Two Population Means Hypothesis Testing and Confidence Intervals For Matched Pairs

Matched Pairs matchedSometimes experiments are conducted in such a way that samples from two populations are matched with something in common so that the i-th sample taken from the first population has something in common with the i-th sample of the second population. –It is the “common element” (same date, same weight, etc.) that is chosen at random and dictates the corresponding observations from each population. –Differences between the sample values (dictated by the “common element”) from each population are computed. –If it can be assumed that the differences have a normal distribution, t-tests can then be performed or t-intervals constructed for the average value of the differences. –Pairing, in general, reduces the variability in the problem.

Hypothesis Tests and Confidence Intervals for Matched Pairs Suppose there a random sample of n elements is taken. For each a corresponding sample from each population is observed. The difference is denoted d i. So there are n observations of differences, d i ’s. Statistics calculated:

Distribution of average difference _ d Distribution of average difference _ d Distribution: t distribution

Hypothesis Tests and Confidence Intervals for Matched Pairs Hypothesis Test: H 0 :  D = v H A :  D > v Test statistic:

Hypothesis Tests and Confidence Intervals for Matched Pairs Confidence Interval: Both the hypothesis test and the confidence interval have n-1 degrees of freedom: DF=n-1

Example Objective: Compare sales at two branch stores, one in Anaheim, the other in Irvine. –Can it be concluded that average daily sales in Anaheim is at least $200 greater than average daily sales in Irvine? –Construct a 95% confidence interval for the average difference in daily sales between the Anaheim and Irvine branches.

Approach 1 Approach 1 Records of sales on seven random dates in Anaheim are selected and seven random dates in Irvine are selected. There is nothing in common between the Anaheim and Irvine samples. Would have to use Difference in Means approach. Probably not the best approach. DateAnaheimDateIrvine 15-Dec Nov Nov85008-May Jun Mar Jul50006-Mar Aug Jun Feb Oct Mar Apr3100

Approach 2 Approach 2 Do not choose the receipts at random, but choose the dates at random and observe the sales at the Anaheim and Irvine branch stores on these dates. These data are paired by the random dates. DateAnaheimIrvine 25-Nov Feb May Aug Apr Jun Dec Difference Calculate Differences _ Calculate statistics: d =400 s D = 258.2

Hypothesis Test Hypothesis Test H 0 :  D = 200 H A :  D > 200 Select α =.05. Reject H 0 (Accept H A ) if t > t.05,6 = > 1.943; thus it can be concluded that average daily sales in Anaheim > $200 more than average daily sales in Irvine.

95% Confidence Interval 400 ±  638.8

ExcelExcel For Matched Pairs Excel Hypothesis Tests –Go to Tools/Data Analysis and select t-Test Paired Two Sample for Means. p-valueLook at p-value for the test. Confidence Intervals –Create a column of differences. Mean ± ConfidenceGo to Tools/Data Analysis and select Descriptive Statistics: Mean ± Confidence

Excel - Hypothesis Test Go Tools Select Data Analysis Select t-Test: Paired Two Sample for Means

Excel: t-Test for Matched Pairs Since H A is  D > 200, enter Column B for Range 1 Column C for Range for Hypothesized Mean Difference Check Labels Designate first cell for output.

Hypothesis Test (Cont’d) p-value for one-tail test Low p-value for 1-tail test (compared to α =.05)! Can conclude average daily sales in Anaheim exceed those in Irvine by > $200 p-value for at two-tail “  ” test

95% Confidence Interval for Matched Pairs =B2-C2 Drag to D3:D8 =I3+I16 =I3-I16 Go to Tools/Data Analysis Descriptive Statistics On Column D. Store output beginning in cell H1.

Review What constitutes “matched pairs” Matched pairs normally reduces variability from difference in means tests Create a set of differences Hypothesis Tests/Confidence Intervals for average difference –By hand –By Excel