lect2_3 Two Means_2012

lect2_3 Two Means_2012 - 1 Statistics 102 Lecture 2-3 L....

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Unformatted text preview: 1 Statistics 102 Lecture 2-3 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 side-by-side 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.

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lect2_3 Two Means_2012 - 1 Statistics 102 Lecture 2-3 L....

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