1. 1 1 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. John Loucks St. Edward’s University...................... SLIDES. BY

2. 2 2 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 12 Comparing Multiple Proportions, Test of Independence and Goodness of Fit n Testing the Equality of Population Proportions for Three or More Populations for Three or More Populations Goodness of Fit Test Goodness of Fit Test Test of Independence Test of Independence

3. 3 3 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Comparing Multiple Proportions, Test of Independence and Goodness of Fit  In this chapter we introduce three additional hypothesis-testing procedures.  The test statistic and the distribution used are based on the chi-square (  2 ) distribution.  In all cases, the data are categorical.

4. 4 4 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Using the notation p 1 = population proportion for population 1 p 1 = population proportion for population 1 p 2 = population proportion for population 2 p 2 = population proportion for population 2 and p k = population proportion for population k and p k = population proportion for population k H 0 : p 1 = p 2 =... = p k H 0 : p 1 = p 2 =... = p k H a : Not all population proportions are equal H a : Not all population proportions are equal Testing the Equality of Population Proportions for Three or More Populations The hypotheses for the equality of population proportions for k > 3 populations are as follows:

5. 5 5 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part.  If H 0 cannot be rejected, we cannot detect a difference among the k population proportions.  If H 0 can be rejected, we can conclude that not all k population proportions are equal.  Further analyses can be done to conclude which population proportions are significantly different from others. Testing the Equality of Population Proportions for Three or More Populations

6. 6 6 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Finger Lakes Homes Finger Lakes Homes manufactures three models of Finger Lakes Homes manufactures three models of prefabricated homes, a two-story colonial, a log cabin, and an A-frame. To help in product-line planning, management would like to compare the customer satisfaction with the three home styles. Testing the Equality of Population Proportions for Three or More Populations p 1 = proportion likely to repurchase a Colonial p 1 = proportion likely to repurchase a Colonial for the population of Colonial owners for the population of Colonial owners p 2 = proportion likely to repurchase a Log Cabin p 2 = proportion likely to repurchase a Log Cabin for the population of Log Cabin owners for the population of Log Cabin owners p 3 = proportion likely to repurchase an A-Frame p 3 = proportion likely to repurchase an A-Frame for the population of A-Frame owners for the population of A-Frame owners

7. 7 7 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. We begin by taking a sample of owners from each of the three populations. Each sample contains categorical data indicating whether the respondents are likely or not likely to repurchase the home. Testing the Equality of Population Proportions for Three or More Populations

8. 8 8 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Home Owner Home Owner Colonial Log A-Frame Total Colonial Log A-Frame Total Likely to Yes 100 81 83 264 Repurchase No 35 20 41 96 Total 135 101 124 360  Observed Frequencies (sample results) Testing the Equality of Population Proportions for Three or More Populations

9. 9 9 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Next, we determine the expected frequencies under the assumption H 0 is correct. If a significant difference exists between the observed and expected frequencies, H 0 can be rejected. Testing the Equality of Population Proportions for Three or More Populations Expected Frequencies Under the Assumption H 0 is True

10. 10 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Testing the Equality of Population Proportions for Three or More Populations Home Owner Home Owner Colonial Log A-Frame Total Colonial Log A-Frame Total Likely to Yes 99.00 74.07 90.93 264 Repurchase No 36.00 26.93 33.07 96 Total 135 101 124 360  Expected Frequencies (computed)

11. 11 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Next, compute the value of the chi-square test statistic. Note: The test statistic has a chi-square distribution with Note: The test statistic has a chi-square distribution with k – 1 degrees of freedom, provided the expected k – 1 degrees of freedom, provided the expected frequency is 5 or more for each cell. frequency is 5 or more for each cell. Note: The test statistic has a chi-square distribution with Note: The test statistic has a chi-square distribution with k – 1 degrees of freedom, provided the expected k – 1 degrees of freedom, provided the expected frequency is 5 or more for each cell. frequency is 5 or more for each cell. f ij = observed frequency for the cell in row i and column j e ij = expected frequency for the cell in row i and column j under the assumption H 0 is true where: Testing the Equality of Population Proportions for Three or More Populations

12. 12 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Obs.Exp. Sqd.Sqd. Diff. / Likely toHomeFreq. Diff. Exp. Freq. Repurch.Owner f ij e ij ( f ij - e ij )( f ij - e ij ) 2 ( f ij - e ij ) 2 / e ij YesColonial9797.50-0.500.25000.0026 YesLog Cab.8372.9410.06101.11421.3862 YesA-Frame8089.56-9.5691.30861.0196 NoColonial3837.500.500.25000.0067 NoLog Cab.1828.06-10.06101.11423.6041 NoA-Frame4434.449.5691.30862.6509 Total360  2 =8.6700 Testing the Equality of Population Proportions for Three or More Populations  Computation of the Chi-Square Test Statistic.

