![© 2003 Prentice-Hall, Inc.
Chap 9-40
p -Value Solution
0 1.80
t35
Reject
(p-Value is between .025 and .05) ≥ (α = 0.01)
Do Not Reject.
p-Value = [.025, .05]
α = 0.01
Test Statistic 1.80 is in the Do Not Reject
Region
2.4377](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/businessstatisticschapter9-160519114719/85/Business-Statistics-Chapter-9-40-320.jpg)
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Business Statistics Chapter 9
- 1. © 2003 Prentice-Hall, Inc. Chap 9-1 Basic Business Statistics (9th Edition) Chapter 9 Fundamentals of Hypothesis Testing: One-Sample Tests
- 2. © 2003 Prentice-Hall, Inc. Chap 9-2 Chapter Topics Hypothesis Testing Methodology Z Test for the Mean ( Known) p-Value Approach to Hypothesis Testing Connection to Confidence Interval Estimation One-Tail Tests t Test for the Mean ( Unknown) Z Test for the Proportion Potential Hypothesis-Testing Pitfalls and Ethical Issues σ σ
- 3. © 2003 Prentice-Hall, Inc. Chap 9-3 What is a Hypothesis? A Hypothesis is a Claim (Assumption) about the Population Parameter Examples of parameters are population mean or proportion The parameter must be identified before analysis I claim the mean GPA of this class is 3.5! © 1984-1994 T/Maker Co. µ =
- 4. © 2003 Prentice-Hall, Inc. Chap 9-4 The Null Hypothesis, H0 States the Assumption (Numerical) to be Tested E.g., The mean GPA is 3.5 Null Hypothesis is Always about a Population Parameter ( ), Not about a Sample Statistic ( ) Is the Hypothesis a Researcher Tries to Reject 0 : 3.5H µ = 0 : 3.5H µ = 0 : 3.5H X =
- 5. © 2003 Prentice-Hall, Inc. Chap 9-5 The Null Hypothesis, H0 Begin with the Assumption that the Null Hypothesis is True Similar to the notion of innocent until proven guilty Refer to the Status Quo Always Contains the “=” Sign The Null Hypothesis May or May Not be Rejected (continued)
- 6. © 2003 Prentice-Hall, Inc. Chap 9-6 The Alternative Hypothesis, H1 Is the Opposite of the Null Hypothesis E.g., The mean GPA is NOT 3.5 ( ) Challenges the Status Quo Never Contains the “=” Sign The Alternative Hypothesis May or May Not Be Accepted (i.e., The Null Hypothesis May or May Not Be Rejected) Is Generally the Hypothesis that the Researcher Claims 1 : 3.5H µ ≠
- 7. © 2003 Prentice-Hall, Inc. Chap 9-7 Error in Making Decisions Type I Error Reject a true null hypothesis When the null hypothesis is rejected, we can say that “We have shown the null hypothesis to be false (with some ‘slight’ probability, i.e. , of making a wrong decision) Has serious consequences Probability of Type I Error is Called level of significance Set by researcher α α
- 8. © 2003 Prentice-Hall, Inc. Chap 9-8 Error in Making Decisions Type II Error Fail to reject a false null hypothesis Probability of Type II Error is The power of the test is Probability of Not Making Type I Error Called the Confidence Coefficient ( )1 α− (continued) ( )1 β− β
- 9. © 2003 Prentice-Hall, Inc. Chap 9-9 Hypothesis Testing Process Identify the Population Assume the population mean GPA is 3.5 ( ) REJECT Take a Sample Null Hypothesis No, not likely! X 2.4 likely ifIs 3.5?µ= = 0 : 3.5H µ = ( )2.4X =
- 10. © 2003 Prentice-Hall, Inc. Chap 9-10 = 3.5 It is unlikely that we would get a sample mean of this value ... ... if in fact this were the population mean. ... Therefore, we reject the null hypothesis that = 3.5. Reason for Rejecting H0 µ Sampling Distribution of 2.4 If H0 is true X X µ
- 11. © 2003 Prentice-Hall, Inc. Chap 9-11 Level of Significance, Defines Unlikely Values of Sample Statistic if Null Hypothesis is True Called rejection region of the sampling distribution Designated by , (level of significance) Typical values are .01, .05, .10 Selected by the Researcher at the Beginning Controls the Probability of Committing a Type I Error Provides the Critical Value(s) of the Test α α
- 12. © 2003 Prentice-Hall, Inc. Chap 9-12 Level of Significance and the Rejection Region H0: µ ≥ 3.5 H1: µ < 3.5 0 0 0 H0: µ ≤ 3.5 H1: µ > 3.5 H0: µ = 3.5 H1: µ ≠ 3.5 α α α/2 Critical Value(s) Rejection Regions
- 13. © 2003 Prentice-Hall, Inc. Chap 9-13 Result Probabilities H0: Innocent The Truth The Truth Verdict Innocent Guilty Decision H0 True H0 False Innocent Correct Error Do Not Reject H0 1 - α Type II Error (β ) Guilty Error Correct Reject H0 Type I Error (α ) Power (1 - β ) Jury Trial Hypothesis Test
- 14. © 2003 Prentice-Hall, Inc. Chap 9-14 Type I & II Errors Have an Inverse Relationship α β Reduce probability of one error and the other one goes up holding everything else unchanged.
