailed test. H 1 :  > k

5. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 H 0 :  = k If you believe  is different from the value stated in H 0, use a two-tailed test. H 1 :   k

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 Hypothesis Testing About a Population Mean  when Sample Evidence Comes From a Large Sample Apply the Central Limit Theorem.

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 Central Limit Theorem Indicates: Since we are working with assumptions concerning a population mean for a large sample, we can assume: 1.The distribution of sample means is (approximately) normal.

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 Since we are working with assumptions concerning a population mean for a large sample, we can assume: 2.The mean of the sampling distribution is the same as the mean of the original distribution. Central Limit Theorem Indicates:

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 Since we are working with assumptions concerning a population mean for a large sample, we can assume: 3.The standard deviation of the sampling distribution = the original standard deviation divided by the square root of the the sample size. Assumptions:

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 Use of the Level of Significance For a one-tailed test,  is the area in the tail (the rejection area). 

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Use of the Level of Significance For a two-tailed test,  is the total area in the two tails. Each tail =  /2.  /2

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Critical z Values for Two- Tailed Test:  = 0.05 – 1.96 0 1.96 If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis The critical regions: z 1.96. H 0 :  = k H 1 :   k

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Critical z Values for Two-Tailed Test:  = 0.01 – 2.58 0 2.58 If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis The critical regions: z 2.58. H 0 :  = k H 1 :   k

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Critical z Value for Right- Tailed Test:  = 0.05 01.645 If test statistic is at, near, or below the claimed mean, we do not reject the Null Hypothesis The critical region: z > 1.645. H 0 :  = k H 1 :  > k

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Critical z Value for Right- Tailed Test:  = 0.01 02.33 If test statistic is at, near, or below the claimed mean, we do not reject the Null Hypothesis The critical region: z > 2.33. H 0 :  = k H 1 :  > k

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 Critical z Value for Left-Tailed Test:  = 0.05 -1.645 0 If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis The critical region: z < – 1.645 H 0 :  = k H 1 :  < k

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 Critical z Value for Left-Tailed Test:  = 0.01 -2.33 0 If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis The critical region: z < – 2.33 H 0 :  = k H 1 :  < k

18. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Hypothesis Test Example Your college claims that the mean age of its students is 28 years. You wish to check the validity of this statistic with a level of significance of  = 0.05. A random sample of 49 students has a mean age of 26 years with a standard deviation of 2.3 years.

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 two Test H 0 :  = 28 Against H 1 :   28 Perform a ________-tailed test. Hypothesis Test Example

20. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 two Using  = 0.05 Critical z value(s) = _________ Test H 0 :  = 28 Against H 1 :   28 Perform a ________-tailed test. Hypothesis Test Example ±1.96

21. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 Critical z Values for Two-Tailed Test:  = 0.05 – 1.96 0 1.96 If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis The critical regions: z 1.96. H 0 :  = 28 H 1 :   28

22. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 Since z < – 1.96, we _________ the null hypothesis. reject Sample Results

23. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 Hypothesis Test Example Your college claims that the mean age of its students is 28 years. You wish to check the validity of this statistic with a level of significance of  = 0.05. A random sample of 49 students has a mean age of 27.5 years with a standard deviation of 2.3 years.

24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 Using  = 0.05 Critical z values =  1.96 Test H 0 :  = 28 Against H 1 :   28 So, perform a two-tailed test. Hypothesis Test Example

25. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 Since the test statistic is neither 1.96, we _______________ the null hypothesis. do not reject Sample Results

26. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Hypothesis Test Example The manufacturer of light bulbs claims that they will burn for 1000 hours. I will test a sample of the bulbs before deciding whether to keep them. The bulbs will be returned to the manufacturer only if my sample indicates that they will burn less than 1000 hours.

27. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 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 Hypothesis Test Example

28. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 left Using  = 0.01 (So, critical z value = _______) Test H 0 :  = 1000 Against H 1 :  < 1000 Perform a ____-tailed test.) Hypothesis Test Example –2.33

29. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 Critical z Value for Left-Tailed Test:  = 0.01 -2.33 0 If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis The critical region: z < – 2.33 H 0 :  = 1000 H 1 :  < 1000

30. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 Since the test statistic is not < – 2.33 we _____________ the null hypothesis. do not reject Sample Results

31. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 Comparison of Critical z Values for Left-Tailed Tests:  = 0.01 and  = 0.05 Comparison of Critical z Values for Left-Tailed Tests:  = 0.01 and  = 0.05 – 2.33 0 – 1.645 0.01.05

32. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 In our last hypothesis test example, we calculated z = – 1.76. Since we were using  = 0.01, the boundary of the critical region was – 2.33. Our conclusion was not to reject the null hypothesis. Had we been using  = 0.05, our conclusion would have been to reject H 0.

33. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 Comparison of Critical z Values for Left-Tailed Tests:  = 0.01 and  = 0.05 Comparison of Critical z Values for Left-Tailed Tests:  = 0.01 and  = 0.05 – 2.33 0 – 1.645 0  =.01  =.05 z = – 1.76 Do not reject H 0. z = – 1.76 Reject H 0.

34. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 Statistical Significance If we reject H 0, we say that the data collected in the hypothesis testing process are statistically significant. If we do not reject H 0, we say that the data collected in the hypothesis testing process are not statistically significant.