Question

Sarah believes that completely cutting caffeine out of a person’s diet will allow him or her more restful sleep at night. In fact, she believes that, on average, adults will have more than two additional nights of restful sleep in a four-week period after removing caffeine from their diets. She randomly selects 8 adults to help her test this theory. Each person is asked to consume two caffeinated beverages per day for 28 days, and then cut back to no caffeinated beverages for the following 28 days. During each period, the participants record the numbers of nights of restful sleep that they had. The following table gives the results of the study. Test Sarah’s claim at the 0.02 level of significance assuming that the population distribution of the paired differences is approximately normal. Let the period before removing caffeine be Population 1 and let the period after removing caffeine be Population 2. Numbers of Nights of Restful Sleep in a Four-Week Period With Caffeine 17 22 19 24 17 22 18 20 Without Caffeine 20 26 23 26 20 24 21 25 Step 2 of 3: Compute the value of the test statistic. Round your answer to three decimal places.

Answer 1

To test Sarah's claim about the effect of cutting out caffeine on restful sleep, we need to compute the test statistic. The test statistic for comparing paired means is the t-statistic, which is calculated as the mean of the paired differences divided by the standard error of the differences. In this case, the value of the test statistic is 8.597.

Step-by-step explanation:

To test Sarah's claim about the effect of cutting out caffeine on restful sleep, we need to compute the test statistic. The test statistic for comparing paired means is the t-statistic, which is calculated as the mean of the paired differences divided by the standard error of the differences.

Let's calculate the paired differences: 20-17=3, 26-22=4, 23-19=4, 26-24=2, 20-17=3, 24-22=2, 21-18=3, 25-20=5.

The mean of the paired differences is (3+4+4+2+3+2+3+5)/8=26/8=3.25. Now, let's calculate the standard deviation of the paired differences: sqrt(((3-3.25)^2+(4-3.25)^2+(4-3.25)^2+(2-3.25)^2+(3-3.25)^2+(2-3.25)^2+(3-3.25)^2+(5-3.25)^2)/7)=1.069.

The standard error of the differences is the standard deviation of the paired differences divided by the square root of the sample size. So, the standard error of the differences is 1.069/sqrt(8)=0.378. Finally, we can calculate the t-statistic as the mean of the paired differences divided by the standard error of the differences: 3.25/0.378=8.597. Therefore, the value of the test statistic is 8.597.