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9-2 Chapter Nine Comparing Population Means McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

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9-3 Statistical Inferences Based on Two Samples 9.1Comparing Two Population Means Using Large Independent Samples 9.2Comparing Two Population Means Using Small Independent Samples 9.3Paired Difference Experiments 9.4Basic Concepts of Experimental Design 9.5One-Way Analysis of Variance

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Sampling Distribution of Normal, if each of the sampled populations is normal and approximately normal if the sample sizes n 1 and n 2 are large If independent random samples are taken from two populations then the sampling distribution of the sample difference in means is Has mean: Has standard deviation:

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9-5 Sampling Distribution of (Continued)

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9-6 Large Sample Confidence Interval, Difference in Mean If two independent samples are from populations that are normal or each of the sample sizes is large, 100(1 - )% confidence interval for 1 - 2 is If 1 and 2 are unknown and the each of the sample sizes is large (n 1, n 2 30), estimate the sample standard deviations by s 1 and s 2 and a 100(1 - )% confidence interval for 1 - 2 is

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9-7 Large Sample Tests about Differences in Means Test Statistic If sampled populations are normal or both samples are large, we can reject H 0 : 1 - 2 = D 0 at the level of significance if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . If population variance unknown and the sample sizes are large, substitute sample variances. Alternative Reject H 0 if:p-Value

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9-8 Example: Large Sample Interval and Test Bank Waiting Times, Current System versus New System 95% Confidence Interval for 1 - 2 Test H 0 : 1 - 2 3 versus H a : 1 - 2 > 3, = 0.05

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Comparing Two Population Means by Using Small, Independent Samples If two independent samples are from populations that are normal with equal variances, 100(1 - )% confidence interval for 1 - 2 is Where s p 2 is the pooled variance And t /2 is based on (n 1 – n 2 – 2) degrees of freedom.

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9-10 Small Sample Tests about Differences in Means When Variances are Equal If sampled populations are both normal with equal variances, we can reject H 0 : 1 - 2 = D 0 at the level of significance if and only if the appropriate rejection point condition holds or, equivalently, if the p- value is less than . Alternative Reject H 0 if:p-Value t , t /2 and p-values are based on (n 1 – n 2 – 2) df Test StatisticPooled Variance

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9-11 Example: Small Sample Difference in Mean Test Catalyst Case, Difference in Mean Hourly Yields? Test H 0 : 1 - 2 = 0 versus H a : 1 - 2 0, = 0.01

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9-12 Small Sample Intervals and Tests about Differences in Means When Variances are Not Equal Test Statistic If sampled populations are both normal, but sample sizes and variances differ substantially, small-sample estimation and testing can be based on the following “unequal variance” procedure. Confidence Interval For both the interval and test, the degrees of freedom are equal to

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Paired Difference Experiments If the sampled population of differences is normally distributed with mean d, then a )100% confidence interval for d is is based on n – 1 degrees of freedom.

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9-14 Paired Difference Test for Difference in Mean Test Statistic If the population of differences is normal, we can reject H 0 : d = D 0 at the level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . t , t /2 and p-values are based on n – 1 degrees of freedom. Alternative Reject H 0 if:p-Value

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9-15 Example: Paired Difference Interval and Test Table % Confidence Interval Excel Test Output

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9-16 Large Sample Interval for the Difference in Proportions If two independent samples are both large, a 100(1 - )% confidence interval for p 1 - p 2 is

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9-17 Large Sample Test for Difference in Proportions Test Statistics If two sampled populations are both large, we can reject H 0 : p 1 - p 2 = D 0 at the level of significance if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . Alternative Reject H 0 if:p-Value

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9-18 Example: Difference Between Proportons: Interval and Test Test H 0 : p 1 - p 2 = 0 versus H a : p 1 - p 2 0 Example 9.11 Advertising Media 95% Confidence Interval for p 1 - p 2

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9-19 Objective: To compare and estimate the effect of different treatments on the response variable. 9.4 Basic Concepts of Experimental Design Example 9.8 The Gasoline Mileage Case Does gasoline mileage vary with gasoline type? Type AType BType C x A1 =34.0x B1 =35.3x C1 =33.3 x A2 =35.0x B2 =36.5x C2 =34.0 x A3 =34.3x B3 =36.4x C3 =34.7 x A4 =35.5x B4 =37.0x C4 =33.0 x A5 =35.8x B5 =37.6x C5 =34.9 Response Variable:Gasoline mileage (in mpg) Treatments:Gasoline types – A, B, C

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One-Way Analysis of Variance Are there differences in the mean response , …, p associated with the p treatments? H 0 : = … = p H a : At least two of the , ,…, p differ Or, is the between- treatment variability large compared to the within-treatment variability?

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9-21 Partitioning the Total Variability in the Response p i= n j= iij p i= ii p n j= ij ii )x(x)xx(n)x Squares of Squares of Squares of SumError SumTreatment Sum Total yVariabilit y Treatment yVariabilit Within Between Total SSE SST = SSTO

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9-22 F Test for Difference Between Treatment Means H 0 : = …= p (no treatment effect) H a : At least two of the , …, p differ Test Statistic: Reject H 0 if F > F or p-value < F is based on p-1 numerator and n-p denominator degrees of freedom.

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9-23 The One-Way Analysis of Variance Table DegreesSum of MeanF Sourceof FreedomSquaresSquaresStatistic Treatmentsp-1SSTMST = SSTF = MST p-1 MSE Errorn-pSSEMSE = SSE n-p Totaln-1SSTO Example 9.12 The Gasoline Mileage Case (Excel Output)

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9-24 Pairwise Comparisons, Individual Intervals Individual 100(1 - )% confidence interval for i - h t is based on n-p degrees of freedom. Example 9.13 The Gasoline Mileage Case (A vs B, = 0.05)

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9-25 Pairwise Comparisons, Simultaneous Intervals Tukey simultaneous 100(1 - )% confidence interval for i - h q is the upper percentage point of the studentized range for p and (n-p) from Table A.9. m denotes common sample size. Example 9.13 The Gasoline Mileage Case (A vs B, = 0.05)

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9-26 Estimation of Individual Treatment Means Individual 100(1 - )% confidence interval for i t is based on n-p degrees of freedom. Example 9.13 The Gasoline Mileage Case (Type B, = 0.05)

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9-27 Statistical Inferences Based on Two Samples Summary: 9.1Comparing Two Population Means Using Large Independent Samples 9.2Comparing Two Population Means Using Small Independent Samples 9.3Paired Difference Experiments 9.4Basic Concepts of Experimental Design 9.5One-Way Analysis of Variance

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