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 a $99 \%$ 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 |
| 53 | 55 |
| 50 | 42 |
| 46 | 54 |
| 44 | 51 |
| 30 | 43 |
| 42 | 34 |
Answer (2)
Calculate the differences in typing speeds: d i = A f t e r i − B e f or e i .
Calculate the mean of the differences: d ˉ = 2.3333 .
Calculate the margin of error: E = 14.3751 .
Calculate the 99% confidence interval: ( − 12.0 , 16.7 ) .
Explanation
Understand the problem and provided data We are asked to construct a 99% confidence interval for the true mean change in typing speeds after completing a learn-to-type software program. We have paired data for 6 individuals, showing their typing speeds before and after using the program.
Calculate the differences First, we need to calculate the difference in typing speeds for each individual by subtracting the 'before' speed from the 'after' speed. This gives us the following differences (After - Before): 2 , − 8 , 8 , 7 , 13 , − 8 .
Calculate the mean of the differences Next, we calculate the mean of these differences, which is the sum of the differences divided by the number of individuals (6). d ˉ = 6 2 + ( − 8 ) + 8 + 7 + 13 + ( − 8 ) = 6 14 = 2.3333
Calculate the standard deviation of the differences Now, we calculate the sample standard deviation of the differences. This involves finding the squared difference between each individual difference and the mean difference, summing these squared differences, dividing by n − 1 (where n is the number of individuals), and then taking the square root. The result of this calculation is s d = 8.7331 .
Find the critical t-value We need to find the critical t-value for a 99% confidence level with n − 1 = 5 degrees of freedom. For a 99% confidence interval, α = 1 − 0.99 = 0.01 , so α /2 = 0.005 . Looking up the t-value for 5 degrees of freedom and α /2 = 0.005 in a t-table, we find t 0.005 , 5 = 4.032 .
Calculate the margin of error Now we calculate the margin of error using the formula: E = t α /2 ⋅ n s d . Plugging in the values, we get: E = 4.032 ⋅ 6 8.7331 = 4.032 ⋅ 2.4495 8.7331 = 4.032 ⋅ 3.5652 = 14.3751
Calculate the confidence interval Finally, we calculate the confidence interval using the formula: ( d ˉ − E , d ˉ + E ) . Plugging in the values, we get: ( 2.3333 − 14.3751 , 2.3333 + 14.3751 ) = ( − 12.0418 , 16.7085 ) Rounding the endpoints to one decimal place, the 99% confidence interval for the true mean change in typing speeds is ( − 12.0 , 16.7 ) .
State the final answer The 99% confidence interval for the true mean change in typing speeds is ( − 12.0 , 16.7 ) .
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 estimating the change in typing speed after using a typing program, which can help determine the program's effectiveness.
We computed the 99% confidence interval for the average change in typing speed after a learn-to-type program. The differences in typing speeds were calculated, leading to a mean change of about 2.3 words per minute and a margin of error of approximately 14.4 words per minute. The final confidence interval is ( − 12.0 , 16.7 ) , indicating the average change could be negative or positive after the program.
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