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 eight of your friends decide to try the program and see what happens. Use the table below to construct a $95 \%$ 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 interval to one decimal place, if necessary.
| | |
| :----- | :----- |
| Typing Speeds (in Words per Minute) | |
| Before | After |
| 30 | 39 |
| 35 | 53 |
| 33 | 32 |
| 45 | 54 |
| 40 | 35 |
| 31 | 30 |
| 40 | 44 |
| 39 | 50 |
| 55 | 55 |
Answer (1)
Calculate the mean difference in typing speeds: d ˉ = 4.89 .
Calculate the standard deviation of the differences: s d = 7.37 .
Find the critical t-value for a 95% confidence level with 8 degrees of freedom: t = 2.306 .
Calculate the 95% confidence interval: ( − 0.8 , 10.6 ) .
Explanation
Understand the problem We are given paired data of typing speeds before and after using a learn-to-type software for 9 individuals. Our goal is to construct a 95% confidence interval for the true mean change in typing speeds.
Calculate the mean difference First, calculate the difference in typing speeds for each person ( d i = A f t e r i − B e f or e i ). Then, calculate the sample mean of the differences: d ˉ = n ∑ i = 1 n d i , where n = 9 .
Calculate the mean difference (details) The differences are: 9, 18, -1, 9, -5, -1, 4, 11, 0. The mean difference is calculated as: d ˉ = 9 9 + 18 − 1 + 9 − 5 − 1 + 4 + 11 + 0 = 9 44 = 4.888888888888889 Rounding to two decimal places, d ˉ ≈ 4.89 .
Calculate the standard deviation of the differences Next, calculate the sample standard deviation of the differences: s d = n − 1 ∑ i = 1 n ( d i − d ˉ ) 2 .
Calculate the standard deviation of the differences (details) The standard deviation of the differences is calculated as 7.37.
Find the critical t-value Determine the critical t-value ( t α /2 ) for a 95% confidence level with n − 1 = 8 degrees of freedom. α = 1 − 0.95 = 0.05 , so α /2 = 0.025 . The critical t-value is 2.306.
Calculate the margin of error Calculate the margin of error: E = t α /2 ⋅ n s d .
Calculate the margin of error (details) The margin of error is calculated as: E = 2.306 ⋅ 9 7.37 = 2.306 ⋅ 3 7.37 = 2.306 ⋅ 2.4566666666666665 ≈ 5.67
Calculate the lower bound Calculate the lower bound of the confidence interval: C I l o w er = d ˉ − E .
Calculate the lower bound (details) The lower bound is calculated as: C I l o w er = 4.89 − 5.67 = − 0.78
Calculate the upper bound Calculate the upper bound of the confidence interval: C I u pp er = d ˉ + E .
Calculate the upper bound (details) The upper bound is calculated as: C I u pp er = 4.89 + 5.67 = 10.56
State the final answer Report the 95% confidence interval as ( C I l o w er , C I u pp er ) , rounded to one decimal place. The 95% confidence interval for the true mean change in typing speeds is ( − 0.8 , 10.6 ) .
Examples
Confidence intervals are used in various fields to estimate population parameters. For example, a marketing team might use a confidence interval to estimate the average increase in sales after launching a new advertising campaign. Similarly, in healthcare, confidence intervals can be used to estimate the effectiveness of a new drug or treatment. In this case, we are using a confidence interval to estimate the true mean change in typing speeds after completing a typing program. This helps us understand the potential benefits of the program.
