 # 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 [–

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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