1 / 54

Learning Objectives

Learning Objectives. Hypothesis Testing. 1. Solve Hypothesis Testing Problems for Two Populations Mean Proportion Variance 2. Distinguish Independent & Related Populations. Who Gets Higher Grades: Males or Females?. Which Courses Are Easier to Learn: Math or Management ?.

selma-terrell
Download Presentation

Learning Objectives

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Learning Objectives Hypothesis Testing 1. Solve Hypothesis Testing Problems for Two Populations • Mean • Proportion • Variance 2. Distinguish Independent & Related Populations

  2. Who Gets Higher Grades: Males or Females? Which CoursesAre Easierto Learn: Mathor Management? Thinking Challenge How Would You Try to Answer These Questions?

  3. Two Population Tests

  4. Testing Two Means Independent Sampling& Paired Difference Experiments

  5. Two Population Tests

  6. 1. Different Data Sources Unrelated Independent 2. Use Difference Between the 2 Sample Means X1 -X2 1. Same Data Source Paired or Matched Repeated Measures(Before/After) 2. Use Difference Between Each Pair of Observations Di= X1i - X2i Independent & Related Populations Independent Related

  7. Two Independent Populations Examples 1. An economist wishes to determine whether there is a difference in mean family income for households in 2 socioeconomic groups. 2. An admissions officer of a small liberal arts college wants to compare the scores of applicants educated in rural high schools & in urban high schools.

  8. These are comparative studies. The general purpose of comparative studies is to establish similarities or to detect and measure differences between populations. The populations can be (1) existing populations or (2) hypothetical populations.

  9. Two Related Populations Examples 1. Nike wants to see if there is a difference in durability of 2 sole materials. One type is placed on one shoe, the other type on the other shoe of the same pair. 2. An analyst for Educational Testing Service wants to compare the scores of students before & after taking a review course.

  10. Thinking Challenge Are They Independent or Paired? 1. Miles per gallon ratings of cars before & after mounting radial tires 2. The life expectancy of light bulbs made in 2 different factories 3. Difference in hardness between 2 metals: one contains an alloy, one doesn’t 4. Tread life of two different motorcycle tires: one on the front, the other on the back

  11. Testing 2 Independent Means

  12. Two Population Tests

  13. Two Independent PopulationsHypotheses for Means

  14. Sampling Distribution

  15. Large-Sample Z Test for 2 Independent Means

  16. Two Population Tests

  17. X  X X  X 1 2 1 2   2 2 2 2   s s 1 2 1 2   n n n n 1 2 1 2 Large-Sample Z Test for 2 Independent Means • 1. Assumptions • Independent, Random Samples • Populations Are Normally Distributed • If Not Normal, Can Be Approximated by Normal Distribution (n1 30 & n2 30 ) 2. Two Independent Sample Z-Test Statistic Z

  18. Large-Sample Z Test Example You want to find out if there is a difference in length of long-distance calls between men and women. You collect the following data: WOMENMENNumber 121 125 Mean 3.27 2.53 Std Dev 1.30 1.16 Is there a difference in averagelength ( = 0.05)?

  19. Large-Sample Z Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2 0 (1 2) 0.05 n1 = 121, n2 = 125 Critical Value(s): Test Statistic: Decision: Conclusion: 3 . 27  2 . 53 z    4 . 69 1 . 69 1 . 35  121 125 Reject H Reject H Reject at  = 0.05 0 0 0.025 0.025 There is Evidence of a Difference in Means z 0 -1.96 1.96

  20. Large-Sample Z Test Thinking Challenge You’re an economist for the Department of Education. You want to find out if there is a difference in spending per pupil between urban & rural high schools. You collect the following: UrbanRural Number 35 35 Mean $ 6,012 $ 5,832 Std Dev $ 602 $ 497 Is there any difference in population means ( = 0.10)?

  21. Large-Sample Z Test Solution H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2 0 (1 2) 0.10 n1 = 35, n2 = 35 Critical Value(s): Test Statistic: Decision: Conclusion: 6012  5832 z    1 . 36 2 2 602 497  35 35 Reject H Reject H Do Not Reject at  = 0.10 0 0 0.05 0.05 There is No Evidence of a Difference in Means z 0 -1.645 1.645

  22. Small-Sample t Test for 2 Independent Means

  23. Two Population Tests

  24. Small-Sample t Test for 2 Independent Means 1. Tests Means of 2 Independent Populations Having Equal Variances 2. Assumptions • Independent, Random Samples • Both Populations Are Normally Distributed • Population Variances Are Unknown But Assumed Equal

  25. Small-Sample t Test Test Statistic

  26. Small-Sample t Test Example You want to find out if there is a difference in length of long-distance calls between men and women. You collect the following data: WOMENMENNumber 21 25 Mean 3.27 2.53 Std Dev 1.30 1.16 Is there a difference in averagelength ( = 0.05)?

