16.57 A small-sample permutation test. For this exercise, perform a permutation test by hand for a small random subset of the DRP data (Example 16.15, page 16-39). Here are the data:

Treatment group 57 67 Control group 53 42 42 37

a. Calculate the difference in means x¯treatment−x¯control between the two groups. This is the observed value of the statistic.

b. Resample: Start with the six scores and choose an SRS of two scores to form the treatment group for the first resample. You can do this by labeling the scores from 1 to 6 and using consecutive random digits from Table B or by rolling a die. Using either method, be sure to skip repeated digits.

A resample is an ordinary SRS, without replacement. The remaining four scores are the control group. What is the difference in group means for this resample?

c. Repeat part

(b) 20 times to get 20 resamples and 20 values of the statistic. Make a histogram of the distribution of these 20 values. This is the permutation distribution for your resamples.

d. What proportion of the 20 statistic values were equal to or greater than the original value in part

(a)? You have just estimated the one-sided P-value for the original 6 observations.

e. For this small data set, there are only 15 possible permutations of the data. As a result, we can calculate the exact P-value by counting the number of permutations with a statistic value greater than or equal to the original value and then dividing by 15. What is the exact P-value here? How close was your estimate?