Question

“I can’t get through my day without coffee” is a common statement from many college students. They assume that the benefits of coffee include staying awake during lectures and remaining more alert during exams and tests. Students in a statistics class designed an experiment to measure memory retention with and without drinking a cup of coffee 1 hour before a test. This experiment took place on two different days in the same week (Monday and Wednesday). Ten students were used. Each student received no coffee or one cup of coffee 1 hour before the test on a particular day. The test consisted of a series of words flashed on a screen, after which the student had to write down as many of the words as possible. On the other day, each student received a different amount of coffee (none or one cup).

a. One of the researchers suggested that all the subjects in the experiment drink no coffee before Monday’s test and one cup of coffee before Wednesday’s test. Explain to the researcher why this is a bad idea and suggest a better method of deciding when each subject receives the two treatments.

b. The researchers actually used the better method of deciding when each subject receives the two treatments that you identified in part (a). For each subject, the number of words recalled when drinking no coffee and when drinking one cup of coffee is recorded in the table. Carry out an appropriate test to determine whether there is convincing evidence that drinking coffee improves memory, on average, for students like the ones in this study.

Short answer

a. As a result, we won't be able to tell if the test results are related to the day of the week or to the treatment.

b. The pattern in the normal quantile plot of the differences (seen below) is essentially linear, hence the normal/large sample is satisfied.

Step-by-step solution

Part (a) Step 1: Given information

We have to explain to the researcher why this is a bad idea and suggest a better method of deciding.

Part (a) Step 2: Explanation

We're giving different treatments on different days, but it's feasible that the day the treatment is given will have an impact on the results.

People may be less fatigued on Monday than on Wednesday, for example, because they had the entire weekend to make up on sleep before Monday. People who are less exhausted, on the other hand, are more likely to perform well on tests, and thus the day of the week may have an impact on the test results.

As a result, we won't be able to tell if the test results are related to the day of the week or to the treatment.

Part (b) Step 1: Given information

We have to determine whether there is convincing evidence that drinking coffee improves memory, on average, for students like the ones in this study.

Part (b) Step 2: Explanation

The table will be as follow:

Student	No cup one cup	Difference
1	24 25	-1
2	30 3 1	-1
3	22 23	-1
4	24 24	0
5	26 27	-1
6	23 25	-1
7	20 28	2
8	27 20	-2
9	28 27	0
10	24 30	-2

The three conditions for conducting a hypothesis test for the population mean difference are as follows:

Normal/Large sample, random, independent (10% condition).

Random: I'm satisfied because the treatments are given in a random order.

Independent: Satisfied because the sample of ten pupils represents less than 10% of the total student population (assuming that there are more than 100 students).

The pattern in the normal quantile plot of the differences (seen below) is essentially linear, hence the normal/large sample is satisfied.