13. 13 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. where  is the significance level and there are k - 1 degrees of freedom p -value approach: Critical value approach: Reject H 0 if p -value <   Rejection Rule Reject H 0 if Testing the Equality of Population Proportions for Three or More Populations

14. 14 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Rejection Rule (using  =.05) Rejection Rule (using  =.05) 22 22 5.991 Do Not Reject H 0 Reject H 0 With  =.05 and k - 1 = 3 - 1 = 2 k - 1 = 3 - 1 = 2 degrees of freedom degrees of freedom Reject H 0 if p -value 5.991 Reject H 0 if p -value 5.991 Testing the Equality of Population Proportions for Three or More Populations

15. 15 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Conclusion Using the p -Value Approach The p -value < . We can reject the null hypothesis. The p -value < . We can reject the null hypothesis. Because  2 = 8.670 is between 9.210 and 7.378, the Because  2 = 8.670 is between 9.210 and 7.378, the area in the upper tail of the distribution is between area in the upper tail of the distribution is between.01 and.025..01 and.025. Area in Upper Tail.10.05.025.01.005  2 Value (df = 2) 4.605 5.991 7.378 9.210 10.597 Actual p -value is.0131 Testing the Equality of Population Proportions for Three or More Populations

16. 16 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Testing the Equality of Population Proportions for Three or More Populations We have concluded that the population proportions for the three populations of home owners are not equal. To identify where the differences between population proportions exist, we will rely on a multiple comparisons procedure.

17. 17 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Comparisons Procedure We begin by computing the three sample proportions. We will use a multiple comparison procedure known as the Marascuillo procedure.

18. 18 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Comparisons Procedure n Marascuillo Procedure We compute the absolute value of the pairwise difference between sample proportions. Colonial and Log Cabin: Colonial and A-Frame: Log Cabin and A-Frame:

19. 19 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Comparisons Procedure n Critical Values for the Marascuillo Pairwise Comparison For each pairwise comparison compute a critical value as follows: For  =.05 and k = 3:  2 = 5.991

20. 20 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Comparisons Procedure CV ij Significant if Pairwise Comparison > CV ij Colonial vs. Log Cabin Colonial vs. A-Frame Log Cabin vs. A-Frame.061.072.133.0923.0971.1034 Not Significant Significant n Pairwise Comparison Tests

21. 21 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Test of Independence 1. Set up the null and alternative hypotheses. 2. Select a random sample and record the observed frequency, f ij, for each cell of the contingency table. frequency, f ij, for each cell of the contingency table. 3. Compute the expected frequency, e ij, for each cell. H 0 : The column variable is independent of the row variable the row variable H 0 : The column variable is independent of the row variable the row variable H a : The column variable is not independent of the row variable of the row variable H a : The column variable is not independent of the row variable of the row variable

22. 22 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Test of Independence 5. Determine the rejection rule. Reject H 0 if p -value <  or. 4. Compute the test statistic. where  is the significance level and, with n rows and m columns, there are ( n - 1)( m - 1) degrees of freedom.

23. 23 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Each home sold by Finger Lakes Homes can be Each home sold by Finger Lakes Homes can be classified according to price and to style. Finger Lakes’ manager would like to determine if the price of the home and the style of the home are independent variables. Test of Independence n Example: Finger Lakes Homes (B)

24. 24 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Price Colonial Log Split-Level A-Frame Price Colonial Log Split-Level A-Frame The number of homes sold for each model and The number of homes sold for each model and price for the past two years is shown below. For convenience, the price of the home is listed as either $99,000 or less or more than $99,000. > $99,000 12 14 16 3 < $99,000 18 6 19 12 Test of Independence n Example: Finger Lakes Homes (B)

25. 25 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Hypotheses Test of Independence H 0 : Price of the home is independent of the style of the home that is purchased style of the home that is purchased H a : Price of the home is not independent of the style of the home that is purchased style of the home that is purchased

26. 26 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Expected Frequencies Test of Independence Price Colonial Log Split-Level A-Frame Total Price Colonial Log Split-Level A-Frame Total < $99K > $99K Total Total 30 20 35 15 100 12 14 16 3 45 18 6 19 12 55

27. 27 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Rejection Rule Test of Independence With  =.05 and (2 - 1)(4 - 1) = 3 d.f., Reject H 0 if p -value 7.815 =.1364 + 2.2727 +... + 2.0833 = 9.149 n Test Statistic

