Business statistics, 6e, 2005, groebner CH12

Business Statistics: A Decision-Making Approach 6th Edition Chapter 12 Goodness-of-Fit Tests and Contingency Analysis Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, Inc Chap 12-1 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 Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 12-2 Chi-Square Goodness-of-Fit Test  Does sample data conform to a hypothesized distribution?  Examples:   Are technical support calls equal across all days of the week? (i.e., calls follow a uniform distribution?) Do measurements from a production process follow a normal distribution? Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 12-3 Chi-Square Goodness-of-Fit Test  (continue d) the Are technical support calls equal across all days of week? (i.e., 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 SundayA Decision192 Business Statistics: Making Approach, 6e © 2005 Prentice-Hall, Inc  = 1722 Chap 12-4 Logic of Goodness-of-Fit Test  If calls are uniformly distributed, the 1722 calls would be expected to be equally divided across the days: 1722 =246 expected calls per day if uniform Chi-Square Goodness-of-Fit Test: test to see if the sample results are consistent with the expected Business Statistics: Aresults Decision Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 12-5 Observed vs Expected Frequencies Observed oi Expected ei 290 250 238 257 265 230 192 246 246 246 246 246 246 246 TOTAL A Decision- 1722 Business Statistics: Making Approach, 6e © 2005 Prentice-Hall, Inc 1722 Monday Tuesday Wednesday Thursday Friday Saturday Sunday Chap 12-6 Chi-Square Test Statistic H0: The distribution of calls is uniform over days of the week HA: The distribution of calls is not uniform  The test statistic is (oi - ei ) c =Â ei 2 (where df =k - 1) where: k = number of categories Business Statistics: Decisionoi =Aobserved cell frequency for category i Making Approach, © 2005cell frequency for category i ei =6e expected Prentice-Hall, Inc Chap 12-7 The Rejection Region H0: The distribution of calls is uniform over days of the week HA: The distribution of calls is not uniform ( o e ) i c =Â i ei  Reject H0 if c >c (with k – degrees of Business Statistics: A Decisionfreedom) Making Approach, 6e © 2005 Prentice-Hall, Inc  2 Do not reject H0  Reject H0  Chap 12-8 Chi-Square Test Statistic H0: The distribution of calls is uniform over days of the week HA: The distribution of calls is not uniform 2 (290 246) (250 246) (192 246) c2 = + + + =23.05 246 246 246 k – = (7 days of the week) so use degrees of freedom: 2.05 = 12.5916 Conclusion: 2 = 23.05 > 2 = 12.5916 so Business A Decisionreject Statistics: H0 and conclude that the Making Approach, 6e © 2005 distribution is not uniform Prentice-Hall, Inc  = 05 Do not reject H0 Reject H0 Chap 2.05 = 12.5916 2 12-9 Normal Distribution Example  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 for each cell)  Compare actual cell frequencies with expected Business Statistics: A Decisioncell frequencies Making Approach, 6e © 2005 Prentice-Hall, Inc Chap 12-10  Normal Distribution Example  What are the expected frequencies for these a normal distribution with μ = 50 and σ = 15? (continue d) for classes Expected Frequency • • • Class Frequency • less than 30 • 30 but < 40 • 40 but < 50 • • 50 but < 60 • • 60 but < 70 • • 70 but < 80 • • • •80 but < 90 Business Statistics: A Decision90 or over Making Approach, 6e â 2005 TOTAL Prentice-Hall, •Inc 10 21 33 • ? 41 26 10 • • 150 Chap 12-12 Expected Frequencies Expected •frequency • • • Value • P(X < value) less than 30 • 0.09121 • 30 but < 40 • 0.16128 • 40 but < 50 • 0.24751 • 37.13 • 50 but < 60 • 0.24751 • 37.13 • 60 but < 70 • 0.16128 • • 70 but < 80 • 0.06846 • 80 but < 90 • 90 or over • • • • • 13.68 24.19 24.19 10.27 0.01892 • 0.00383 • Business Statistics: A •Decision• TOTAL 1.00000 Making Approach, 6e © 2005 Prentice-Hall, Inc 2.84 0.57 150.00 Expected frequencies in a sample of size n=150, from a normal distribution with μ=50, σ=15 Example: Ê 30 - 50 ˆ P(x 3.841, reject H0, otherwise, not reject H0  = 0.05 2.05 = 3.841 Business Do Statistics: AH Decisionnot reject Reject H0 Making Approach, 6e © 2005 Prentice-Hall, Inc 2 Here, 2 = 0.6848 < 3.841, so we not reject H0 and conclude that gender and hand preference are independent Chap 12-25 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 Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 12-26 ... distribution  Set up a contingency analysis table and perform a chi-square test of independence Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 12-2 Chi-Square... uniform distribution?) Do measurements from a production process follow a normal distribution? Business Statistics: A DecisionMaking Approach, 6e © 2005 Prentice-Hall, Inc Chap 12-3 Chi-Square... day: Monday 290 Tuesday 250 Wednesday 238 Thursday 257 Friday 265 Saturday 230 SundayA Decision192 Business Statistics: Making Approach, 6e © 2005 Prentice-Hall, Inc  = 1722 Chap 12-4 Logic of Goodness-of-Fit