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Basic Business Statistics 12th Edition Chapter 10 Two-Sample Tests Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-1 Learning Objectives In this chapter, you learn how to use hypothesis testing for comparing the difference between: The means of two independent populations The means of two related populations The proportions of two independent populations The variances of two independent populations by testing the ratio of the two variances Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-2 Two-Sample Tests DCOVA Two-Sample Tests Population Means, Independent Samples Population Means, Related Samples Population Proportions Population Variances Examples: Group 1 vs. Group 2 Same group before vs. after treatment Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Proportion 1 vs. Proportion 2 Variance 1 vs. Variance 2 Chap 10-3 Difference Between Two Means DCOVA Population means, independent samples * σ1 and σ2 unknown, assumed equal σ1 and σ2 unknown, not assumed equal Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Goal: Test hypothesis or form a confidence interval for the difference between two population means, μ1 – μ2 The point estimate for the difference is X1 – X2 Chap 10-4 Difference Between Two Means: Independent Samples DCOVA Population means, independent samples * Different data sources Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population σ1 and σ2 unknown, assumed equal Use Sp to estimate unknown σ. Use a Pooled-Variance t test. σ1 and σ2 unknown, not assumed equal Use S1 and S2 to estimate unknown σ1 and σ2. Use a Separate-Variance t test. Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-5 Hypothesis Tests for Two Population Means DCOVA Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 μ2 H1: μ1 < μ2 H0: μ1 ≤ μ2 H1: μ1 > μ2 H0: μ1 = μ2 H1: μ1 ≠ μ2 i.e., i.e., i.e., H0: μ1 – μ2 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-6 Hypothesis tests for μ1 – μ2 DCOVA Two Population Means, Independent Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 – μ2 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 a a -ta Reject H0 if tSTAT < -ta Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall ta Reject H0 if tSTAT > ta a/2 -ta/2 a/2 ta/2 Reject H0 if tSTAT < -ta/2 or tSTAT > ta/2 Chap 10-7 Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and assumed equal DCOVA Assumptions: Population means, independent samples σ1 and σ2 unknown, assumed equal σ1 and σ2 unknown, not assumed equal Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Samples are randomly and independently drawn * Populations are normally distributed or both sample sizes are at least 30 Population variances are unknown but assumed equal Chap 10-8 Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and assumed equal (continued) DCOVA • The pooled variance is: Population means, independent samples σ1 and σ2 unknown, assumed equal 2 2 n 1 S n 1 S 1 2 2 S2 1 p * (n 1 1) (n 2 1) • The test statistic is: t STAT X X μ 1 2 1 μ2 1 1 S n1 n 2 2 p σ1 and σ2 unknown, not assumed equal Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall • Where tSTAT has d.f. = (n1 + n2 – 2) Chap 10-9 Confidence interval for µ1 - µ2 with σ1 and σ2 unknown and assumed equal DCOVA Population means, independent samples σ1 and σ2 unknown, assumed equal σ1 and σ2 unknown, not assumed equal Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall * The confidence interval for μ1 – μ2 is: X 1 X 2 ta/2 1 1 S n1 n 2 2 p Where tα/2 has d.f. = n1 + n2 – 2 Chap 10-10 Pooled-Variance t Test Example DCOVA You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQ Number 21 25 Sample mean 3.27 2.53 Sample std dev 1.30 1.16 Assuming both populations are approximately normal with equal variances, is there a difference in mean yield (a = 0.05)? Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-11 Pooled-Variance t Test Example: Calculating the Test Statistic (continued) H0: μ1 - μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) DCOVA The test statistic is: t STAT X 1 X 2 μ1 μ 2 1 1 S n1 n 2 2 p 3.27 2.53 0 1 1 1.5021 21 25 2 2 2 2 n 1 S n 1 S 21 1 1.30 25 1 1.16 1 2 2 S2 1 P (n1 1) (n 2 1) Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall (21 - 1) ( 25 1) 2.040 1.5021 Chap 10-12 Pooled-Variance t Test Example: Hypothesis Test Solution DCOVA H0: μ1 - μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) a = 0.05 df = 21 + 25 - 2 = 44 Critical Values: t = ± 2.0154 Reject H0 .025 -2.0154 Reject H0 .025 0 2.0154 t 2.040 Test Statistic: Decision: 3.27 2.53 t STAT 2.040 Reject H0 at a = 0.05 1 1 Conclusion: 1.5021 21 25 There is evidence of a difference in means. Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-13 Excel Pooled-Variance t test Comparing NYSE & NASDAQ DCOVA Pooled-Variance t Test for the Difference Between Two Means (assumes equal population variances) Data Hypothesized Difference Level of Significance Population 1 Sample Sample Size Sample Mean Sample Standard Deviation Population 2 Sample Sample Size Sample Mean Sample Standard Deviation Intermediate Calculations Population 1 Sample Degrees of Freedom Population 2 Sample Degrees of Freedom Total Degrees of Freedom Pooled Variance Standard Error Difference in Sample Means t Test Statistic 0 0.05 21 =COUNT(DATA!$A:$A) 3.27 =AVERAGE(DATA!$A:$A) 1.3 =STDEV(DATA!$A:$A) 25 =COUNT(DATA!$B:$B) 2.53 =AVERAGE(DATA!$B:$B) 1.16 =STDEV(DATA!$B:$B) Decision: Reject H0 at a = 0.05 Conclusion: There is evidence of a difference in means. 20 =B7 - 1 24 =B11 - 1 44 =B16 + B17 1.502 =((B16 * B9^2) + (B17 * B13^2)) / B18 0.363 =SQRT(B19 * (1/B7 + 1/B11)) 0.74 =B8 - B12 2.040 =(B21 - B4) / B20 Two-Tail Test Lower Critical Value Upper Critical Value p-value Reject the null hypothesis Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall -2.015 =-TINV(B5, B18) 2.015 =TINV(B5, B18) 0.047 =TDIST(ABS(B22),B18,2) =IF(B27<B5,"Reject the null hypothesis", "Do not reject the null hypothesis") Chap 10-14 Minitab Pooled-Variance t test Comparing NYSE & NASDAQ DCOVA Two-Sample T-Test and CI Sample 1 2 N Mean StDev SE Mean 21 3.27 1.30 0.28 25 2.53 1.16 0.23 Difference = mu (1) - mu (2) Estimate for difference: 0.740 95% CI for difference: (0.009, 1.471) T-Test of difference = 0 (vs not =): T-Value = 2.04 P-Value = 0.047 DF = 44 Both use Pooled StDev = 1.2256 Decision: Reject H0 at a = 0.05 Conclusion: There is evidence of a difference in means. Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-15 Pooled-Variance t Test Example: Confidence Interval for µ1 - µ2 DCOVA Since we rejected H0 can we be 95% confident that µNYSE > µNASDAQ? 95% Confidence Interval for µNYSE - µNASDAQ X X t 1 2 a/2 1 1 S 0.74 2.0154 0.3628 (0.009, 1.471) n1 n 2 2 p Since 0 is less than the entire interval, we can be 95% confident that µNYSE > µNASDAQ Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-16 Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown, not assumed equal DCOVA Assumptions: Population means, independent samples Samples are randomly and independently drawn Populations are normally distributed or both sample sizes are at least 30 σ1 and σ2 unknown, assumed equal σ1 and σ2 unknown, not assumed equal Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall * Population variances are unknown and cannot be assumed to be equal Chap 10-17 Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and not assumed equal (continued) DCOVA Population means, independent samples σ1 and σ2 unknown, assumed equal The test statistic is: t STAT X 1 X 2 μ1 μ 2 S12 S 22 n1 n 2 tSTAT has d.f. ν = 2 σ1 and σ2 unknown, not assumed equal Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall * S1 2 S 2 2 n n 1 2 2 2 S1 2 S22 n n 1 2 n1 1 n2 1 Chap 10-18 Related Populations The Paired Difference Test DCOVA Tests Means of 2 Related Populations Related samples Paired or matched samples Repeated measures (before/after) Use difference between paired values: Di = X1i - X2i Eliminates Variation Among Subjects Assumptions: Both Populations Are Normally Distributed Or, if not Normal, use large samples Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-19 Related Populations The Paired Difference Test DCOVA (continued) The ith paired difference is Di , where Related samples Di = X1i - X2i The point estimate for the paired difference population mean μD is D : D n D i 1 i n n The sample standard deviation is SD SD 2 (D D ) i i 1 n1 n is the number of pairs in the paired sample Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-20 The Paired Difference Test: Finding tSTAT DCOVA Paired samples The test statistic for μD is: t STAT D μD SD n Where tSTAT has n - 1 d.f. Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-21 The Paired Difference Test: Possible Hypotheses DCOVA Paired Samples Lower-tail test: Upper-tail test: Two-tail test: H0: μD 0 H1: μD < 0 H0: μD ≤ 0 H1: μD > 0 H0: μD = 0 H1: μD ≠ 0 a a -ta ta Reject H0 if tSTAT < -ta Reject H0 if tSTAT > ta Where tSTAT has n - 1 d.f. Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall a/2 -ta/2 a/2 ta/2 Reject H0 if tSTAT < -ta/2 or tSTAT > ta/2 Chap 10-22 The Paired Difference Confidence Interval Paired samples DCOVA The confidence interval for μD is D ta / 2 SD n n where Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall SD (D D) i1 2 i n 1 Chap 10-23 Paired Difference Test: Example DCOVA Assume you send your salespeople to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data: Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, Di C.B. T.F. M.H. R.K. M.O. 6 20 3 0 4 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall 4 6 2 0 0 - 2 -14 - 1 0 - 4 -21 D = Di n = -4.2 SD 2 (D D ) i n 1 5.67 Chap 10-24 Paired Difference Test: DCOVA Solution Has the training made a difference in the number of complaints (at the 0.01 level)? H0: μD = 0 H1: μD 0 a = .01 D = - 4.2 t0.005 = ± 4.604 d.f. = n - 1 = 4 Test Statistic: t STAT D μ D 4.2 0 1.66 SD / n 5.67/ 5 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Reject Reject a/2 a/2 - 4.604 4.604 - 1.66 Decision: Do not reject H0 (tstat is not in the reject region) Conclusion: There is insufficient evidence there is significant change in the number of complaints. Chap 10-25 Paired t Test In Excel DCOVA Paired t Test Data Hypothesized Mean Diff. Level of Significance Intermediate Calculations Sample Size Dbar Degrees of Freedom SD Standard Error t-Test Statistic Two-Tail Test Lower Critical Value Upper Critical Value p-value Do not reject the null Hypothesis 0 0.05 Since -2.776 < -1.66 < 2.776 we do not reject the null hypothesis. Or 5 =COUNT(I2:I6) -4.2 =AVERAGE(I2:I6) 4 =B8 - 1 5.67 =STDEV(I2:I6) 2.54 =B11/SQRT(B8) -1.66 =(B9 - B4)/B12 Since p-value = 0.173 > 0.05 we do not reject the null hypothesis. Thus we conclude that there is Insufficient evidence to conclude there is a difference in the average number of complaints. -2.776 =-TINV(B5,B10) 2.776 =TINV(B5,B10) 0.173 =TDIST(ABS(B13),B10,2) =IF(B18<B5,"Reject the null hypothesis", "Do not reject the null hypothesis") Data not shown is in column I Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-26 Paired t Test In Minitab Yields The Same Conclusions Paired T-Test and CI: After, Before DCOVA Paired T for After - Before After Before Difference N 5 5 5 Mean 2.40 6.60 -4.20 StDev 2.61 7.80 5.67 SE Mean 1.17 3.49 2.54 95% CI for mean difference: (-11.25, 2.85) T-Test of mean difference = 0 (vs not = 0): T-Value = -1.66 P-Value = 0.173 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-27 Two Population Proportions DCOVA Goal: test a hypothesis or form a confidence interval for the difference between two population proportions, π1 – π2 Population proportions The point estimate for the difference is Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall p1 p2 Chap 10-28 Two Population Proportions DCOVA Population proportions In the null hypothesis we assume the null hypothesis is true, so we assume π1 = π2 and pool the two sample estimates The pooled estimate for the overall proportion is: X1 X2 p n1 n 2 where X1 and X2 are the number of items of interest in samples 1 and 2 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-29 Two Population Proportions (continued) The test statistic for π1 – π2 is a Z statistic: