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 \(\boxed{4.930}\).