Ce diaporama a bien été signalé.
Le téléchargement de votre SlideShare est en cours. ×

# chap08.pptx

Publicité
Publicité
Publicité
Publicité
Publicité
Publicité
Publicité
Publicité
Publicité
Publicité
Publicité
Chargement dans…3
×

## Consultez-les par la suite

1 sur 38 Publicité

# chap08.pptx

statistics

statistics

Publicité
Publicité

## Plus De Contenu Connexe

Publicité

### chap08.pptx

1. 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 4th Edition
2. 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. 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. 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, H0 3 μ : H0  3 μ : H0  3 X : H0 
5. 5. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-5 The Null Hypothesis, H0  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. 6. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-6 The Alternative Hypothesis, H1  Is the opposite of the null hypothesis  e.g.: The average number of TV sets in U.S. homes is not equal to 3 ( H1: 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. 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. 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 H0: μ = 50 ) X
9. 9. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-9 Do not reject H0 Reject H0 Reject H0  There are two cutoff values (critical values), defining the regions of rejection Sampling Distribution of /2 0 H0: μ = 50 H1: μ  50 /2 Lower critical value Upper critical value 50 X X 20 Likely Sample Results
10. 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. 11. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-11 Level of Significance and the Rejection Region H0: μ ≥ 3 H1: μ < 3 0 H0: μ ≤ 3 H1: μ > 3   Represents critical value Lower tail test Level of significance =  0 Upper tail test Two tailed test Rejection region is shaded /2 0  /2  H0: μ = 3 H1: μ ≠ 3
12. 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. 13. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-13 Example The Truth Verdict Innocent No error Type II Error Guilty Type I Error Possible Jury Trial Outcomes Guilty Innocent No Error
14. 14. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-14 Outcomes and Probabilities Actual Situation Decision Do Not Reject H0 No error (1 - )  Type II Error ( β ) Reject H0 Type I Error ( )  Possible Hypothesis Test Outcomes H0 False H0 True Key: Outcome (Probability) No Error ( 1 - β )
15. 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 H0 is true  Type II error can only occur if H0 is false If Type I error probability (  ) , then Type II error probability ( β )
16. 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 H0 is true  Also called observed level of significance
17. 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 H0  If p-value   , do not reject H0 X (continued)
18. 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, H0 2. State the alternative hypotheses, H1 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 7 Compute the p-value for the test statistic from the sample result 8. Make the statistical decision: Reject H0 if the p-value is less than alpha 9. Express the conclusion in the context of the problem
19. 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. 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 H0: μ = 3 H1: 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. 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 2.0 .08 .16 100 0.8 3 2.84 n s μ X t         Hypothesis Testing Example (continued) X
22. 22. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-22 Reject H0 Do not reject H0  8. Is the test statistic in the rejection region?  = .05/2 -t= -1.98 0 Reject H0 if p is < alpha; otherwise do not reject H0 Hypothesis Testing Example (continued)  = .05/2 Reject H0 +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. 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. 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 H0: μ ≥ 3 H1: μ < 3 H0: μ ≤ 3 H1: 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. 25. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-25 Reject H0 Do not reject H0  There is only one critical value, since the rejection area is in only one tail Lower Tail Tests  -t 3 H0: μ ≥ 3 H1: μ < 3 Critical value X 
26. 26. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-26 Reject H0 Do not reject H0 Upper Tail Tests  tα 3 H0: μ ≤ 3 H1: μ > 3  There is only one critical value, since the rejection area is in only one tail Critical value t X 
27. 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. 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. 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 ps   When both np and n(1-p) are at least 5, ps can be approximated by a normal distribution with mean and standard deviation  size sample sample in successes of number n X ps   p μ s p  n p) p(1 σ s p   (continued)
30. 30. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-30  The sampling distribution of ps is approximately normal, so the test statistic is a Z value: Hypothesis Tests for Proportions n ) p ( p p p Z s    1 np  5 and n(1-p)  5 Hypothesis Tests for p np < 5 or n(1-p) < 5 Not discussed in this chapter
31. 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 ) p 1 ( np np X Z    X  5 and n-X  5 Hypothesis Tests for X X < 5 or n-X < 5 Not discussed in this chapter
32. 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: np = (500)(.08) = 40 n(1-p) = (500)(.92) = 460 
33. 33. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-33 Z Test for Proportion: Solution  = .05 n = 500, ps = .05 p-value for -2.27 is .0134 Decision: Reject H0 at  = .05 H0: p = .08 H1: p  .08 Critical Values: ± 1.96 Test Statistic: Conclusion: z 0 Reject Reject .025 .025 1.96 -2.47 There is sufficient evidence to reject the company’s claim of 8% response rate. 2.47 500 .08) .08(1 .08 .05 n p) p(1 p p Z s         -1.96
34. 34. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-34 Using PHStat Options
35. 35. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-35 Sample PHStat Output Input Output
36. 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. 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. 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: