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Lesson: 11.3 Hypothesis Testing: Two Pop...
JEFFERY RIPKA
Question 1 of 6, Step 2 of 3
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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 0.05 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.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline \multicolumn{8}{|c|}{ Numbers of Nights of Restful Sleep in a Four-Week Period } \\
\hline With Caffeine & 17 & 24 & 23 & 18 & 16 & 21 & 22 & 18 \\
\hline Without Caffeine & 17 & 27 & 24 & 22 & 20 & 25 & 25 & 20 \\
\hline
\end{tabular}
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Step 2 of 3: Compute the value of the test statistic. Round your answer to three decimal places.
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Rounding to three decimal places, the value of the test statistic is \(\boxed{4.930}\).
Step 1 ：Given the data for the number of nights of restful sleep with caffeine and without caffeine, we first need to calculate the differences for each person. The data for sleep with caffeine is [17, 24, 23, 18, 16, 21, 22, 18] and without caffeine is [17, 27, 24, 22, 20, 25, 25, 20]. The differences are [0, 3, 1, 4, 4, 4, 3, 2].
Step 2 ：Next, we calculate the mean (\(\bar{d}\)) and standard deviation (\(s_d\)) of these differences. The mean of the differences is 2.625 and the standard deviation is 1.5059406173077154.
Step 3 ：We are performing a paired t-test, so we use the formula for the test statistic: \(t = \frac{\bar{d} - \mu_0}{s_d / \sqrt{n}}\), where \(\mu_0\) is the hypothesized mean difference (which is 0 in this case), and \(n\) is the number of pairs (which is 8 in this case).
Step 4 ：Substituting the values into the formula, we get \(t = \frac{2.625 - 0}{1.5059406173077154 / \sqrt{8}} = 4.930221761155702\).
Step 5 ：Rounding to three decimal places, the value of the test statistic is \(\boxed{4.930}\).
