To review these concepts, go to Confidence Intervals for Comparing Two Treatment or Population Means Study Guide.

### Practice

A teacher became curious as to whether or not caffeine had an affect on a person's memory. He began to consider this when a former student of his commented that drinking caffeinated beverages while studying for a test helped him remember the facts for the test. He decided to do a study and randomly selected 100 of his students to participate. He then randomly assigned 50 students to ingest one glass of fruit juice with 45 mg of caffeine and 50 students to drink one glass of fruit juice with no caffeine added. The students did not know whether they drank the juice with caffeine or the juice without it. Thirty minutes after the students drank the juice, the professor had them perform a memory test. Out of a possible 15 points, the students performed as shown in Table 19.2.

- This study has a two-group design. Explain why this statement is true.
- Estimate the mean and standard deviation for each treatment.
- Is it reasonable to assume the variance is the same for both populations? If so, estimate the variance common to both.
- Estimate the difference in the treatment means and find its standard error.
- Is the assumption that the memory scores are normally distributed reasonable for each treatment?
- To which population may inference be drawn from this study?
- Consider the caffeine study in the previous set of practice problems. Set a 95% confidence interval on the mean difference in memory test scores after ingesting caffeine compared to not ingesting caffeine. Is there support for the student's belief that ingesting caffeine while studying improves memory?

### Solutions

- Half of the students were randomly chosen to be given caffeine and the other half were randomly assigned not to receive caffeine.
- The mean and standard deviation for students given caffeine are estimated to be 11.73 and 1.87, respectively. The mean score for students who did not receive caffeine is estimated to be 11.65 with a standard deviation of 2.52.
- The standard deviations for the two treatment groups are very close, therefore it is reasonable to assume that they are estimating a common variance. The estimate of that common variance = 4.92 points
^{2}. - The estimated difference in the mean score of a memory test taken by students who had consumed caffeine and students who had not is 0.08. The standard error of this estimate is 0.44 points.
- The group of students from which the teacher made his random selection
- We estimate that the mean of students' scores who consumed caffeine is 0.08 points greater than the mean of students' scores who did not have caffeine, and we are 95% confident this estimate is within 0.88 points of the difference in these two population means. There is such a small difference in the population means that not much support seems to exist for student's belief that ingesting caffeine while studying improves memory

Because the data are discrete, it is impossible for the memory scores to be normally distributed. However, the sample size is large enough that the sampling distribution of can reasonably be assumed to be normal by the Central Limit Theorem

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