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# Procedure for Hypothesis Testing

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1. Procedure for Hypothesis Testing 1. Establish the null hypothesis, H0. 2. Establish the alternate hypothesis: H1. 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. Null Hypothesis Claim about  or historical value of  H0:  = k

3. H0:  = k If you believe  is less than the value stated in H0, use a left-tailed test. H1:  < k

4. H0:  = k If you believe  is more than the value stated in H0, use a right-tailed test. H1:  > k

5. H0:  = k If you believe  is different from the value stated in H0, use a two-tailed test. H1:  k

6. Hypothesis Testing About a Population Mean when Sample Evidence Comes From a Large Sample Apply the Central Limit Theorem.

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. Central Limit Theorem Indicates: 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.

9. Assumptions: 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.

10. Use of the Level of Significance • For a one-tailed test,  is the area in the tail (the rejection area). 

11. Use of the Level of Significance • For a two-tailed test,  is the total area in the two tails. • Each tail =  /2. /2 /2

12. Critical z Values for Two-Tailed Test:  = 0.05 H0:  = k H1:  k If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis – 1.96 0 1.96 The critical regions: z < – 1.96 with z > 1.96.

13. Critical z Values for Two-Tailed Test:  = 0.01 H0:  = k H1:  k If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis – 2.58 0 2.58 The critical regions: z < – 2.58 with z > 2.58.

14. Critical z Value for Right-Tailed Test:  = 0.05 H0:  = k H1:  > k If test statistic is at, near, or below the claimed mean, we do not reject the Null Hypothesis 0 1.645 The critical region: z > 1.645.

15. Critical z Value for Right-Tailed Test:  = 0.01 H0:  = k H1:  > k If test statistic is at, near, or below the claimed mean, we do not reject the Null Hypothesis 0 2.33 The critical region: z > 2.33.

16. Critical z Value for Left-Tailed Test:  = 0.05 H0:  = k H1:  < k If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis -1.645 0 The critical region: z < – 1.645

17. Critical z Value for Left-Tailed Test:  = 0.01 H0:  = k H1:  < k If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis -2.33 0 The critical region: z < – 2.33

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. Hypothesis Test Example Test H0:  = 28 Against H1:   28 Perform a ________-tailed test. two

20. Hypothesis Test Example Test H0:  = 28 Against H1:   28 Perform a ________-tailed test. two Using  = 0.05 Critical z value(s) = _________ ±1.96

21. Critical z Values for Two-Tailed Test:  = 0.05 H0:  = 28 H1:  28 If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis – 1.96 0 1.96 The critical regions: z < – 1.96 with z > 1.96.

22. Sample Results reject Since z < – 1.96, we _________ the null hypothesis.

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. Hypothesis Test Example Test H0:  = 28 Against H1:   28 So, perform a two-tailed test. Using  = 0.05 Critical z values = 1.96

25. Sample Results Since the test statistic is neither < – 1.96 nor > 1.96 , we _______________ the null hypothesis. do not reject

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. Hypothesis Test Example 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

28. Hypothesis Test Example Test H0:  = 1000 Against H1:  < 1000 Perform a ____-tailed test.) left –2.33 Using  = 0.01 (So, critical z value = _______)

29. Critical z Value for Left-Tailed Test:  = 0.01 H0:  = 1000 H1:  < 1000 If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis -2.33 0 The critical region: z < – 2.33

30. Sample Results Since the test statistic is not < – 2.33 we _____________ the null hypothesis. do not reject

31. .05 .01 – 1.645 0 – 2.33 0 Comparison of Critical z Values for Left-Tailed Tests:  = 0.01 and  = 0.05

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 H0.

33.  =.01  = .05 – 1.645 0 – 2.33 0 Comparison of Critical z Values for Left-Tailed Tests:  = 0.01 and  = 0.05 z = – 1.76 Reject H0. z = – 1.76 Do not reject H0.

34. Statistical Significance • If we reject H0, we say that the data collected in the hypothesis testing process are statistically significant. • If we do not reject H0, we say that the data collected in the hypothesis testing process are not statistically significant.