The following table shows the number of bacteria colonies present in each of several petri dishes, after E. coli bacteria were added to the dishes and they were incubated for 24 hours. The "soap" dishes contained a solution prepared from ordinary soap; the "control" dishes contained a solution of sterile water. (These data are a subset of the larger data set seen in Exercise 6.6.9.) The sample means are 44 for the control group and 39.7 for the soap group; thus the difference in sample means is 4.3, with the control mean being larger, as would be expected if the soap were effective. There are 20 possible randomizations of the six observations into two groups, each of size 3.
(a) Create a list of these 20 randomizations (one of which is the original assignment of observations to the two groups) and for each case calculate the control mean minus the soap mean.
(b) How many of the 20 randomizations produce a difference in means at least as large as 4.3?
(c) Is there evidence that the soap inhibits E. coli growth? Justify your answer using the randomization results.