Population proportions Z STAT where Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall DCOVA p1 p2 π1 π2 1 1 p (1 p ) n1 n2 X1 X 2 X1 p , p1 n1 n2 n1 X2 , p2 n2 Chap 10-30 Hypothesis Tests for Two Population Proportions DCOVA Population proportions Lower-tail test: Upper-tail test: Two-tail test: H0: π1 π2 H1: π1 < π2 H0: π1 ≤ π2 H1: π1 > π2 H0: π1 = π2 H1: π1 ≠ π2 i.e., i.e., i.e., H0: π1 – π2 0 H1: π1 – π2 < 0 H0: π1 – π2 ≤ 0 H1: π1 – π2 > 0 H0: π1 – π2 = 0 H1: π1 – π2 ≠ 0 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-31 Hypothesis Tests for Two Population Proportions (continued) Population proportions DCOVA Lower-tail test: Upper-tail test: Two-tail test: H0: π1 – π2 0 H1: π1 – π2 < 0 H0: π1 – π2 ≤ 0 H1: π1 – π2 > 0 H0: π1 – π2 = 0 H1: π1 – π2 ≠ 0 a a -za Reject H0 if ZSTAT < -Za Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall za Reject H0 if ZSTAT > Za a/2 -za/2 a/2 za/2 Reject H0 if ZSTAT < -Za/2 or ZSTAT > Za/2 Chap 10-32 Hypothesis Test Example: Two population Proportions DCOVA Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A? In a random sample, 36 of 72 men and 35 of 50 women indicated they would vote Yes Test at the .05 level of significance Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-33 Hypothesis Test Example: Two population Proportions (continued) The hypothesis test is: DCOVA H0: π1 – π2 = 0 (the two proportions are equal) H1: π1 – π2 ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: p1 = 36/72 = 0.50 Women: p2 = 35/50 = 0.70 The pooled estimate for the overall proportion is: X 1 X 2 36 35 71 p 0 .582 n1 n2 72 50 122 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-34 Hypothesis Test Example: Two population Proportions DCOVA (continued) The test statistic for π1 – π2 is: z STAT p1 p2 π1 π2 1 1 p ( 1 p) n1 n2 .50 .70 0 1 1 .582 ( 1 .582 ) 72 50 Critical Values = ±1.96 For a = .05 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Reject H0 Reject H0 .025 .025 -1.96 -2.20 2 .20 1.96 Decision: Reject H0 Conclusion: There is evidence of a difference in proportions who will vote yes between men and women. Chap 10-35 Two Proportion Test In Excel DCOVA Z Test for Differences in Two Proportions Data Hypothesized Difference Level of Significance Group 1 Number of items of interest Sample Size Group 2 Number of items of interest Sample Size Intermediate Calculations Group 1 Proportion Group 2 Proportion Difference in Two Proportions Average Proportion Z Test Statistic Two-Tail Test Lower Critical Value Upper Critical Value p-value Reject the null hypothesis 0 0.05 36 72 35 50 Since -2.20 < -1.96 Or Since p-value = 0.028 < 0.05 We reject the null hypothesis 0.5 =B7/B8 0.7 =B10/B11 -0.2 =B14 - B15 Decision: 0.582 =(B7 + B10)/(B8 + B11) -2.20 =(B16-B4)/SQRT((B17*(1-B17))*(1/B8+1/B11)) -1.96 =NORMSINV(B5/2) 1.96 =NORMSINV(1 - B5/2) 0.028 =2*(1 - NORMSDIST(ABS(B18))) =IF(B23 < B5,"Reject the null hypothesis", "Do not reject the null hypothesis") Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Reject H0 Conclusion: There is evidence of a difference in proportions who will vote yes between men and women. Chap 10-36 Two Proportion Test In Minitab Shows The Same Conclusions DCOVA Test and CI for Two Proportions Sample X 1 36 2 35 N 72 50 Sample p 0.500000 0.700000 Difference = p (1) - p (2) Estimate for difference: -0.2 95% CI for difference: (-0.371676, -0.0283244) Test for difference = 0 (vs not = 0): Z = -2.28 P-Value = 0.022 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-37 Confidence Interval for Two Population Proportions DCOVA Population proportions The confidence interval for π1 – π2 is: p1 p 2 Za/2 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall p1 (1 p1 ) p 2 (1 p 2 ) n1 n2 Chap 10-38 Testing for the Ratio Of Two Population Variances DCOVA Tests for Two Population Variances F test statistic * Hypotheses H0: σ12 = σ22 H1: σ12 ≠ σ22 H0: σ12 ≤ σ22 H1: σ12 > σ22 FSTAT 2 1 2 2 S S Where: S12 = Variance of sample 1 (the larger sample variance) n1 = sample size of sample 1 S 22 = Variance of sample 2 (the smaller sample variance) n2 = sample