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Test of (µ 1 – µ 2 ), 1 = 2, Populations Normal Test Statistic and df = n 1 + n 2 – 2 2– 21 2 2 )1– 2 ( 2 1 )1– 1 ( 2 where 2 1 1 1 2 0 ] 2 – 1 [– ] 2 – 1 [ nn snsn p s nn p s xx t © 2008 Thomson South-Western

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Test Statistic Test of (µ 1 – µ 2 ), Unequal Variances, Independent Samples © 2008 Thomson South-Western

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Test of Independent Samples (µ 1 – µ 2 ), 1 2, n 1 and n 2 30 Test Statistic –with s 1 2 and s 2 2 as estimates for 1 2 and 2 2 z [x 1 –x 2 ]–[ 1 – 2 ] 0 s 1 2 n 1 s 2 2 n 2 © 2008 Thomson South-Western

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Test of Dependent Samples (µ 1 – µ 2 ) = µ d Test Statistic –whered = ( x 1 – x 2 ) = d/n, the average difference n = the number of pairs of observations s d = the standard deviation of d df = n – 1 n d s d t © 2008 Thomson South-Western

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Test of ( 1 – 2 ), where n 1 p 1 5, n 1 (1–p 1 ) 5, n 2 p 2 5, and n 2 (1–p 2 ) Test Statistic –where p 1 = observed proportion, sample 1 p 2 = observed proportion, sample 2 n 1 = sample size, sample 1 n 2 = sample size, sample 2 p n 1 p 1 n 2 p 2 n 1 n 2 © 2008 Thomson South-Western

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Test of 1 2 = 2 2 If 1 2 = 2 2, then 1 2 / 2 2 = 1. So the hypotheses can be worded either way. Test Statistic: whichever is larger The critical value of the F will be F( /2, 1, 2 ) –where = the specified level of significance 1 = ( n – 1), where n is the size of the sample with the larger variance 2 = ( n – 1), where n is the size of the sample with the smaller variance 2 1 2 2 or 2 2 2 1 s s s s F © 2008 Thomson South-Western

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Confidence Interval for (µ 1 – µ 2 ) The (1 – )% confidence interval for the difference in two means: –Equal-variances t -interval –Unequal-variances t -interval 2 1 1 1 2 2 ) 2 – 1 ( nn p stxx 2 2 2 1 2 1 2 ) 2 – 1 ( n s n s txx © 2008 Thomson South-Western

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Confidence Interval for (µ 1 – µ 2 ) The (1 – )% confidence interval for the difference in two means: –Known-variances z -interval © 2008 Thomson South-Western

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Confidence Interval for ( 1 – 2 ) The (1 – )% confidence interval for the difference in two proportions: –when sample sizes are sufficiently large. (p 1 –p 2 ) z 2 p 1 (1–p 1 ) n 1 p 2 (1–p 2 ) n 2 © 2008 Thomson South-Western

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One-Way ANOVA, cont. Format for data: Data appear in separate columns or rows, organized by treatment groups. Sample size of each group may differ. Calculations: –SST = SSTR + SSE (definitions follow) –Sum of squares total (SST) = sum of squared differences between each individual data value (regardless of group membership) minus the grand mean,, across all data... total variation in the data (not variance). 2 )–( SST x ij x x © 2008 Thomson South-Western

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One-Way ANOVA, cont. Calculations, cont.: –Sum of squares treatment (SSTR) = sum of squared differences between each group mean and the grand mean, balanced by sample size... between- groups variation (not variance). –Sum of squares error (SSE) = sum of squared differences between the individual data values and the mean for the group to which each belongs... within- group variation (not variance). 2 )–( SSTRx j x j n SSE (x ij –x j ) 2 © 2008 Thomson South-Western

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One-Way ANOVA, cont. Calculations, cont.: –Mean square treatment (MSTR) = SSTR/( t – 1) where t is the number of treatment groups... between- groups variance. –Mean square error (MSE) = SSE/( N – t ) where N is the number of elements sampled and t is the number of treatment groups... within-groups variance. – F -Ratio = MSTR/MSE, where numerator degrees of freedom are t – 1 and denominator degrees of freedom are N – t. © 2008 Thomson South-Western

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Goodness-of-Fit Tests Test Statistic: where O j = Actual number observed in each class E j = Expected number, j n j E j E j O 2 )–( 2 © 2008 Thomson South-Western

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Chi-Square Tests of Independence Hypotheses: – H 0 : The two variables are independent. – H 1 : The two variables are not independent. Rejection Region: –Degrees of freedom = ( r – 1) ( k – 1) Test Statistic: ij E E O 2 )–( 2 © 2008 Thomson South-Western

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Chi-Square Tests of Multiple ’s Rejection Region: Degrees of freedom: df = ( k – 1) Test Statistic: 2 ( O ij – E ) 2 E © 2008 Thomson South-Western

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Determining the Least Squares Regression Line Least Squares Regression Line: – Slope – y -intercept ˆ y b 0 b 1 x 1 b 1 (x i y i ) – n x y (x i 2 ) – n x 2 © 2008 Thomson South-Western

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To Form Interval Estimates The Standard Error of the Estimate, s y,x –The standard deviation of the distribution of the »data points above and below the regression line, »distances between actual and predicted values of y, »residuals, of –The square root of MSE given by ANOVA 2– 2 ) ˆ –(, n y i y xy s © 2008 Thomson South-Western

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Equations for the Interval Estimates Confidence Interval for the Mean of y Prediction Interval for the Individual y n i x i x xvaluex n xy sty 2 )( – ) 2 ( 2 )– ( 1 ), ( 2 ˆ ˆ y t 2 (s y,x ) 1 1 n (x – x) 2 (x i 2 ) – (x i ) 2 n © 2008 Thomson South-Western

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Coefficient of Correlation, r and Coefficient of Determination, r 2

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Three Tests for Linearity 1. Testing the Coefficient of Correlation H 0 : = 0 There is no linear relationship between x and y. H 1 : 0 There is a linear relationship between x and y. Test Statistic: 2. Testing the Slope of the Regression Line H 0 : = 0 There is no linear relationship between x and y. H 1 : 0 There is a linear relationship between x and y. Test Statistic: t r 1 – r 2 n – 2 © 2008 Thomson South-Western

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Three Tests for Linearity 3. The Global F -test H 0 : There is no linear relationship between x and y. H 1 : There is a linear relationship between x and y. Test Statistic: Note: At the level of simple linear regression, the global F -test is equivalent to the t -test on 1. When we conduct regression analysis of multiple variables, the global F - test will take on a unique function. F MSR MSE SSR 1 SSE (n – 2) © 2008 Thomson South-Western

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A General Test of 1 Testing the Slope of the Population Regression Line Is Equal to a Specific Value. H 0 : = The slope of the population regression line is . H 1 : The slope of the population regression line is not . Test Statistic: 2 )(– 2, 10 – 1 xnx xy s b t © 2008 Thomson South-Western

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