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STA291

In this lecture, we explore a common statistical challenge: making inferences about the difference between two comparable parameters using matched-pair experimental designs. We will discuss practical examples, such as comparing average ROI from two investment firms, tracking candidate favorability changes, and assessing pain relief efficacy of analgesics. Key topics include assumptions for normality, sampling, and the importance of using matched pairs to control for variability. We will also walk through a specific hypothesis testing example involving discounts offered by sales representatives at a car dealership, highlighting the calculation of mean differences and confidence intervals.

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STA291

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  1. STA291 Statistical Methods Lecture 22

  2. Really common problem Want to make an inference (estimation or hypothesis test) about the difference between two (comparable) parameters: • Compare average ROI for two investment firms • See whether a candidate’s favorability increased or decreased after a particular occurrence • Find which analgesic provides the best pain relief

  3. Matched-Pair Experimental Design Common analysis situation: variability among subjects (much) greater than difference between treatments To control for variability among subjects, we use each subject as her/his/its own control – boils down to observation of difference as variable of interest

  4. Assumptions (Simple) random sample Especially with small n, normality of differences Paired data – natural connection between individual observations in the two data sets

  5. Notation***** ith matched pair ith difference (what will be ourithobsn.) d-bar, average difference standard deviation of the differences Some texts introduce new notation: (y1i , y2i) di = y1i – y2i

  6. More notation********** Null hypothesis Test statistic; with df = n-1 Confidence interval for the mean difference Even more new notation: H0: md = D0

  7. Example: Car Dealership • Suppose your are the owner of a car dealership and you want to test the average difference two of your sales people, Tyrone and Shannon, are willing to give in discounts per car to customers. You randomly select 30 cars and ask each sales representative how much he would give in discounts to each of the 30 cars. You find the sample difference to be $64.40 with a standard deviation of $146.74. Test the hypothesis that the mean discount each sales rep. would give is the same and give a 95% percent confidence interval for the difference.

  8. Looking back • Two-sample problems in general • Difference of means: matched-pair data • Assumptions • Hypothesis testing • Confidence intervals

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