28. 28 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Conclusion Using the p -Value Approach The p -value < . We can reject the null hypothesis. The p -value < . We can reject the null hypothesis. Because  2 = 9.145 is between 7.815 and 9.348, the Because  2 = 9.145 is between 7.815 and 9.348, the area in the upper tail of the distribution is between area in the upper tail of the distribution is between.05 and.025..05 and.025. Area in Upper Tail.10.05.025.01.005  2 Value (df = 3) 6.251 7.815 9.348 11.345 12.838 Test of Independence Actual p -value is.0274

29. 29 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Conclusion Using the Critical Value Approach Test of Independence We reject, at the.05 level of significance, We reject, at the.05 level of significance, the assumption that the price of the home is independent of the style of home that is purchased.  2 = 9.145 > 7.815

30. 30 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Goodness of Fit Test: Multinomial Probability Distribution 1. State the null and alternative hypotheses. H 0 : The population follows a multinomial distribution with specified probabilities distribution with specified probabilities for each of the k categories for each of the k categories H 0 : The population follows a multinomial distribution with specified probabilities distribution with specified probabilities for each of the k categories for each of the k categories H a : The population does not follow a multinomial distribution with specified multinomial distribution with specified probabilities for each of the k categories probabilities for each of the k categories H a : The population does not follow a multinomial distribution with specified multinomial distribution with specified probabilities for each of the k categories probabilities for each of the k categories

31. 31 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Goodness of Fit Test: Multinomial Probability Distribution 2. Select a random sample and record the observed frequency, f i, for each of the k categories. frequency, f i, for each of the k categories. 3. Assuming H 0 is true, compute the expected frequency, e i, in each category by multiplying the frequency, e i, in each category by multiplying the category probability by the sample size. category probability by the sample size.

32. 32 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Goodness of Fit Test: Multinomial Probability Distribution 4. Compute the value of the test statistic. Note: The test statistic has a chi-square distribution Note: The test statistic has a chi-square distribution with k – 1 df provided that the expected frequencies with k – 1 df provided that the expected frequencies are 5 or more for all categories. are 5 or more for all categories. Note: The test statistic has a chi-square distribution Note: The test statistic has a chi-square distribution with k – 1 df provided that the expected frequencies with k – 1 df provided that the expected frequencies are 5 or more for all categories. are 5 or more for all categories. f i = observed frequency for category i e i = expected frequency for category i k = number of categories where:

33. 33 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Goodness of Fit Test: Multinomial Probability Distribution where  is the significance level and there are k - 1 degrees of freedom p -value approach: Critical value approach: Reject H 0 if p -value <  5. Rejection rule: Reject H 0 if

34. 34 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multinomial Distribution Goodness of Fit Test n Example: Finger Lakes Homes (A) Finger Lakes Homes manufactures four models of Finger Lakes Homes manufactures four models of prefabricated homes, a two-story colonial, a log cabin, a split-level, and an A-frame. To help in production planning, management would like to determine if previous customer purchases indicate that there is a preference in the style selected.

35. 35 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Split- A- Split- A- Model Colonial Log Level Frame # Sold 30 20 35 15 The number of homes sold of each model for 100 The number of homes sold of each model for 100 sales over the past two years is shown below. Multinomial Distribution Goodness of Fit Test n Example: Finger Lakes Homes (A)

36. 36 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Hypotheses Multinomial Distribution Goodness of Fit Test where: p C = population proportion that purchase a colonial p C = population proportion that purchase a colonial p L = population proportion that purchase a log cabin p L = population proportion that purchase a log cabin p S = population proportion that purchase a split-level p S = population proportion that purchase a split-level p A = population proportion that purchase an A-frame p A = population proportion that purchase an A-frame H 0 : p C = p L = p S = p A =.25 H a : The population proportions are not p C =.25, p L =.25, p S =.25, and p A =.25 p C =.25, p L =.25, p S =.25, and p A =.25

37. 37 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Rejection Rule 22 22 7.815 Do Not Reject H 0 Reject H 0 Multinomial Distribution Goodness of Fit Test With  =.05 and k - 1 = 4 - 1 = 3 k - 1 = 4 - 1 = 3 degrees of freedom degrees of freedom if p -value 7.815. Reject H 0 if p -value 7.815.