  27. Small-Sample t Test Solution

  28. Small-Sample t Test Solution Test Statistic: Decision: Conclusion: H0:1 - 2 = 0 (1 = 2) Ha:1 - 2 0 (1 2) 0.05 df 21 + 25 - 2 = 44 Critical Value(s): Reject at  = 0.05 There is evidence of a difference in means

  29. Small-Sample t Test Thinking Challenge You’re a research analyst for General Motors. Assuming equal variances, is there a difference in the average miles per gallon (mpg) of two car models ( = 0.05)? You collect the following: SedanVanNumber 15 11 Mean 22.00 20.27 Std Dev 4.77 3.64

  30. Small-Sample t Test Solution Test Statistic: Decision: Conclusion: H0:1 - 2 = 0 (1 = 2) Ha:1 - 2 0 (1 2) 0.05 df 15 + 11 - 2 = 24 Critical Value(s): Do not reject at  = 0.05 There is no evidence of a difference in means

  31. Small-Sample t Test Solution

  32. Paired-Sample t Test Paired Difference Experiments

  33. Two Population Tests

  34. Paired-Sample t Test for Mean Difference 1. Tests Means of 2 Related Populations • Paired or Matched • Repeated Measures (Before/After) 2. Eliminates Variation Among Subjects 3. Assumptions • The Paired Differences Are Normally Distributed • If Not Normal, Mean Can Be Approximated by Normal Distribution

  35. Paired-Sample t Test Hypotheses Research Questions No Difference Pop 1 Pop 2 Pop 1 Pop 2   Hypothesis Any Difference Pop 1 < Pop 2 Pop 1 > Pop 2    H   = 0 0 0 0 D D D    H  0 < 0 > 0 1 D D D Note: Di = X1i - X2ifor ith observation

  36. Paired-Sample t Test Data Collection Table Observation Group 1 Group 2 Difference 1 x x D = x -x 11 21 1 11 21 2 x x D = x -x 12 22 2 12 22     i x x D = x - x 1i 2i i 1i 2i     n x x D = x - x 1n 2n n 1n 2n

  37. Paired-Sample t Test Test Statistic xD t  df  n  1 D S D n D Sample Mean Sample Standard Deviation n n   D (Di - xD)2 i i  1 i  1 x  S  D D n n  1 D D

  38. Paired-Sample t TestExample You work in Human Resources. You want to see if a training program is effective. You collect the following test score data: NameBefore (1)After (2) Sam 85 94 Tamika 94 87 Brian 78 79 Mike 87 88 At the 0.10 level, was the training effective?

  39. Computation Table Observation Before After Difference Sam 85 94 -9 Tamika 94 87 7 Brian 78 79 -1 Mike 87 88 -1 Total - 4

  40. Null HypothesisSolution 1. Was the training effective? 2. Effective means ‘After’ > ‘Before’. 3. Statistically, this means A > B. 4. Rearranging terms gives B - A0 . 5. Defining D = B - A & substituting into (4) gives D . 6. The alternative hypothesis is Ha: D 0.

  41. Paired-Sample t Test Solution H0: D = 0 (D = B - A) Ha: D < 0  =0.10 df = 4 - 1 = 3 Critical Value(s): Test Statistic: Decision: Conclusion: x  1 D t     0,306 S 6 . 53 D n 4 D Reject Do Not Reject at  = 0.10 0.10 There Is No Evidence Training Was Effective t -2,35 0

  42. You’re a marketing research analyst. You want to compare a client’s calculator to a competitor’s. You sample 8 retail stores. At the 0.01 level, does your client’s calculator sell for less than their competitor’s? (1) (2)StoreClientCompetitor 1 $ 10 $ 11 2 8 11 3 7 10 4 9 12 5 11 11 6 10 13 7 9 12 8 8 10 Paired-Sample t Test Thinking Challenge

  43. Paired-Sample t Test Solution* H0: D = 0 (D = 1 - 2) Ha: D < 0  =0.01 df = 8 - 1 = 7 Critical Value(s): Test Statistic: Decision: Conclusion: x  2 . 25 D t     5 . 486 S 1 . 16 D n 8 D Reject Reject at  = 0.01 0.01 There Is Evidence Client’s Brand (1) Sells for Less t -3,499 0

  44. Z Test for Differences in Two Proportions

  45. Two Population Tests

  46. Z Test for Difference in Two Proportions 1. Assumptions • Populations Are Independent • Normal Approximation Can Be Used • Does Not Contain 0 or n • Z-Test Statistic for Two Proportions

  47. Hypotheses for Two Proportions

  48. Z Test for Two Proportions Example As personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. At the 0.01 level, is there a difference in perceptions?

  49. Z Test for Two Proportions Solution

  50. Z Test for Two Proportions Solution Test Statistic: Decision: Conclusion: H0: p1 - p2 = 0 Ha: p1 - p2 0  = 0.01 n1 = 78 n1 = 82 Critical Value(s): Reject at  = 0.01 There is evidence of a difference in proportions

More Related