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Unformatted text preview: 1 Statistics 102 Lecture 23 L. Brown & N. Zhang Tests and Confidence Intervals for Two Means Read: Sections 2.7 and 2.8 of Dielman & Do advertisements help to increase store sales? & Data from two independent samples Analysis assuming equal variances Analysis allowing variances to be different & From paired samples 2 Example: The Effect of an Ad Campaign on Store Sales A national chain of clothing stores wishes to investigate the effect of an intensive in store ad campaign on store sales. They begin with a RANDOM sample of 28 stores. In 13 of these stores they run the ad campaign. In the remaining 15 they do not. Here are sidebyside boxplots for the weekly sales (in $1,000) in these stores. Sales 30 40 50 60 70 80 90 100 with Campaign without Campaign ID 3 Summary Statistics from JMP Sample Measure With Campaign Without Campaign Mean Y 1 = 62.85 Y 2 = 60.35 Std Dev s 1 = 20.03 s 2 = 18.39 n n 1 = 13 n 2 = 15 Std Error Mean 5.55 4.75 Upper 95% Mean 75.0 70.5 Lower 95% Mean 50.7 50.2 Formulas: Y 1 = 1 n 1 Y 1 i i = 1 n 1 ! and s 1 2 = 1 n 1 ! 1 Y 1 i ! Y 1 ( ) 2 i = 1 n 1 " , etc. Need to use sample means Y 1 and Y 2 to test H that two population means are equal ie , H : 1 = 2 Notice the two population standard deviations ! 1 and ! 2 are unknown too. 4 Basic Statistical Setting: & Two random samples  Populations assumed to be normal : With population means 1 and 2 With population standard deviations ! 1 and ! 2 Independent samples with sample sizes n 1 and n 2 Statistics computed from the samples: Sample means Y 1 and Y 2 Sample standard deviations s 1 and s 2 & Goal = comparisons of the two population means  primarily a. Tests of H : 1 = 2 vs H a : 1 ! 2 [or of H : 1 ! 2 or of H : 1 ! 2 ], or b. Confidence intervals for the difference 1 ! 2 Note: H can also be expressed as H : 2 ! 1 = . 4 Basic Statistical Setting: & Two random samples  Populations assumed to be normal : With population means 1 and 2 With population standard deviations ! 1 and ! 2 Independent samples with sample sizes n 1 and n 2 Statistics computed from the samples: Sample means Y 1 and Y 2 Sample standard deviations s 1 and s 2 & Goal = comparisons of the two population means  primarily a. Tests of H : 1 = 2 vs H a : 1 ! 2 [or of H : 1 ! 2 or of H : 1 ! 2 ], or b. Confidence intervals for the difference 1 ! 2 Note: H can also be expressed as H : 2 ! 1 = . 5 Fact : Y 1 ! Y 2 is a good estimator of 1 ! 2 . We also need the standard deviation of Y 1 ! Y 2 . This is SD Y 1 ! Y 2 ( ) = " 1 2 n 1 + " 2 2 n 2 ....
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This note was uploaded on 04/04/2012 for the course STAT 102 taught by Professor Shaman during the Spring '08 term at UPenn.
 Spring '08
 SHAMAN
 Statistics, Variance

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