Method

Resampling

If the sample of a controlled experiment is too small, the control group and the treatment group can be randomly combined to create new control and treatment groups. This is done to compare the significance of the experiment result versus taking a random sample. This process is called resampling. As an example, consider that a car company is testing a new production method.

Each observation is the number of cars produced at a random hour. The control group is the number of the cars produced using the old production method. The treatment group is the number of the cars produced using the new method. The observations are listed on the following table. The mean of each group is also provided.

	Cars Produced										Mean
Control Group	$35$	$38$	$26$	$41$	$58$	$35$	$20$	$55$	$41$	$50$	$39.9$
Treatment Group	$41$	$26$	$44$	$53$	$47$	$53$	$70$	$55$	$55$	$50$	$49.4$

To compare the results of each group, the mean of the control group

μ_{c}

is subtracted from the mean of the treatment group

μ_{t} .

μ_{t} - μ_{c} = 49.4 - 39.9 ⇓ μ_{t} - μ_{c} = 9.5

Looking at the difference between the means, it can be noted that the new production method seems to be more productive than the older method. However, this result can be caused by random chance since there are so few observations. To test this, a resampling will be done.

Combining the Measurements

The first thing to do is to combine the measurements by assigning a number to each observation. First, the data in the control group are numbered from $1$ to $10 .$

Control Group	$35$	$38$	$26$	$41$	$58$	$35$	$20$	$55$	$41$	$50$
Assigned Number	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$

Then, the numbers from $11$ to $20$ are assigned to the treatment group.

Treatment Group	$41$	$26$	$44$	$53$	$47$	$53$	$70$	$55$	$55$	$50$
Assigned Number	$11$	$12$	$13$	$14$	$15$	$16$	$17$	$18$	$19$	$20$

Now there is a combined list of observations numbered from $1$ to $20 .$

Creating New Control and Treatment Groups

To create new control and treatment groups, a random number generator is used to generate $20$ numbers from $1$ to $20$ without repeating a number. In a graphing calculator, this is done by pressing $MATH,$ going to section PRB, and selecting the function randIntNoRep( with input $(1, 20) .$

The table below shows the numbers obtained.

$20$	$5$	$19$	$6$	$12$	$3$	$7$	$14$	$17$	$1$
$9$	$10$	$8$	$13$	$2$	$11$	$15$	$4$	$16$	$18$

The associated results of the first ten observations make the new control group, while the new treatment group is made of the others. For example, $50$ is recorded in the new control group since the $20 th$ observation is associated with $50$ cars produced. The new groups and their respective means are shown in the following table.

	Cars Produced										Mean
New Control Group	$50$	$58$	$55$	$35$	$26$	$26$	$20$	$53$	$70$	$35$	$42.8$
New Treatment Group	$41$	$50$	$55$	$44$	$38$	$41$	$47$	$41$	$53$	$55$	$46.5$

Compare the New Means

Now the means of the new groups are compared in the same way that the means of the original groups were compared. The mean of the new control group

μ_{n c}

is subtracted from the mean of the new treatment group

μ_{n t} .

μ_{n t} - μ_{n c} = 46.5 - 42.8 ⇓ μ_{n t} - μ_{n c} = 3.7

The new difference is less than half the difference obtained from subtracting the means of the original groups. This indicates that the results obtained from the original groups are significantly different than making the groups at random, so the new method can be determined to be better than the old one.

It is important to remember that the same data can be resampled more than once to verify the results.

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