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 
34 33
30 35
50 36
53 46
33 35
47 43

To construct a 99% confidence interval for the true mean change in typing speeds before and after completing the program, we follow these steps:

### Step 1: Define the Differences
For each individual, calculate the difference between the typing speeds before and after the program. Let \( d_i \) be the difference for each person, where \( d_i = \text{Before} - \text{After} \).

| Participant | Before | After | Difference (\( d \)) |
|-------------|--------|-------|---------------------|
| 1           | 34     | 33    | 34 - 33 = 1         |
| 2           | 30     | 35    | 30 - 35 = -5        |
| 3           | 50     | 36    | 50 - 36 = 14        |
| 4           | 53     | 46    | 53 - 46 = 7         |
| 5           | 33     | 35    | 33 - 35 = -2        |
| 6           | 47     | 43    | 47 - 43 = 4         |

So, the differences (\( d \)) are: 1, -5, 14, 7, -2, 4.

### Step 2: Calculate the Sample Mean and Standard Deviation of the Differences
1. **Mean of the differences (\( \bar{d} \))**:
   \[
   \bar{d} = \frac{\sum d_i}{n} = \frac{1 + (-5) + 14 + 7 + (-2) + 4}{6} = \frac{19}{6} \approx 3.17
   \]

2. **Standard Deviation of the differences (\( s_d \))**:
   First, calculate the variance (\( s_d^2 \)) and then take the square root to get the standard deviation:
   \[
   s_d^2 = \frac{\sum (d_i - \bar{d})^2}{n-1}
   \]
   \[
   s_d = \sqrt{\frac{(1 - 3.17)^2 + (-5 - 3.17)^2 + (14 - 3.17)^2 + (7 - 3.17)^2 + (-2 - 3.17)^2 + (4 - 3.17)^2}{5}}
   \]
   Let's calculate this step-by-step:
   
   - \( (1 - 3.17)^2 \approx 4.71 \)
   - \( (-5 - 3.17)^2 \approx 66.69 \)
   - \( (14 - 3.17)^2 \approx 118.47 \)
   - \( (7 - 3.17)^2 \approx 14.66 \)
   - \( (-2 - 3.17)^2 \approx 26.47 \)
   - \( (4 - 3.17)^2 \approx 0.69 \)

   Sum of squared differences:
   \[
   \sum (d_i - \bar{d})^2 \approx 231.69
   \]
   Then, divide by \( n - 1 \):
   \[
   s_d^2 \approx \frac{231.69}{5} = 46.34
   \]
   \[
   s_d \approx \sqrt{46.34} \approx 6.81
   \]

### Step 3: Determine the Critical Value
For a 99% confidence interval with \( n = 6 \), degrees of freedom (\( df \)) is \( n - 1 = 5 \). Using a t-table, the critical value (\( t^* \)) for 99% confidence and 5 degrees of freedom is approximately 4.032.

### Step 4: Calculate the Margin of Error (ME)
\[
\text{ME} = t^* \times \frac{s_d}{\sqrt{n}} = 4.032 \times \frac{6.81}{\sqrt{6}} \approx 4.032 \times 2.78 \approx 11.2
\]

### Step 5: Construct the Confidence Interval
\[
\text{Confidence Interval} = \bar{d} \pm \text{ME}
\]
\[
= 3.17 \pm 11.2
\]
\[
\approx (-8.0, 14.4)
\]

### Final Answer
The 99% confidence interval for the true mean change in typing speeds is approximately:
\[
(-8.0, 14.4)
\]

This interval means that we are 99% confident that the true mean change in typing speed after using the program falls between -8.0 and 14.4 words per minute.

Would you like further details or have any questions?

### Related Questions:
1. How do you determine which t-distribution value to use for a specific confidence interval?
2. What is the impact of sample size on the width of a confidence interval?
3. How would the confidence interval change if we used a 95% confidence level instead?
4. Why is it important to use the differences between paired data for this type of problem?
5. How can we interpret a confidence interval that includes zero?

### Tip:
The width of a confidence interval is influenced by the variability of the data, sample size, and the confidence level. A higher confidence level results in a wider interval.

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 34 33 30 35 50 36 53 46 33 35 47 43

Learn how to construct a 99% confidence interval for the true mean change in typing speed after using a typing program. This example covers paired data analysis, confidence interval calculations, and the t-distribution for a small sample of 6 participants. Suitable for high school and introductory college students.

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