Chapter 11.6, Problem 15E

To determine

To explain: the number of resampling of the treatment and control groups which are theoretically possible

Answer to Problem 15E

Total 12870 resampling's of the treatment and control groups are possible.

Explanation of Solution

Given:

Circumference (inches)
Control group	34.6	35.4	33	34.6
Treatment group	31.4	33	32.4	32.6

Circumference (inches)
Control group	35.2	35.2	36.2	35
Treatment group	33.4	33.4	34.8	33

Calculation:

Resampling is a method in which new samples are repeatedly drawn from the data set. One way to resample: combine the measurements from both the control and treatment groups. and repeatedly create new “control” and “treatment” groups at random from the measurements without repeats.

Now, , the total number of measurements is 16, 8 measurements of the control group and 8 of the treatment group.

Now resampling is done by taking different combinations of this data set to create new control and new treatment groups.

Now, the total number of different possible resampling is the number of combinations of 16 items in groups of 8. That is

  $\begin{array}{l}{16}^{C_8}=\frac{16!}{\left(16-8\right)!\left(8\right)!}\\ =\frac{16!}{\left(8\right)!\left(8\right)!}\\ =\frac{16\cdot 15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9}{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\\ =\boxed{12870}\end{array}$

Hence, total 12870 resampling's of the treatment and control groups are possible.

Conclusion:

Total 12870 resampling's of the treatment and control groups are possible.