- 15. © 2003 Prentice-Hall, Inc. Chap 9-15 Factors Affecting Type II Error True Value of Population Parameter increases when the difference between the hypothesized parameter and its true value decrease Significance Level increases when decreases Population Standard Deviation increases when increases Sample Size increases when n decreases β β α β σ β n α β β β σ
- 16. © 2003 Prentice-Hall, Inc. Chap 9-16 How to Choose between Type I and Type II Errors Choice Depends on the Cost of the Errors Choose Smaller Type I Error When the Cost of Rejecting the Maintained Hypothesis is High A criminal trial: convicting an innocent person The Exxon Valdez: causing an oil tanker to sink Choose Larger Type I Error When You Have an Interest in Changing the Status Quo A decision in a startup company about a new piece of software A decision about unequal pay for a covered group
- 17. © 2003 Prentice-Hall, Inc. Chap 9-17 Critical Values Approach to Testing Convert Sample Statistic (e.g., ) to Test Statistic (e.g., Z, t or F –statistic) Obtain Critical Value(s) for a Specified from a Table or Computer If the test statistic falls in the critical region, reject H0 Otherwise, do not reject H0 X α
- 18. © 2003 Prentice-Hall, Inc. Chap 9-18 p-Value Approach to Testing Convert Sample Statistic (e.g., ) to Test Statistic (e.g., Z, t or F –statistic) Obtain the p-value from a table or computer p-value: probability of obtaining a test statistic as extreme or more extreme ( or ) than the observed sample value given H0 is true Called observed level of significance Smallest value of that an H0 can be rejected Compare the p-value with If p-value , do not reject H0 If p-value , reject H0 X ≤ ≥ < ≥ α α α α
- 19. © 2003 Prentice-Hall, Inc. Chap 9-19 General Steps in Hypothesis Testing E.g., Test the Assumption that the True Mean # of TV Sets in U.S. Homes is at Least 3 ( Known)σ 1. State the H0 2. State the H1 3. Choose 4. Choose n 5. Choose Test 0 1 : 3 : 3 =.05 100 Z H H n test µ µ α ≥ < = α
- 20. © 2003 Prentice-Hall, Inc. Chap 9-20 100 households surveyed Computed test stat =-2, p-value = .0228 Reject null hypothesis The true mean # TV set is less than 3 (continued) Reject H0 α -1.645 Z 6. Set up critical value(s) 7. Collect data 8. Compute test statistic and p-value 9. Make statistical decision 10.Express conclusion General Steps in Hypothesis Testing
- 21. © 2003 Prentice-Hall, Inc. Chap 9-21 One-Tail Z Test for Mean ( Known) Assumptions Population is normally distributed If not normal, requires large samples Null hypothesis has or sign only is known Z Test Statistic σ ≤ ≥ / X X X X Z n µ µ σ σ − − = = σ
- 22. © 2003 Prentice-Hall, Inc. Chap 9-22 Rejection Region Z0 Reject H0 Z0 Reject H0 H0: µ ≥ µ0 H1: µ < µ0 H0: µ ≤ µ0 H1: µ > µ0 Z must be significantly below 0 to reject H0 Small values of Z don’t contradict H0 ; don’t reject H0 ! αα
- 23. © 2003 Prentice-Hall, Inc. Chap 9-23 Example: One-Tail Test Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified σ to be 15 grams. Test at the α = 0.05 level. 368 gm. H0: µ ≤ 368 H1: µ > 368 X
- 24. © 2003 Prentice-Hall, Inc. Chap 9-24 Reject and Do Not Reject Regions Z0 1.645 .05 Reject 1.50 X 368XXµ µ= = 372.5 0 : 368H µ ≤ 1 : 368H µ > Do Not Reject
- 25. © 2003 Prentice-Hall, Inc. Chap 9-25 Finding Critical Value: One-Tail Z .04 .06 1.6 .9495 .9505 .9515 1.7 .9591 .9599 .9608 1.8 .9671 .9678 .9686 .9738 .9750 Z0 1.645 .05 1.9 .9744 Standardized Cumulative Normal Distribution Table (Portion) What is Z given α = 0.05? α = .05 Critical Value = 1.645 .95 1Zσ =
