Question

Two samples or paired data? In each of the following settings, decide whether you should use two-sample t procedures to perform inference about a difference in means or paired t procedures to perform inference about a mean difference. Explain your choice.

a. To compare the average weight gain of pigs fed two different diets, nine pairs of pigs were used. The pigs in each pair were littermates. A coin toss was used to decide which pig in each pair got Diet A and which got Diet B.

b. Separate random samples of male and female college professors are taken. We wish to compare the average salaries of male and female teachers.

c. To test the effects of a new fertilizer, $100$ plots are treated with the new fertilizer, and $100$ plots are treated with another fertilizer. A computer’s random number generator is used to determine which plots get which fertilizer.

Short answer

Part(a) We should use paired t procedures to perform inference about a mean difference.

Part(b) We should use two-sample t procedures to perform inference about a difference in means.

Part(c) We should use two-sample t procedures to perform inference about a difference in means.

Step-by-step solution

Part(a) Step 1 : Given information

We need to decide whether we should use two-sample t procedures to perform inference about a difference in means or paired t procedures to perform inference about a mean difference.

Part(a) Step 2 : Simplify

If the two samples contain the same individuals or if the subjects in one sample are connected to the subjects in the other sample, we must employ paired t methods.
If the subjects in the two samples are fully unrelated, we must employ two sample t methods.
Each pair of pigs was made up entirely of littermates.
Diet A was given to one pig in each pair, whereas diet B was given to the other.
The first sample comprises all of the pigs who were fed diet A, while the second sample contains all of the pigs who were fed diet B.
The pigs in the two samples are connected because all of the piglets in the first sample are littermates of a pig in the second sample.

Part(b) Step 1 : Given information

We need to decide whether we should use two-sample t procedures to perform inference about a difference in means or paired t procedures to perform inference about a mean difference.

Part(b) Step 2 : Simplify

If the two samples contain the same individuals or if the subjects in one sample are connected to the subjects in the other sample, we must employ paired t methods.
If the subjects in the two samples are fully unrelated, we must employ two sample t methods.
Male and female college professors were randomly sampled separately, resulting in the first sample being a random sample of male college professors and the second sample being a random sample of female college professors.
Because the professors in the two samples are absolutely independent because they are separate random samples, we should utilise the two-sample t method.

Part(c) Step 1 : Given information

We need to decide whether we should use two-sample t procedures to perform inference about a difference in means or paired t procedures to perform inference about a mean difference.

Part(c) Step 2 : Simplify

If the two samples contain the same individuals or if the subjects in one sample are connected to the subjects in the other sample, we must employ paired t methods.
If the subjects in the two samples are fully unrelated, we must employ two sample t methods.
$100$plots are treated with new fertilizers, and $100$plots are treated with other fertilizers, resulting in the first sample consisting of $100$plots treated with new fertilizers and the second sample consisting of $100$ plots treated with other fertilizers.
Because the plots were allocated fertilizers at random, the plots in the two samples will be completely unrelated, hence the two-sample t test is suitable.