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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 12 Goodness-of-Fit Tests and Contingency Analysis

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-2 Chapter Goals After completing this chapter, you should be able to: Use the chi-square goodness-of-fit test to determine whether data fits a specified distribution Set up a contingency analysis table and perform a chi-square test of independence

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-3 Does sample data conform to a hypothesized distribution? Examples: Are technical support calls equal across all days of the week? (i.e., do calls follow a uniform distribution?) Do measurements from a production process follow a normal distribution? Chi-Square Goodness-of-Fit Test

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-4 Are technical support calls equal across all days of the week? (i.e., do calls follow a uniform distribution?) Sample data for 10 days per day of week: Sum of calls for this day: Monday 290 Tuesday 250 Wednesday 238 Thursday 257 Friday 265 Saturday 230 Sunday 192 Chi-Square Goodness-of-Fit Test (continued) = 1722

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-5 Logic of Goodness-of-Fit Test If calls are uniformly distributed, the 1722 calls would be expected to be equally divided across the 7 days: Chi-Square Goodness-of-Fit Test: test to see if the sample results are consistent with the expected results

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-6 Observed vs. Expected Frequencies Observed o i Expected e i Monday Tuesday Wednesday Thursday Friday Saturday Sunday 290 250 238 257 265 230 192 246 TOTAL1722

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-7 Chi-Square Test Statistic The test statistic is where: k = number of categories o i = observed cell frequency for category i e i = expected cell frequency for category i H 0 : The distribution of calls is uniform over days of the week H A : The distribution of calls is not uniform

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-8 The Rejection Region Reject H 0 if H 0 : The distribution of calls is uniform over days of the week H A : The distribution of calls is not uniform 0 22 Reject H 0 Do not reject H 0 (with k – 1 degrees of freedom) 22

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-9 Chi-Square Test Statistic H 0 : The distribution of calls is uniform over days of the week H A : The distribution of calls is not uniform 0 =.05 Reject H 0 Do not reject H 0 22 k – 1 = 6 (7 days of the week) so use 6 degrees of freedom: 2.05 = 12.5916 Conclusion: 2 = 23.05 > 2 = 12.5916 so reject H 0 and conclude that the distribution is not uniform

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-10 Do measurements from a production process follow a normal distribution with μ = 50 and σ = 15? Process: Get sample data Group sample results into classes (cells) (Expected cell frequency must be at least 5 for each cell) Compare actual cell frequencies with expected cell frequencies Normal Distribution Example

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-11 Normal Distribution Example 150 Sample Measurements 80 65 36 66 50 38 57 77 59 …etc… ClassFrequency less than 3010 30 but < 4021 40 but < 5033 50 but < 6041 60 but < 7026 70 but < 8010 80 but < 907 90 or over2 TOTAL150 (continued) Sample data and values grouped into classes:

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-12 What are the expected frequencies for these classes for a normal distribution with μ = 50 and σ = 15? (continued) ClassFrequency Expected Frequency less than 30 10 30 but < 40 21 40 but < 50 33 ? 50 but < 60 41 60 but < 70 26 70 but < 80 10 80 but < 90 7 90 or over 2 TOTAL150 Normal Distribution Example

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-13 Expected Frequencies ValueP(X < value) Expected frequency less than 300.0912113.68 30 but < 400.1612824.19 40 but < 500.2475137.13 50 but < 600.2475137.13 60 but < 700.1612824.19 70 but < 800.0684610.27 80 but < 900.018922.84 90 or over0.003830.57 TOTAL1.00000150.00 Expected frequencies in a sample of size n=150, from a normal distribution with μ=50, σ=15 Example:

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-14 The Test Statistic Class Frequency (observed, o i ) Expected Frequency, e i less than 3010 13.68 30 but < 4021 24.19 40 but < 5033 37.13 50 but < 6041 37.13 60 but < 7026 24.19 70 but < 8010 10.27 80 but < 907 2.84 90 or over2 0.57 TOTAL150 150.00 Reject H 0 if The test statistic is (with k – 1 degrees of freedom)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-15 The Rejection Region H 0 : The distribution of values is normal with μ = 50 and σ = 15 H A : The distribution of calls does not have this distribution 0 =.05 Reject H 0 Do not reject H 0 22 8 classes so use 7 d.f.: 2.05 = 14.0671 Conclusion: 2 = 12.097 < 2 = 14.0671 so do not reject H 0 2.05 = 14.0671

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-16 Contingency Tables Situations involving multiple population proportions Used to classify sample observations according to two or more characteristics Also called a crosstabulation table.

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-17 Contingency Table Example H 0 : Hand preference is independent of gender H A : Hand preference is not independent of gender Left-Handed vs. Gender Dominant Hand: Left vs. Right Gender: Male vs. Female

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-18 Contingency Table Example Sample results organized in a contingency table: (continued) Gender Hand Preference LeftRight Female12108120 Male24156180 36264300 120 Females, 12 were left handed 180 Males, 24 were left handed sample size = n = 300:

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-19 Logic of the Test If H 0 is true, then the proportion of left-handed females should be the same as the proportion of left-handed males The two proportions above should be the same as the proportion of left-handed people overall H 0 : Hand preference is independent of gender H A : Hand preference is not independent of gender

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-20 Finding Expected Frequencies Overall: P(Left Handed) = 36/300 =.12 120 Females, 12 were left handed 180 Males, 24 were left handed If independent, then P(Left Handed | Female) = P(Left Handed | Male) =.12 So we would expect 12% of the 120 females and 12% of the 180 males to be left handed… i.e., we would expect (120)(.12) = 14.4 females to be left handed (180)(.12) = 21.6 males to be left handed

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-21 Expected Cell Frequencies Expected cell frequencies: (continued) Example:

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-22 Observed v. Expected Frequencies Observed frequencies vs. expected frequencies: Gender Hand Preference LeftRight Female Observed = 12 Expected = 14.4 Observed = 108 Expected = 105.6 120 Male Observed = 24 Expected = 21.6 Observed = 156 Expected = 158.4 180 36264300

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-23 The Chi-Square Test Statistic where: o ij = observed frequency in cell (i, j) e ij = expected frequency in cell (i, j) r = number of rows c = number of columns The Chi-square contingency test statistic is:

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-24 Observed v. Expected Frequencies Gender Hand Preference LeftRight Female Observed = 12 Expected = 14.4 Observed = 108 Expected = 105.6 120 Male Observed = 24 Expected = 21.6 Observed = 156 Expected = 158.4 180 36264300

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-25 Contingency Analysis 2.05 = 3.841 Reject H 0 = 0.05 Decision Rule: If 2 > 3.841, reject H 0, otherwise, do not reject H 0 Do not reject H 0 Here, 2 = 0.6848 < 3.841, so we do not reject H 0 and conclude that gender and hand preference are independent

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 12-26 Chapter Summary Used the chi-square goodness-of-fit test to determine whether data fits a specified distribution Example of a discrete distribution (uniform) Example of a continuous distribution (normal) Used contingency tables to perform a chi-square test of independence Compared observed cell frequencies to expected cell frequencies

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