Ch21ComparingMeansBetween2IndependentGroups - STAT 200...

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STAT 200 - Elementary Statistics for Applications Comparing Means between Two Independent Groups Camila Casquilho The University of British Columbia, Department of Statistics Adapted notes from: Eugenia Yu, The University of British Columbia, Department of Statistics 1 / 18
Comparing Means between Two Independent Groups I Objective: to compare the means of two independent populations I We draw a random sample from each of the two independent populations: y 11 , y 12 , · · · , y 1 n 1 (sample size n 1 ) from a population with mean μ 1 and standard deviation σ 1 y 21 , y 22 , · · · , y 2 n 2 (sample size n 2 ) from a population with mean μ 2 and standard deviation σ 2 I Two samples are said to be independent if the individuals selected for one sample do not dictate which individuals are to be in a second sample. I Two samples are said to be dependent or paired when the individuals selected to be in one sample determine the individuals to be included in the second sample. 2 / 18
Distinguishing between independent and dependent samples I Let’s consider two scenarios: 1 You want to compare the mean IQ between males and females. To test for a difference, you randomly select 20 females and 20 males. The two sets of IQ scores (one per gender group) are independent of each other. 2 You want to compare the mean IQ between the older and younger siblings of twin pairs. To test for a difference, you randomly select 20 twin pairs. One sample of IQ scores come from the older siblings of the 20 twin pairs, and the other sample come from the younger siblings of the same 20 twin pairs. There is a paired structure between the two samples. 3 / 18
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Sampling distribution for difference in means of two independent populations I To estimate μ 1 - μ 2 , we use y 1 - y 2 where y 1 and y 2 are sample means from the two samples. I Given that the 2 random samples are independent, the sampling distribution of y 1 - y 2 will have mean : μ 1 - μ 2 (unbiased) standard deviation : SD ( y 1 - y 2 ) = q σ 2 1 n 1 + σ 2 2 n 2 (See Appendix for the mathematical derivation of the expression for the standard deviation.) I As long as the two samples are independent (their sizes n 1 and n 2 also need to be sufficiently large if the distributions of y 1 and y 2 are unknown), the sampling distribution of y 1 - y 2 will follow the Normal model. 5 / 18

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