size of sample 2 n1 –1 = numerator degrees of freedom n2 – 1 = denominator degrees of freedom Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-39 The F Distribution DCOVA The F critical value is found from the F table There are two degrees of freedom required: numerator and denominator The larger sample variance is always the numerator S S When FSTAT In the F table, 2 1 2 2 df1 = n1 – 1 ; df2 = n2 – 1 numerator degrees of freedom determine the column denominator degrees of freedom determine the row Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-40 Finding the Rejection Region DCOVA H0: σ12 = σ22 H1: σ12 ≠ σ22 H0: σ12 ≤ σ22 H1: σ12 > σ22 a/2 0 Do not reject H0 Fα/2 Reject H0 Reject H0 if FSTAT > Fα/2 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall a F 0 Do not reject H0 Fα Reject H0 F Reject H0 if FSTAT > Fα Chap 10-41 F Test: An Example DCOVA 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 21 25 Mean 3.27 2.53 Std dev 1.30 1.16 Is there a difference in the variances between the NYSE & NASDAQ at the a = 0.05 level? Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-42 F Test: Example Solution DCOVA Form the hypothesis test: (there is no difference between variances) H 0: σ12 σ 22 (there is a difference between variances) H 1: σ12 σ 22 Find the F critical value for a = 0.05: Numerator d.f. = n1 – 1 = 21 –1 = 20 Denominator d.f. = n2 – 1 = 25 –1 = 24 Fα/2 = F.025, 20, 24 = 2.33 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-43 F Test: Example Solution DCOVA (continued) The test statistic is: FSTAT H0: σ12 = σ22 H1: σ12 ≠ σ22 S12 1.30 2 2 1.256 2 S 2 1.16 a/2 = .025 0 Do not reject H0 FSTAT = 1.256 is not in the rejection region, so we do not reject H0 Conclusion: There is insufficient evidence of a difference in variances at a = .05 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Reject H0 F F0.025=2.33 Chap 10-44 Two Variance F Test In Excel DCOVA F Test for Differences in Two Variables Data Level of Significance Larger-Variance Sample Sample Size Sample Variance Smaller-Variance Sample Sample Size Sample Variance 0.05 21 1.6900 =1.3^2 25 1.3456 =1.16^2 Intermediate Calculations F Test Statistic Population 1 Sample Degrees of Freedom Population 2 Sample Degrees of Freedom Conclusion: There is insufficient evidence of a difference in variances at a = .05 because: 1.256 1.255945 20 =B6 - 1 24 =B9 - 1 F statistic =1.256 < 2.327 Fα/2 or p-value = 0.589 > 0.05 = α. Two-Tail Test Upper Critical Value p-value Do not reject the null hypothesis Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall 2.327 =FINV(B4/2,B14,B15) 0.589 =2*FDIST(B13,B14,B15) =IF(B19<B4,"Reject the null hypothesis", "Do not reject the null hypothesis") Chap 10-45 Two Variance F Test In Minitab Yields The Same Conclusion DCOVA Test and CI for Two Variances Null hypothesis Sigma(1) / Sigma(2) = 1 Alternative hypothesis Sigma(1) / Sigma(2) not = 1 Significance level Alpha = 0.05 Statistics Sample 1 2 N 21 25 StDev 1.300 1.160 Variance 1.690 1.346 Ratio of standard deviations = 1.121 Ratio of variances = 1.256 95% Confidence Intervals Distribution of Data Normal CI for StDev Ratio (0.735, 1.739) CI for Variance Ratio (0.540, 3.024) Tests Test Method DF1 DF2 Statistic P-Value F Test (normal) 20 24 1.26 0.589 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-46 Chapter Summary Compared two independent samples Performed pooled-variance t test for the difference in two means Performed separate-variance t test for difference in two means Formed confidence intervals for the difference between two means Compared two related samples (paired samples) Performed paired t test for the mean difference Formed confidence intervals for the mean difference Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-47 Chapter Summary (continued) Compared two population proportions Formed confidence intervals for the difference between two population proportions Performed Z-test for two population proportions Performed F test for the ratio of two population variances Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 10-48