- 26. © 2003 Prentice-Hall, Inc. Chap 9-26 Example Solution: One-Tail Test α = 0.5 n = 25 Critical Value: 1.645 Conclusion: Do Not Reject at α = .05. Insufficient Evidence that True Mean is More Than 368. Z0 1.645 .05 Reject H0: µ ≤ 368 H1: µ > 368 1.50 X Z n µ σ − = = 1.50
- 27. © 2003 Prentice-Hall, Inc. Chap 9-27 p -Value Solution Z0 1.50 p-Value =.0668 Z Value of Sample Statistic From Z Table: Lookup 1.50 to Obtain .9332 Use the alternative hypothesis to find the direction of the rejection region. 1.0000 - .9332 .0668 p-Value is P(Z ≥ 1.50) = 0.0668
- 28. © 2003 Prentice-Hall, Inc. Chap 9-28 p -Value Solution (continued) 0 1.50 Z Reject (p-Value = 0.0668) ≥ (α = 0.05) Do Not Reject. p Value = 0.0668 α = 0.05 Test Statistic 1.50 is in the Do Not Reject Region 1.645
- 29. © 2003 Prentice-Hall, Inc. Chap 9-29 One-Tail Z Test for Mean ( Known) in PHStat PHStat | One-Sample Tests | Z Test for the Mean, Sigma Known … Example in Excel Spreadsheet σ
- 30. © 2003 Prentice-Hall, Inc. Chap 9-30 Example: Two-Tail Test Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified σ to be 15 grams and the distribution to be normal. Test at the α = 0.05 level. 368 gm. H0: µ = 368 H1: µ ≠ 368 X
- 31. © 2003 Prentice-Hall, Inc. Chap 9-31 Reject and Do Not Reject Regions Z0 1.96 .025 Reject -1.96 .025 1.50 X 368XXµ µ= = 372.5 Reject 0 : 368H µ = 1 : 368H µ ≠
- 32. © 2003 Prentice-Hall, Inc. Chap 9-32 372.5 368 1.50 15 25 X Z n µ σ − − = = =α = 0.05 n = 25 Critical Value: ±1.96 Example Solution: Two-Tail Test Test Statistic: Decision: Conclusion: Do Not Reject at α = .05. Z0 1.96 .025 Reject -1.96 .025 H0: µ = 368 H1: µ ≠ 368 1.50 Insufficient Evidence that True Mean is Not 368.
- 33. © 2003 Prentice-Hall, Inc. Chap 9-33 p-Value Solution (p-Value = 0.1336) ≥ (α = 0.05) Do Not Reject. 0 1.50 Z Reject α = 0.05 1.96 p-Value = 2 x 0.0668 Test Statistic 1.50 is in the Do Not Reject Region Reject
- 34. © 2003 Prentice-Hall, Inc. Chap 9-34 PHStat | One-Sample Tests | Z Test for the Mean, Sigma Known … Example in Excel Spreadsheet Two-Tail Z Test for Mean ( Known) in PHStatσ
- 35. © 2003 Prentice-Hall, Inc. Chap 9-35 ( ) ( ) For 372.5, 15 and 25, the 95% confidence interval is: 372.5 1.96 15/ 25 372.5 1.96 15/ 25 or 366.62 378.38 X nσ µ µ = = = − ≤ ≤ + ≤ ≤ Connection to Confidence Intervals We are 95% confident that the population mean is between 366.62 and 378.38. If this interval contains the hypothesized mean (368), we do not reject the null hypothesis. It does. Do not reject.
- 36. © 2003 Prentice-Hall, Inc. Chap 9-36 t Test: Unknown Assumption Population is normally distributed If not normal, requires a large sample is unknown t Test Statistic with n-1 Degrees of Freedom σ / X t S n µ− = σ
- 37. © 2003 Prentice-Hall, Inc. Chap 9-37 Example: One-Tail t Test Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, and s = 15. Test at the α = 0.01 level. 368 gm. H0: µ ≤ 368 H1: µ > 368 σ is not given
- 38. © 2003 Prentice-Hall, Inc. Chap 9-38 Reject and Do Not Reject Regions t35 0 2.4377 .01 Reject 1.80 X 368XXµ µ= = 372.5 0 : 368H µ ≤ 1 : 368H µ > Do Not Reject
- 39. © 2003 Prentice-Hall, Inc. Chap 9-39 Example Solution: One-Tail α = 0.01 n = 36, df = 35 Critical Value: 2.4377 Test Statistic: Decision: Conclusion: Do Not Reject at a = .01. t35 0 2.4377 .01 Reject H0: µ ≤ 368 H1: µ > 368 372.5 368 1.80 15 36 X t S n µ− − = = = 1.80 Insufficient Evidence that True Mean is More Than 368.
