A learn-to-type software program claims that it can improve your typing skills. To test the claim and possibly help yourself out, you and five of your friends decide to try the program and see what happens. Use the table below to construct an $80 \%$ confidence interval for the true mean change in the typing speeds for people who have completed the typing program. Let Population 1 be the typing speed before taking the program and Population 2 be the typing speed after taking the program. Round the endpoints of the intrival to one decimal place, if necessary. \begin{tabular}{|c|c|} \hline \multicolumn{2}{|c|}{ Typing Speeds (in Words per Minute) } \\ \hline Before & After \\ \hline 38 & 54 \\ \hline 34 & 44 \\ \hline 53 & 50 \\ \hline 31 & 51 \\ \hline 55 & 35 \\ \hline 54 & 45 \\ \hline \end{tabular} Copy Data

Step 1 ：First, calculate the change in typing speed for each person by subtracting the speed before the program from the speed after the program. The changes are 16, 10, -3, 20, -20, and -9 words per minute.

Step 2 ：Next, calculate the mean change in typing speed by adding up all the changes and dividing by the number of people. The mean change is approximately \(2.33\) words per minute.

Step 3 ：Then, calculate the standard deviation of the changes. The standard deviation is approximately \(15.58\) words per minute.

Step 4 ：Use the t-distribution to find the t-value for an 80% confidence interval with 5 degrees of freedom. The t-value is approximately \(1.48\).

Step 5 ：Finally, calculate the lower and upper bounds of the confidence interval using the formula: mean ± (t-value * standard deviation / √n). The lower bound is approximately \(-7.05\) words per minute and the upper bound is approximately \(11.72\) words per minute.

Step 6 ：Final Answer: The 80% confidence interval for the true mean change in typing speeds for people who have completed the typing program is approximately \(\boxed{(-7.1, 11.7)}\) words per minute.