# 1 n 2 this does not necessarily mean that the data is

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1 = n 2 , this does not necessarily mean that the data is paired. For example, if the first appraiser examined ten works of art, and the second appraiser examined ten different works of art, this would be two independent samples rather than paired data, and we would then use the techniques from Sets 25 and 26. 35
Notation: For the pair of observations ( x i , y i ) , we define D i to be the difference between the observations: D i = x i - y i We will be working under the assumption that these differences are nor- mally distributed. Parameter of interest: μ D , the population mean difference between paired observations. Point estimate: D , the sample mean of the observed differences. 36
Suppose we have n pairs of observations, and have computed the n dif- ferences. If the sample size is small, then the test statistic for hypothesis testing we will use is: t n - 1 = D - μ D s D / n Note: s D is the sample standard deviation of the n differences. (If we had a large sample size, then the test statistic would have standard normal distribution.) 37
Example: A new cancer drug is being tested on 6 patients. The size of their tumor (in mm ) is measured before and after receiving the medica- tion. Is there evidence that the cancer drug shrinks the tumors? Patient 1 Patient 2 Patient 3 Patient 4 Patient 5 Patient 6 Before 16.5 17.9 27.8 26.1 22.5 18.9 After 15.2 18.5 24.1 23.4 22.2 17.0 Let μ D be the mean difference in the tumor size, where the differences are the “before” measurement minus the “after” measurement. Example: Construct a 90% confidence interval for the mean of the dif- ferences, μ D . 38
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