- 40. © 2003 Prentice-Hall, Inc. Chap 9-40 p -Value Solution 0 1.80 t35 Reject (p-Value is between .025 and .05) ≥ (α = 0.01) Do Not Reject. p-Value = [.025, .05] α = 0.01 Test Statistic 1.80 is in the Do Not Reject Region 2.4377
- 41. © 2003 Prentice-Hall, Inc. Chap 9-41 PHStat | One-Sample Tests | t Test for the Mean, Sigma Known … Example in Excel Spreadsheet t Test: Unknown in PHStatσ
- 42. © 2003 Prentice-Hall, Inc. Chap 9-42 Proportion Involves Categorical Variables Two Possible Outcomes “Success” (possesses a certain characteristic) and “Failure” (does not possess a certain characteristic) Fraction or Proportion of Population in the “Success” Category is Denoted by p
- 43. © 2003 Prentice-Hall, Inc. Chap 9-43 Proportion 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 (continued) Number of Successes Sample Size s X p n = = sp pµ = (1 ) sp p p n σ − =
- 44. © 2003 Prentice-Hall, Inc. Chap 9-44 Example: Z Test for Proportion ( ) ( ) ( ) Check: 500 .04 20 5 1 500 1 .04 480 5 np n p = = ≥ − = − = ≥ A marketing company claims that a survey will have a 4% response rate. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the α = .05 significance level.
- 45. © 2003 Prentice-Hall, Inc. Chap 9-45 Reject and Do Not Reject Regions Z0 1.96 .025 Reject -1.96 .025 1.1411 0.04SP pµ = = 0.05 Reject 0 : 0.04H p = 1 : 0.04H p ≠ SP
- 46. © 2003 Prentice-Hall, Inc. Chap 9-46 0.05 Critical Values: ± 1.96 1.1411 ( ) ( ) .05 .04 1.1411 1 .04 1 .04 500 Sp p Z p p n − − ≅ = = − − Z Test for Proportion: Solution α = .05 n = 500 Do not reject at α = .05. H0: p = .04 H1: p ≠ .04 Test Statistic: Decision : Conclusion : Z 0 Reject Reject .025.025 1.96-1.96 We do not have sufficient evidence to reject the company’s claim of 4% response rate. SP 0.04
- 47. © 2003 Prentice-Hall, Inc. Chap 9-47 p -Value Solution (p-Value = 0.2538) ≥ (α = 0.05) Do Not Reject. 0 1.1411 Z Reject α = 0.05 1.96 p-Value = 2 x .1269 Test Statistic 1.1411 is in the Do Not Reject Region Reject
- 48. © 2003 Prentice-Hall, Inc. Chap 9-48 Z Test for Proportion in PHStat PHStat | One-Sample Tests | Z Test for the Proportion … Example in Excel Spreadsheet
- 49. © 2003 Prentice-Hall, Inc. Chap 9-49 Potential Pitfalls and Ethical Issues Data Collection Method is Not Randomized to Reduce Selection Biases Treatment of Human Subjects are Manipulated Without Informed Consent Data Snooping is Used to Choose between One-Tail and Two-Tail Tests, and to Determine the Level of Significance
- 50. © 2003 Prentice-Hall, Inc. Chap 9-50 Potential Pitfalls and Ethical Issues Data Cleansing is Practiced to Hide Observations that do not Support a Stated Hypothesis Fail to Report Pertinent Findings (continued)
- 51. © 2003 Prentice-Hall, Inc. Chap 9-51 Chapter Summary Addressed Hypothesis Testing Methodology Performed Z Test for the Mean ( Known) Discussed p –Value Approach to Hypothesis Testing Made Connection to Confidence Interval Estimation σ
- 52. © 2003 Prentice-Hall, Inc. Chap 9-52 Chapter Summary Performed One-Tail and Two-Tail Tests Performed t Test for the Mean ( Unknown) Performed Z Test for the Proportion Discussed Potential Pitfalls and Ethical Issues (continued) σ