1. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-1 Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests Statistics for Managers Using Microsoft ® Excel 4 th Edition

2. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-2 Chapter Goals After completing this chapter, you should be able to:  Formulate null and alternative hypotheses for applications involving a single population mean or proportion  Formulate a decision rule for testing a hypothesis  Know how to use the p-value approaches to test the null hypothesis for both mean and proportion problems  Know what Type I and Type II errors are

3. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-3 What is a Hypothesis?  A hypothesis is a claim (assumption) about a population parameter:  population mean  population proportion Example: The mean monthly cell phone bill of this city is μ = $42 Example: The proportion of adults in this city with cell phones is p =.68

4. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-4  States the assumption to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( )  Is always about a population parameter, not about a sample statistic The Null Hypothesis, H 0

5. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-5 The Null Hypothesis, H 0  Begins with the assumption that the null hypothesis is true  Similar to the notion of innocent until proven guilty  Refers to the status quo  Always contains “=”, “≤” or “  ” sign  May or may not be rejected (continued)

6. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-6 The Alternative Hypothesis, H 1  Is the opposite of the null hypothesis  e.g.: The average number of TV sets in U.S. homes is not equal to 3 ( H 1 : μ ≠ 3 )  Challenges the status quo  Never contains the “=”, “≤” or “  ” sign  Is generally the hypothesis that is believed (or needs to be supported) by the researcher

7. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-7 Hypothesis Testing  We assume the null hypothesis is true  If the null hypothesis is rejected we have proven the alternate hypothesis  If the null hypothesis is not rejected we have proven nothing as the sample size may have been to small

8. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Population Claim: the population mean age is 50. (Null Hypothesis: REJECT Suppose the sample mean age is 20: X = 20 Sample Null Hypothesis 20 likely if μ = 50?  Is Hypothesis Testing Process If not likely, Now select a random sample H 0 : μ = 50 ) X

9. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-9 Do not reject H 0 Reject H 0  There are two cutoff values (critical values), defining the regions of rejection Sampling Distribution of  /2 0 H 0 : μ = 50 H 1 : μ  50  /2 Lower critical value Upper critical value 50 X 20 Likely Sample Results

10. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-10 Level of Significance,   Defines the unlikely values of the sample statistic if the null hypothesis is true  Defines rejection region of the sampling distribution  Is designated by , (level of significance)  Typical values are.01,.05, or.10  Is the compliment of the confidence coefficient  Is selected by the researcher before sampling  Provides the critical value of the test

11. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-11 Level of Significance and the Rejection Region H 0 : μ ≥ 3 H 1 : μ < 3 0 H 0 : μ ≤ 3 H 1 : μ > 3   Represents critical value Lower tail test Level of significance =  0 Upper tail test Two tailed test Rejection region is shaded /2 0   H 0 : μ = 3 H 1 : μ ≠ 3

12. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-12  Type I Error  When a true null hypothesis is rejected  The probability of a Type I Error is   Called level of significance of the test  Set by researcher in advance  Type II Error  Failure to reject a false null hypothesis  The probability of a Type II Error is β Errors in Making Decisions

13. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-13 Example The Truth Verdict InnocentNo error Type II Error GuiltyType I Error Possible Jury Trial Outcomes Guilty Innocent No Error

14. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-14 Outcomes and Probabilities Actual Situation Decision Do Not Reject H 0 No error (1 - )  Type II Error ( β ) Reject H 0 Type I Error ( )  Possible Hypothesis Test Outcomes H 0 False H 0 True Key: Outcome (Probability) No Error ( 1 - β )

15. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-15 Type I & II Error Relationship  Type I and Type II errors can not happen at the same time  Type I error can only occur if H 0 is true  Type II error can only occur if H 0 is false If Type I error probability (  ), then Type II error probability ( β )

16. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-16 p-Value Approach to Testing  p-value: Probability of obtaining a test statistic more extreme ( ≤ or  ) than the observed sample value given H 0 is true  Also called observed level of significance

17. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-17 p-Value Approach to Testing  Convert Sample Statistic (e.g. ) to Test Statistic (e.g. t statistic )  Obtain the p-value from a table or computer  Compare the p-value with   If p-value < , reject H 0  If p-value  , do not reject H 0 X (continued)

18. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-18 9 Steps in Hypothesis Testing 1. State the null hypothesis, H 0 2.State the alternative hypotheses, H 1 3. Choose the level of significance, α 4. Choose the sample size, n 5. Determine the appropriate test statistic to use 6. Collect the data 7Compute the p-value for the test statistic from the sample result 8. Make the statistical decision: Reject H 0 if the p-value is less than alpha 9.Express the conclusion in the context of the problem

19. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-19 Hypothesis Tests for the Mean  Known  Unknown Hypothesis Tests for 

20. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-20 Hypothesis Testing Example Test the claim that the true mean # of TV sets in U.S. homes is equal to 3.  1-2. State the appropriate null and alternative hypotheses H 0 : μ = 3 H 1 : μ ≠ 3 (This is a two tailed test)  3. Specify the desired level of significance Suppose that  =.05 is chosen for this test  4. Choose a sample size Suppose a sample of size n = 100 is selected

21. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-21  5. Determine the appropriate Test σ is unknown so this is a t test  6. Collect the data Suppose the sample results are n = 100, = 2.84 s = 0.8  7. So the test statistic is: The p value for n=100,  =.05, t=-2 is.048 Hypothesis Testing Example (continued)

22. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-22 Reject H 0 Do not reject H 0  8. Is the test statistic in the rejection region?  =.05/2 -t= -1.98 0 Reject H 0 if p is < alpha; otherwise do not reject H 0 Hypothesis Testing Example (continued)  =.05/2 Reject H 0 +t= +1.98 Here, t = -2.0 < -1.98, so the test statistic is in the rejection region The p-value.048 is < alpha.05, we reject the null hypothesis

23. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-23  9. Express the conclusion in the context of the problem Since The p-value.048 is < alpha.05, we have rejected the null hypothesis Thereby proving the alternate hypothesis Conclusion: There is sufficient evidence that the mean number of TVs in U.S. homes is not equal to 3 Hypothesis Testing Example (continued) If we had failed to reject the null hypothesis the conclusion would have been: There is not sufficient evidence to reject the claim that the mean number of TVs in U.S. home is 3

24. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-24 One Tail Tests  In many cases, the alternative hypothesis focuses on a particular direction H 0 : μ ≥ 3 H 1 : μ < 3 H 0 : μ ≤ 3 H 1 : μ > 3 This is a lower tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 This is an upper tail test since the alternative hypothesis is focused on the upper tail above the mean of 3

25. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-25 Reject H 0 Do not reject H 0  There is only one critical value, since the rejection area is in only one tail Lower Tail Tests  -t 3 H 0 : μ ≥ 3 H 1 : μ < 3 Critical value

26. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-26 Reject H 0 Do not reject H 0 Upper Tail Tests  tαtα 3 H 0 : μ ≤ 3 H 1 : μ > 3  There is only one critical value, since the rejection area is in only one tail Critical value t

27. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-27 Assumptions of the One-Sample t Test  The data is randomly selected  The population is normally distributed or the sample size is over 30 and the population is not highly skewed

28. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-28 Hypothesis Tests for Proportions  Involves categorical values  Two possible outcomes  “Success” (possesses a certain characteristic)  “Failure” (does not possesses that characteristic)  Fraction or proportion of the population in the “success” category is denoted by p

29. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-29 Proportions  Sample proportion in the success category is denoted by p s   When both np and n(1-p) are at least 5, p s can be approximated by a normal distribution with mean and standard deviation  (continued)

30. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-30  The sampling distribution of p s is approximately normal, so the test statistic is a Z value: Hypothesis Tests for Proportions np  5 and n(1-p)  5 Hypothesis Tests for p np < 5 or n(1-p) < 5 Not discussed in this chapter

31. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-31  An equivalent form to the last slide, but in terms of the number of successes, X: Z Test for Proportion in Terms of Number of Successes X  5 and n-X  5 Hypothesis Tests for X X < 5 or n-X < 5 Not discussed in this chapter

32. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-32 Example: Z Test for Proportion A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the  =.05 significance level. Check: n p = (500)(.08) = 40 n(1-p) = (500)(.92) = 460 

33. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-33 Z Test for Proportion: Solution  =.05 n = 500, p s =.05 p-value for -2.27 is.0134 Decision: Reject H 0 at  =.05 H 0 : p =.08 H 1 : p .08 Critical Values: ± 1.96 Test Statistic: Conclusion: z 0 Reject.025 1.96 -2.47 There is sufficient evidence to reject the company’s claim of 8% response rate. -1.96

34. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-34 Using PHStat Options

35. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-35 Sample PHStat Output Input Output

36. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-36 Potential Pitfalls and Ethical Considerations  Use randomly collected data to reduce selection biases  Do not use human subjects without informed consent  Choose the level of significance, α, before data collection  Do not employ “data snooping” to choose between one- tail and two-tail test, or to determine the level of significance  Do not practice “data cleansing” to hide observations that do not support a stated hypothesis  Report all pertinent findings

37. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-37 Chapter Summary  Addressed hypothesis testing methodology  Discussed critical value and p–value approaches to hypothesis testing  Discussed type 1 and Type2 errors  Performed two tailed t test for the mean (σ unknown)  Performed Z test for the proportion  Discussed one-tail and two-tail tests  Addressed pitfalls and ethical issues

38. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-38 Answer Sheet for All Problems  ___________ Null Hypothesis  ___________ Alternate Hypothesis  ___________ Alpha  ___________ p-value  ___________ Decision (reject or do not reject)  Conclusion: