Athletes performing in bright sunlight often smear black eye grease under their eyes to reduce glare. Does eye grease work? In one study, 16 student subjects took a test of sensitivity to contrast after three hours facing into bright sun, both with and without eye grease. (Greater sensitivity to contrast improves vision, and glare reduces sensitivity to contrast.) This is a matched pairs design. The differences in sensitivity, with eye grease minus without eye grease, are given in the table. \begin{tabular}{|l|c|c|c|c|c|c|c|} \hline 0.07 & 0.64 & -0.12 & -0.05 & -0.18 & 0.14 & -0.16 & 0.03 \\ \hline 0.05 & 0.02 & 0.43 & 0.24 & -0.11 & 0.28 & 0.05 & 0.29 \\ \hline \end{tabular} Click to download the data in your preferred format. CSV Excel JMP Mac-Text Minitab14-18 Minitab18+ PC-Text R SPSS TI CrunchIt! How much more sensitive to contrast are athletes with eye grease than without eye grease? Give a $95 \%$ confidence interval to answer this question. Give your answers to four decimal places. lower bound: upper bound:

Step 1 ：The problem is asking for a 95% confidence interval for the difference in sensitivity to contrast with and without eye grease. This is a problem of estimating the mean of a population based on a sample. The sample in this case is the differences in sensitivity to contrast with and without eye grease.

Step 2 ：The confidence interval for the mean of a population is given by the formula: \(\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}\) where: \(\bar{x}\) is the sample mean, \(t_{\alpha/2, n-1}\) is the t-score for a two-tailed test with \(\alpha\) level of significance and \(n-1\) degrees of freedom, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step 3 ：In this case, \(\alpha = 0.05\) (since we want a 95% confidence interval), and the sample size \(n = 16\).

Step 4 ：First, we need to calculate the sample mean and the sample standard deviation. Then, we can use these values to calculate the confidence interval.

Step 5 ：The differences in sensitivity to contrast with and without eye grease are [0.07, 0.64, -0.12, -0.05, -0.18, 0.14, -0.16, 0.03, 0.05, 0.02, 0.43, 0.24, -0.11, 0.28, 0.05, 0.29].

Step 6 ：The sample mean is 0.10125 and the sample standard deviation is 0.2263294059551255.

Step 7 ：The t-score for a two-tailed test with 0.05 level of significance and 15 degrees of freedom is 2.131449545559323.

Step 8 ：Substituting these values into the formula, we get the lower bound of the confidence interval as -0.01935242736744093 and the upper bound as 0.22185242736744093.

Step 9 ：Final Answer: The 95% confidence interval for the difference in sensitivity to contrast with and without eye grease is \(\boxed{[-0.0194, 0.2219]}\) to four decimal places. This means we are 95% confident that the true mean difference in sensitivity to contrast with and without eye grease is between -0.0194 and 0.2219.