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# The Matched-Pairs Design for Comparing Two Treatment Means Practice Questions

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Updated on Oct 5, 2011

To review these concepts, go to The Matched-Pairs Design for Comparing Two Treatment Means Study Guide.

## The Matched-Pairs Design for Comparing Two Treatment Means Practice Questions

### Practice

A dermatologist wants to know which of two creams is better for curing hand eczema. For the experiment, she will apply the hand creams and determine which one works better. Forty patients with hand eczema volunteer for the experiment.

1. Describe how to conduct the study using a matched-pairs design.
2. Describe how to conduct the study using a twogroup design.
3. Which of the two designs would you use for the study? Explain.

A high school chemistry teacher has given his students an in-class assignment. He has given each of his students two liquids, A and B, and told the students that one of the liquids is saltwater and the other is plain water, but not which one is which. The students have learned that, when adding salt to water, the boiling point increases. The teacher has asked his students to record each liquid's boiling point and, based on this information, determine which liquid is saltwater and which is plain water. Fifteen students are in the class, and each student has a burner with two side-by-side hot plates on it. Each student flips a coin. If the coin lands heads up, the student places liquid A on the right hot plate. If the coin lands tails up, the student places liquid B on the right hot plate. The students use glass thermometers to record the temperatures of the liquids at the exact time they begin to boil. The temperatures were recorded in degrees Celsius and are given in Table 18.3.

1. Explain why this study has a matched-pairs design. Include a clear statement describing what constitutes a pair.
2. Find the difference in observations from each pair.
3. Estimate the mean and standard deviation of the differences in the boiling points of each liquid.
4. Find the standard error of the estimated mean of the differences in the boiling points of each liquid.
5. Is the assumption that the differences are normally distributed reasonable?
6. To which population may inferences be drawn from this study?
7. Look again at the study of the temperatures at which liquids A and B boiled in the previous practice problem. Find a 95% confidence interval for the mean difference in the temperatures at which liquid A and liquid B boiled. Be sure to interpret the interval in the context of the problem.
8. Once again, consider the study of the differences in the boiling point of saltwater and plain water. Based on the student's data, is the mean temperature at which liquid A boils significantly different from the mean temperature at which liquid B boils? Recall that the boiling point of saltwater is greater than that of plain water. Use this information to answer the science teacher's question, "Which of these two liquids is saltwater?"

#### Solutions

1. All patients would have one cream applied to one of their hands and the other cream applied to the other hand. Which hand the creams are applied to would be randomly decided.
2. The dermatologist would randomly select 20 of her patients to use one type of cream on both of their hands; the other 20 patients would use the other type of cream on both of their hands.
3. The matched-pairs design would be best for this study. People vary in the types of jobs and activities they perform in a day, causing some people to use their hands more than others. This might affect the results. With the matched-pairs design we can eliminate this source of variation.
4. The two treatments are liquid A and liquid B. Each treatment is given to a student. Thus, liquid A and liquid B are paired by the student. A pair consists of the boiling points for the two liquids given to each student. Which side of the hot plate would be used to boil liquids A and B was determined randomly.
5.

6. Estimated mean = 7.22; estimated standard deviation = 1.050
7. 0.271
8.

Because there are very few data values, it is difficult to fully asses normality. However, looking at the data we have, we see that there may be some hint of skew to the right, but over all it is fairly symmetric. Therefore, it is not unreasonable to assume the data are normally distributed.

9. It depends on the source of saltwater and plain water. For example, if the teacher had a gallon of distilled water and used it to provide all the plain water and to mix the saltwater, inference can only be made to the water used in the study.
10. We estimate that, on average, liquid A boiled at 7.22 degrees Celsius higher than liquid B, and we are 95% confident that this estimate is within 0.58 degrees of the true mean difference in the temperatures at which both liquids began to boil.
11. The test statistic = 26.64 and the p-value < 2(0.001) = 0.002. The mean temperature at which liquid B began to boil was significantly less than the mean temperature at which liquid A began to boil. From this we can guess that liquid A is saltwater and liquid B is plain water.

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