A personal computer manufacturer is interested in comparing assembly times for two keyboard assembly processes. To do this, the company selects 10 workers at random and asks each of them to use both assembly processes. The assembly time (in minutes) for Process 1 is recorded for each of these workers. Then, the assembly time for Process 2 is recorded for each of the same workers. The data and the differences (Process 1 minus Process 2) are shown in the table below. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline Worker & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline Process 1 & 32 & 64 & 17 & 28 & 26 & 75 & 73 & 20 & 55 & 5 \\ \hline Process 2 & 34 & 67 & 47 & 32 & 41 & 89 & 75 & 33 & 66 & 38 \\ \hline \begin{tabular}{c} Difference \\ (Process 1 - Process 2) \end{tabular} & -2 & -3 & -30 & -4 & -15 & -14 & -2 & -13 & -11 & -33 \\ \hline \end{tabular} Send data to calculator Assume that the population of these differences in assembly times (Process 1 minus Process 2 ) is approximately normally distributed. Construct a $90 \%$ confidence interval for $\mu_{d}$, the population mean difference in assembly times for the two processes. Then find the lower and upper limits of the $90 \%$ confidence interval. Carry your intermediate computations to three or more decimal places. Round your answers to two or more decimal places. (If necessary, consult a list of formulas.) Lower limit: Upper limit:
Solution
Step 1 ：Given the differences in assembly times for the two processes, we have the following data: -2, -3, -30, -4, -15, -14, -2, -13, -11, -33. The sample size (n) is 10.
Step 2 ：Calculate the mean of the differences. The mean difference (\(\bar{x}\)) is -12.7.
Step 3 ：Calculate the standard deviation of the differences. The standard deviation (s) is approximately 11.156.
Step 4 ：The z-score corresponding to a 90% confidence level is 1.645.
Step 5 ：Substitute these values into the formula for a confidence interval: \(\bar{x} \pm z \frac{s}{\sqrt{n}}\).
Step 6 ：Calculate the lower limit of the confidence interval: -12.7 - 1.645 * (11.156 / \(\sqrt{10}\)) = -18.50.
Step 7 ：Calculate the upper limit of the confidence interval: -12.7 + 1.645 * (11.156 / \(\sqrt{10}\)) = -6.90.
Step 8 ：The 90% confidence interval for the population mean difference in assembly times for the two processes is \(\boxed{-18.50}\) to \(\boxed{-6.90}\).