38. 38 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Expected Frequencies n Test Statistic Multinomial Distribution Goodness of Fit Test e 1 =.25(100) = 25 e 2 =.25(100) = 25 e 3 =.25(100) = 25 e 4 =.25(100) = 25 e 3 =.25(100) = 25 e 4 =.25(100) = 25 = 1 + 1 + 4 + 4 = 10

39. 39 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multinomial Distribution Goodness of Fit Test n Conclusion Using the p -Value Approach The p -value < . We can reject the null hypothesis. The p -value < . We can reject the null hypothesis. Because  2 = 10 is between 9.348 and 11.345, the Because  2 = 10 is between 9.348 and 11.345, the area in the upper tail of the distribution is between area in the upper tail of the distribution is between.025 and.01..025 and.01. Area in Upper Tail.10.05.025.01.005  2 Value (df = 3) 6.251 7.815 9.348 11.345 12.838

40. 40 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Conclusion Using the Critical Value Approach Multinomial Distribution Goodness of Fit Test We reject, at the.05 level of significance, We reject, at the.05 level of significance, the assumption that there is no home style preference.  2 = 10 > 7.815

41. 41 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Goodness of Fit Test: Normal Distribution 1. State the null and alternative hypotheses. 3.Compute the expected frequency, e i, for each interval. (Multiply the sample size by the probability of a (Multiply the sample size by the probability of a normal random variable being in the interval. normal random variable being in the interval. 2. Select a random sample and a. Compute the mean and standard deviation. b. Define intervals of values so that the expected frequency is at least 5 for each interval. frequency is at least 5 for each interval. c. For each interval, record the observed frequencies H 0 : The population has a normal distribution H a : The population does not have a normal distribution

42. 42 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 4. Compute the value of the test statistic. Goodness of Fit Test: Normal Distribution 5. Reject H 0 if (where  is the significance level and there are k - 3 degrees of freedom). and there are k - 3 degrees of freedom).

43. 43 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: IQ Computers IQ Computers (one better than HP?) manufactures IQ Computers (one better than HP?) manufactures and sells a general purpose microcomputer. As part of a study to evaluate sales personnel, management wants to determine, at a.05 significance level, if the annual sales volume (number of units sold by a salesperson) follows a normal probability distribution. Goodness of Fit Test: Normal Distribution

44. 44 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A simple random sample of 30 of the salespeople was taken and their numbers of units sold are listed below. n Example: IQ Computers (mean = 71, standard deviation = 18.54) 33 43 44 45 52 52 56 58 63 64 64 65 66 68 70 72 73 73 74 75 83 84 85 86 91 92 94 98 102 105 Goodness of Fit Test: Normal Distribution

45. 45 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Hypotheses H a : The population of number of units sold does not have a normal distribution with does not have a normal distribution with mean 71 and standard deviation 18.54. mean 71 and standard deviation 18.54. H 0 : The population of number of units sold has a normal distribution with mean 71 has a normal distribution with mean 71 and standard deviation 18.54. and standard deviation 18.54. Goodness of Fit Test: Normal Distribution

46. 46 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Interval Definition To satisfy the requirement of an expected To satisfy the requirement of an expected frequency of at least 5 in each interval we will divide the normal distribution into 30/5 = 6 equal probability intervals. Goodness of Fit Test: Normal Distribution

47. 47 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Interval Definition Areas = 1.00/6 =.1667 Areas = 1.00/6 =.1667 71 53.02 71 .43(18.54) = 63.03 78.97 88.98 = 71 +.97(18.54) Goodness of Fit Test: Normal Distribution

48. 48 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Observed and Expected Frequencies 1-2 1 0 1 5 5 5 5 5 530 6 3 6 5 4 630 Less than 53.02 53.02 to 63.03 53.02 to 63.03 63.03 to 71.00 63.03 to 71.00 71.00 to 78.97 71.00 to 78.97 78.97 to 88.98 78.97 to 88.98 More than 88.98 i f i e i f i - e i i f i e i f i - e i Total Goodness of Fit Test: Normal Distribution

49. 49 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Test Statistic With  =.05 and k - p - 1 = 6 - 2 - 1 = 3 d.f. With  =.05 and k - p - 1 = 6 - 2 - 1 = 3 d.f. (where k = number of categories and p = number of population parameters estimated), Reject H 0 if p -value 7.815. n Rejection Rule Goodness of Fit Test: Normal Distribution

50. 50 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Conclusion Using the p -Value Approach The p -value > . We cannot reject the null hypothesis. There is little evidence to support rejecting the assumption the population is normally distributed with  = 71 and  = 18.54. The p -value > . We cannot reject the null hypothesis. There is little evidence to support rejecting the assumption the population is normally distributed with  = 71 and  = 18.54. Because  2 = 1.600 is between.584 and 6.251 in the Chi-Square Distribution Table, the area in the upper tail Because  2 = 1.600 is between.584 and 6.251 in the Chi-Square Distribution Table, the area in the upper tail of the distribution is between.90 and.10. Area in Upper Tail.90.10.05.025.01  2 Value (df = 3).584 6.251 7.815 9.348 11.345 Goodness of Fit Test: Normal Distribution Actual p -value is.6594

51. 51 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 12