1. 1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach

2. 2 2 Slide IS 310 – Business Statistics Additional Uses of Chi-Square Distribution In Chapter 11, we discussed the use of Chi-Square distribution in testing population variances. Here in chapter 12, we will cover two additional testing procedures by using the Chi-Square distribution. o Goodness of Fit Test o Goodness of Fit Test o Test of Independence o Test of Independence

3. 3 3 Slide IS 310 – Business Statistics Additional Uses of Chi-Square Distribution Goodness of Fit Test How close are sample results to the expected results? How close are sample results to the expected results? Example: In tossing a coin, you expect half heads and half tails. You tossed a coin 100 times. You expected 50 heads and 50 tails. However, you obtained 48 heads and 52 tails. Are 48 heads and 52 tails close enough to call the coin fair?

4. 4 4 Slide IS 310 – Business Statistics Additional Uses of Chi-Square Distribution Test of Independence Are two variables of interest independent of each other? Are two variables of interest independent of each other?Examples: Is starting salary of fresh graduates independent of graduates’ field of study? Is beer preference independent of the gender of the beer drinker?

5. 5 5 Slide IS 310 – Business Statistics Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population 1. Set up the null and alternative hypotheses. 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.

6. 6 6 Slide IS 310 – Business Statistics Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population 4. Compute the value of the test statistic. Note: The test statistic has a chi-square distribution with k – 1 df provided that the expected frequencies are 5 or more for all categories. Note: The test statistic has a chi-square distribution with k – 1 df provided that the expected frequencies 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:

7. 7 7 Slide IS 310 – Business Statistics Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population 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

8. 8 8 Slide IS 310 – Business Statistics Multinomial Distribution Goodness of Fit Test n Example: Finger Lakes Homes (A) Finger Lakes Homes manufactures Finger Lakes Homes manufactures four models of prefabricated homes, four models of prefabricated homes, a two-story colonial, a log cabin, a a two-story colonial, a log cabin, a split-level, and an A-frame. To help split-level, and an A-frame. To help in production planning, management in production planning, management would like to determine if previous would like to determine if previous customer purchases indicate that there customer purchases indicate that there is a preference in the style selected. is a preference in the style selected.

9. 9 9 Slide IS 310 – Business Statistics Split- A- Split- A- Model Colonial Log Level Frame # Sold 30 20 35 15 The number of homes sold of each 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)

10. 10 Slide IS 310 – Business Statistics 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

11. 11 Slide IS 310 – Business Statistics Hypotheses H : There is no preference in the home styles or all o home styles have equal preferences o home styles have equal preferences H : All home styles do not have equal preferences a

12. 12 Slide IS 310 – Business Statistics 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.

13. 13 Slide IS 310 – Business Statistics 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

14. 14 Slide IS 310 – Business Statistics 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 Note: A precise p -value can be found using Note: A precise p -value can be found using Minitab or Excel. Minitab or Excel. Note: A precise p -value can be found using Note: A precise p -value can be found using Minitab or Excel. Minitab or Excel.

15. 15 Slide IS 310 – Business Statistics 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

16. 16 Slide IS 310 – Business Statistics Sample Problem Problem # 3 (10-Page 462; 11-Page 477) H : Viewing audience proportions are the same 0 H : Viewing audience proportions are not the same a Category Frequencies Frequencies Differences Squared Observed Expected /Expected Frequencies Observed Expected /Expected Frequencies ABC 95 87 0.74 CBA 70 84 2.33 NBC 89 75 2.61 Independents 46 54 1.19

17. 17 Slide IS 310 – Business Statistics Sample Problem Continued Decision Rule: Reject Null Hypothesis if 2 2 2 2 Χ -statistic > Χ Χ -statistic > Χ 0.05, 3 0.05, 3 2 2 2 2 Χ -statistic = 6.87 Χ = 7.815 0.05,3 0.05,3 Decision: Do not reject the null hypothesis Interpretation: Viewing audience proportions have not changed.

18. 18 Slide IS 310 – Business Statistics Test of Independence: Contingency Tables 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.

19. 19 Slide IS 310 – Business Statistics Test of Independence: Contingency Tables 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.

20. 20 Slide IS 310 – Business Statistics Each home sold by Finger Lakes 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. Contingency Table (Independence) Test n Example: Finger Lakes Homes (B)

21. 21 Slide IS 310 – Business Statistics Price Colonial Log Split-Level A-Frame Price Colonial Log Split-Level A-Frame The number of homes sold for 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 Contingency Table (Independence) Test n Example: Finger Lakes Homes (B)

22. 22 Slide IS 310 – Business Statistics n Hypotheses Contingency Table (Independence) Test 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

23. 23 Slide IS 310 – Business Statistics n Expected Frequencies Contingency Table (Independence) Test 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

24. 24 Slide IS 310 – Business Statistics n Rejection Rule Contingency Table (Independence) Test 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

25. 25 Slide IS 310 – Business Statistics 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 Contingency Table (Independence) Test Note: A precise p -value can be found using Note: A precise p -value can be found using Minitab or Excel. Minitab or Excel. Note: A precise p -value can be found using Note: A precise p -value can be found using Minitab or Excel. Minitab or Excel.

26. 26 Slide IS 310 – Business Statistics n Conclusion Using the Critical Value Approach Contingency Table (Independence) Test 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

27. 27 Slide IS 310 – Business Statistics Sample Problem Problem # 12 (10-Page 469; 11-Page 484) H : Method of payment is independent of age group 0 H : Method of payment is NOT independent of age a Payment Age Observed Expected Squared Squared Difference/ Exp. Method Group Frequency Frequency Difference Frequency Plastic 18-24 21 15.5 30.25 1.95 Plastic 25-34 27 23.3 13.69 0.59 Plastic 35-44 27 25.5 2.25 0.09 Plastic 45-up 36 46.6 112.36 2.41 Cash/Chk 18-24 21 26.4 29.2 1.1 Cash/Chk 25-34 36 39.7 13.69 0.34 Cash/Chk 35-44 42 43.5 2.25 0.05 Cash/Chk 45-up 90 79.4 112.36 1.42

28. 28 Slide IS 310 – Business Statistics Sample Problem Contd 2 Χ -statistic = 7.95 2 Critical Χ = 7.815.05,3.05,3 2 2 2 2 Since Χ –statistic > Critical Χ, we reject the Null Hypothesis Interpretation: Method of payment is NOT independent of age group. p-value is between 0.025 and 0.05 The age group 18-24 uses plastic more than any other group

29. 29 Slide IS 310 – Business Statistics End of Chapter 12