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 test the wear characteristics of two tire brands, A and B, each of $50$cars of the same make and model is randomly assigned Brand A tires or Brand B tires.

b. To test the effect of background music on productivity, factory workers are observed. For one month, each subject works without music. For another month, the subject works while listening to music on an MP3 player. The month in which each subject listens to music is determined by a coin toss.

c. How do young adults look back on adolescent romance? Investigators interviewed a random sample of $40$couples in their mid-twenties. The female and male partners were interviewed separately. Each was asked about his or her current relationship and also about a romantic relationship that lasted at least $2$months when they were aged $15$or $16$. One response variable was a measure on a numerical scale of how much the attractiveness of the adolescent partner mattered. You want to find out how much men and women differ on this measure.

Short answer

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

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

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

Step-by-step solution

Part(a) Step 1 : Given information

We need to decide whether to use two-sample t procedures or paired t procedures to check inference about a mean difference.

Part(a) Step 2 : Simplify

We must use paired t techniques if the two samples contain the same individuals or if the subjects in one sample are related to the subjects in the other sample.
We must use two sample t techniques if the subjects in the two samples are completely unrelated.
Automobiles are randomly assigned to either brand A or brand B in this scenario, resulting in a first sample of brand A cars and a second sample of brand B cars.
Because the cars were randomly assigned to one of the samples, the cars in the two samples will be completely unrelated, making the two sample t methods suitable.

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.
For one month, each topic worked with music and for one month, each subject worked without music.
The data for all participants who worked with music for one month is the first sample, while the data for all subjects who worked without music for one month is the second sample.
We should utilise paired t methods because the two samples are the same subjects.

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.
We questioned the male and female partners separately in each of the $40$couples.
The male partners are in the first sample, while the female partners are in the second.
Because all of the participants in the first sample are male spouses of a subject in the second sample, we should utilise the paired t methods to compare the two samples.