Business Statistics, 4e by Ken Black Chapter 12 Analysis of Categorical Data Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

Learning Objectives Understand the 2 goodness-of-fit test and how to use it. Analyze data using the 2 test of independence. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

2 Goodness-of-Fit Test The 2 goodness-of-fit test compares expected (theoretical) frequencies of categories from a population distribution to the observed (actual) frequencies from a distribution to determine whether there is a difference between what was expected and what was observed. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

2 Goodness-of-Fit Test Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

Milk Sales Data for Demonstration Problem 12.1 Month Gallons January 1,610 February 1,585 March 1,649 April 1,590 May 1,540 June 1,397 July 1,410 August 1,350 September 1,495 October 1,564 November 1,602 December 1,655 18,447 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

Hypotheses and Decision Rules for Demonstration Problem 12.1 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

Calculations for Demonstration Problem 12.1 Month fo fe (fo - fe)2/fe January 1,610 1,537.25 3.44 February 1,585 1.48 March 1,649 8.12 April 1,590 1.81 May 1,540 0.00 June 1,397 12.80 July 1,410 10.53 August 1,350 22.81 September 1,495 1.16 October 1,564 0.47 November 1,602 2.73 December 1,655 9.02 18,447 18,447.00 74.38 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

Demonstration Problem 12.1: Conclusion 0.01 df = 11 24.725 Non Rejection region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

Bank Customer Arrival Data for Demonstration Problem 12.2 Number of Arrivals Observed Frequencies 7 1 18 2 25 3 17 4 12 5 5 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

Hypotheses and Decision Rules for Demonstration Problem 12.2 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

Calculations for Demonstration Problem 12 Calculations for Demonstration Problem 12.2: Estimating the Mean Arrival Rate Number of Arrivals X Observed Frequencies f f·X 7 1 18 2 25 50 3 17 51 4 12 48 5 5 192 Mean Arrival Rate Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

Calculations for Demonstration Problem 12 Calculations for Demonstration Problem 12.2: Poisson Probabilities for  = 2.3 Number of Arrivals X Expected Probabilities P(X) Frequencies n·P(X) 0.1003 8.42 1 0.2306 19.37 2 0.2652 22.28 3 0.2033 17.08 4 0.1169 9.82  0.0838 7.04 Poisson Probabilities for  = 2.3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

2 Calculations for Demonstration Problem 12.2 Number of Arrivals X Observed Frequencies f Expected nP(X) (fo - fe)2 fe 1 2 3 4 5 7 8.42 18 19.37 25 22.28 17 17.08 12 9.82 5 7.04 84 84.00 0.24 0.10 0.33 0.00 0.48 0.59 1.74 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

Demonstration Problem 12.2: Conclusion 0.05 df = 4 9.488 Non Rejection region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

Using a 2 Goodness-of-Fit Test to Test a Population Proportion Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

Using a 2 Goodness-of-Fit Test to Test a Population Proportion: Calculations   2 33 16 167 184 18.0625 + 1.5707 19.6332      o e f = fo fe Defects Nondefects 200 n = Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

Using a 2 Goodness-of-Fit Test to Test a Population Proportion: Conclusion 0.05 df = 1 3.841 Non Rejection region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

Qualitative Variables 2 Test of Independence Qualitative Variables Nominal Data Used to analyze the frequencies of two variables with multiple categories to determine whether the two variables are independent. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

2 Test of Independence: Investment Example In which region of the country do you reside? A. Northeast B. Midwest C. South D. West Which type of financial investment are you most likely to make today? E. Stocks F. Bonds G. Treasury bills Type of financial Investment E F G A O13 nA Geographic B nB Region C nC D nD nE nF nG N Contingency Table Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

2 Test of Independence: Investment Example Type of Financial Investment E F G A e12 nA Geographic B nB Region C nC D nD nE nF nG N Contingency Table Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

2 Test of Independence: Formulas   ij i j e n N where  : = the row the column the total of row i of column of all fr equencies 2    o f df (r - 1)(c 1) r the number r of rows c r of columns Expected Frequencies Calculated  (Observed ) Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 24

2 Test of Independence: Gasoline Preference Versus Income Category Type of Gasoline Income Regular Premium Extra Less than $30,000 $30,000 to $49,999 $50,000 to $99,000 At least $100,000 r = 4 c = 3 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

Gasoline Preference Versus Income Category: Observed Frequencies Type of Gasoline Income Regular Premium Extra Less than $30,000 85 16 6 107 $30,000 to $49,999 102 27 13 142 $50,000 to $99,000 36 22 15 73 At least $100,000 23 25 63 238 88 59 385 Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

Gasoline Preference Versus Income Category: Expected Frequencies Type of Gasoline Income Regular Premium Extra Less than $30,000 (66.15) (24.46) (16.40) 85 16 6 107 $30,000 to $49,999 (87.78) (32.46) (21.76) 102 27 13 142 $50,000 to $99,000 (45.13) (16.69) (11.19) 36 22 15 73 At least $100,000 (38.95) (14.40) (9.65) 23 25 63 238 88 59 385   ij i j e n N  11 12 66 24 46 40 . Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

Gasoline Preference Versus Income Category: 2 Calculation   2 88 66 15 16 24 46 6 40 102 87 78 27 32 13 21 76 36 45 22 69 11 19 38 95 23 14 25 9 65 70      o e f . Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

Gasoline Preference Versus Income Category: Conclusion 0.01 df = 6 16.812 Non rejection region Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29