Business statistics a decision making approach 6th edition ch10ppln

Business Statistics: A Decision-Making Approach 6th Edition Chapter 10 Hypothesis Tests for One and Two Population Variances Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-1 Chapter Goals After completing this chapter, you should be able to:  Formulate and complete hypothesis tests for a single population variance  Find critical chi-square distribution values from the chisquare table  Formulate and complete hypothesis tests for the difference between two population variances  Use the F table to find critical F values Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-2 Hypothesis Tests for Variances Hypothesis Tests for Variances Tests for a Single Population Variances Tests for Two Population Variances Chi-Square test statistic F test statistic Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-3 Single Population Hypothesis Tests for Variances Tests for a Single Population Variances * Chi-Square test statistic Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc H0: σ2 = σ02 HA: σ2 ≠ σ02 Two tailed test H0: σ2 ≥ σ02 HA: σ2 < σ02 Lower tail test H0: σ2 ≤ σ02 HA: σ2 > σ02 Upper tail test Chap 10-4 Chi-Square Test Statistic Hypothesis Tests for Variances The chi-squared test statistic for a Single Population Variance is: Tests for a Single Population Variances Chi-Square test statistic (n − 1)s χ = σ2 * where χ2 = standardized chi-square variable n = sample size s2 = sample variance σ2 = hypothesized variance Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-5 The Chi-square Distribution  The chi-square distribution is a family of distributions, depending on degrees of freedom:  d.f = n - 12 16 20 24 28 d.f = χ2 12 16 20 24 28 d.f = Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc χ2 12 16 20 24 28 χ2 d.f = 15 Chap 10-6 Finding the Critical Value  The critical value, chi-square table , is found from the χ 2α Upper tail test: H0: σ2 ≤ σ02 HA: σ2 > σ02 α χ2 Do not reject H0 χ Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Reject H0 α Chap 10-7 Example  A commercial freezer must hold the selected temperature with little variation Specifications call for a standard deviation of no more than degrees (or variance of 16 degrees2) A sample of 16 freezers is tested and yields a sample variance of s2 = 24 Test to see whether the standard deviation specification is exceeded Use α = 05 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-8 Finding the Critical Value  The the chi-square table to find the critical value: χ 2α = 24.9958 (α = 05 and 16 – = 15 d.f.) The test statistic is: (n − 1)s (16 − 1)24 χ = = = 22.5 σ 16 Since 22.5 < 24.9958, not reject H0 There is not significant evidence at the α = 05 level that the standard deviation specification is exceeded α = 05 χ2 Do not reject H0 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc χ 2α Reject H0 = 24.9958 Chap 10-9 Lower Tail or Two Tailed Chi-square Tests Lower tail test: Two tail test: H0: σ2 ≥ σ02 HA: σ2 < σ02 H0: σ2 = σ02 HA: σ2 ≠ σ02 α α/2 α/2 χ2 Reject χ Do not reject H0 χ2 Reject 1-α Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Do not reject H0 χ 21-α/2 Reject χ 2α/2 Chap 10-10 F Test for Difference in Two Population Variances Hypothesis Tests for Variances H0: σ12 – σ22 = HA: σ12 – σ22 ≠ Two tailed test H0: σ12 – σ22 ≥ HA: σ12 – σ22 < Lower tail test H0: σ12 – σ22 ≤ HA: σ12 – σ22 > * Tests for Two Population Variances F test statistic Upper tail test Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-11 F Test for Difference in Two Population Variances Hypothesis Tests for Variances The F test statistic is: (Place the larger sample variance in the numerator) 2 s F= s Tests for Two Population Variances * s12 = Variance of Sample n1 - = numerator degrees of freedom s22 = Variance of Sample n2 - = denominator degrees of freedom Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc F test statistic Chap 10-12 The F Distribution  The F critical value is found from the F table  The are two appropriate degrees of freedom: numerator and denominator s12 F= s2  where df1 = n1 – ; df2 = n2 – In the F table,  numerator degrees of freedom determine the row  denominator degrees of freedom determine the column Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-13 Finding the Critical Value H0: σ12 – σ22 ≥ HA: σ12 – σ22 < H0: σ12 – σ22 = HA: σ12 – σ22 ≠ H0: σ12 – σ22 ≤ HA: σ12 – σ22 > α Do not reject H0 Fα Reject H0 α/2 F rejection region for a one-tail test is  s12 F = > Fα s2 Do not reject H0 Fα/2 Reject H0 F rejection region for a two-tailed test is  s12 F = > Fα / s2 (when the larger sample variance in the numerator) Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-14 F Test: An Example You are a financial analyst for a brokerage firm You want to compare dividend yields between stocks listed on the NYSE & NASDAQ You collect the following data: NYSE NASDAQ Number 2125 Mean 3.272.53 Std dev 1.301.16 Is there a difference in the variances between the NYSE & NASDAQ at the α = 0.05 level? Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-15 F Test: Example Solution   Form the hypothesis test: H0: σ21 – σ22 = (there is no difference between variances) HA: σ21 – σ22 ≠ (there is a difference between variances) Find the F critical value for α = 05:  Numerator:   df1 = n1 – = 21 – = 20 Denominator:  df2 = n2 – = 25 – = 24 F.05/2, 20, 24 = 2.327 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-16 F Test: Example Solution (continued)  The test statistic is: H0: σ12 – σ22 = HA: σ12 – σ22 ≠ s12 1.30 F= = = 1.256 s2 1.16 α/2 = 025  F = 1.256 is not greater than the critical F value of 2.327, so we not reject H0  Conclusion: There is no evidence of a difference in variances at α = 05 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Do not reject H0 Reject H0 Fα/2 =2.327 Chap 10-17 Using EXCEL and PHStat EXCEL  F test for two variances:  Tools | data analysis | F-test: two sample for variances PHStat  Chi-square test for the variance:   PHStat | one-sample tests | chi-square test for the variance F test for two variances:  PHStat | two-sample tests | F test for differences in two variances Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-18 Chapter Summary  Performed chi-square tests for the variance  Used the chi-square table to find chi-square critical values  Performed F tests for the difference between two population variances  Used the F table to find F critical values Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-19 ... the larger sample variance in the numerator) Business Statistics: A Decision- Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-14 F Test: An Example You are a financial analyst for a brokerage... standard deviation specification is exceeded Use α = 05 Business Statistics: A Decision- Making Approach, 6e © 2010 PrenticeHall, Inc Chap 10-8 Finding the Critical Value  The the chi-square table... Population Variances Chi-Square test statistic (n − 1)s χ = σ2 * where χ2 = standardized chi-square variable n = sample size s2 = sample variance σ2 = hypothesized variance Business Statistics: A Decision- Making