CH-12 PPT - Comparing Two Groups Comparing two means...

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Comparing Two Groups
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Comparing two means Two-sample z statistic Two-samples t procedures Two-sample t significance test Two-sample t confidence interval Robustness Details of the two-sample t procedures (various cases)
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Two-sample z statistic We have two independent SRSs (simple random samples) possibly coming from two distinct populations with ( μ1,σ1 ) and ( μ2,σ2 ). We use 1 and 2 to estimate the unknown μ1 and μ2 . When both populations are normal, the 2 2 2 1 2 1 n n σ + 2 2 2 1 2 1 2 1 2 1 ) ( ) ( n n x x z μ + - - - = x x x x
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Two independent samples t distribution We have two independent SRSs (simple random samples) possibly coming from two distinct populations with ( μ1,σ1 ) and ( μ2,σ2 ) unknown. We use ( 1, s 1) and ( 2, s 2) to estimate ( ) and ( ), respectively. x x
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SE = s 1 2 n 1 + s 2 2 n 2 s 1 2 n 1 + s 2 2 n 2 df μ 1- 2 x 1 - x 2 The two-sample t statistic follows approximately the t distribution with a standard error SE (spread) reflecting variation from both samples: Conservatively, the degrees of freedom is equal to the smallest of ( n 1 − 1, n 2 − 1).
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t = ( x 1 - x 2 ) - ( μ 1 2 ) SE Two-sample t significance test The null hypothesis is that both population means μ1 and μ2 are equal, thus their difference is equal to zero. H 0: = μ2 < = μ2 = 0 with either a one-sided or a two-sided alternative hypothesis. We find how many standard errors (SE) away from ( ) is ( 1− 2) by standardizing with t : t = x 1 - x 2 s 1 2 n 1 + s 2 2 n 2 x x
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A company that makes and sells frozen pizza would like to know if their product sells at different levels across geographic locations. To answer this question, we will compare sales volume data (in pounds) in Dallas and Denver, to see if sales differ. Weekly sales volume during the same time period in these two cities over 156 weeks yields average sales volume of 51,738 lbs in Dallas and 45,743 pounds in Denver. 1
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CH-12 PPT - Comparing Two Groups Comparing two means...

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