part c is incorrect, it is suppose to be finding the confidence interval

Let's correct the confidence interval calculation for part (c).

### Part (c): Confidence Interval Calculation

To calculate a 95% confidence interval for the difference between the means:

Given dataLet's correct the confidence interval calculation for part (c).

### Part (c): Confidence Interval Calculation

To calculate a 95% confidence interval for the difference between the means:

Given 
- \( \bar{x}_1 = 1391 \)
- \( \bar{x}_2 = 1249 \)
- \( s_p = 235.23 \) (from the previous calculation)
- \( n_1 = n_2 = 11 \)
- Degrees of Freedom (\( df = 20 \))
- Critical value for 95% confidence interval with \( df = 20 \): \( t_{0.025, 20} \approx 2.086 \)

1. **Difference of means (\( \bar{x}_1 - \bar{x}_2 \))**:
\[ \bar{x}_1 - \bar{x}_2 = 1391 - 1249 = 142 \]

2. **Margin of Error**:
\[ \text{Margin of Error} = t_{0.025, 20} \cdot s_p \cdot \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} \]
\[ \text{Margin of Error} = 2.086 \cdot 235.23 \cdot \sqrt{\frac{1}{11} + \frac{1}{11}} \]
\[ \text{Margin of Error} = 2.086 \cdot 235.23 \cdot \sqrt{\frac{2}{11}} \]
\[ \text{Margin of Error} = 2.086 \cdot 235.23 \cdot 0.4264 \]
\[ \text{Margin of Error} = 2.086 \cdot 100.32 \]
\[ \text{Margin of Error} = 209.23 \]

3. **Confidence Interval**:
\[ \text{CI} = (\bar{x}_1 - \bar{x}_2) \pm \text{Margin of Error} \]
\[ \text{CI} = 142 \pm 209.23 \]

So, the confidence interval is:
\[ \text{CI} = [142 - 209.23, 142 + 209.23] \]
\[ \text{CI} = [-67.23, 351.23] \]

### Interpretation:
The 95% confidence interval for the difference between the mean weekly sales of type A and type B training ranges from -67.23 to 351.23. Since the interval includes 0, we do not have strong evidence to conclude that there is a significant difference between the two training methods.

Would you like more details or have any questions?

Here are 5 questions you can ask next:
1. How do you calculate the p-value for a given t-score?
2. What is the difference between independent and dependent samples in hypothesis testing?
3. How can we test for equal variances in two samples?
4. What are Type I and Type II errors in hypothesis testing?
5. How do you interpret the results of a confidence interval?

**Tip:** Always check the assumptions of your statistical tests, such as normality and equal variances, to ensure the validity of your conclusions.

Learn how to correctly calculate a confidence interval for the difference between means in statistics. This problem covers key statistical concepts including margin of error calculation and interpretation of confidence intervals. Suitable for college-level students studying statistics.

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