To calculate the 95% confidence interval for the difference of the sample means, we use the formula:

$\text{Confidence Interval} = \bar{X}_1 - \bar{X}_2 \pm Z \left( \frac{\sigma_d}{\sqrt{n}} \right)$

where:

• $\bar{X}_1 - \bar{X}_2$ is the difference of the sample means
• $Z$ is the Z-score corresponding to the desired confidence level (for 95%, $Z \approx 1.96$)
• $\sigma_d$ is the standard deviation of the difference of the sample means
• $n$ is the sample size (not given, but usually large enough to approximate the normal distribution)

Given the information:

• Difference of the sample means $\bar{X}_1 - \bar{X}_2 = 34.6$
• Standard deviation of the difference $\sigma_d = 11.9$

The confidence interval calculation simplifies to:

$\text{Confidence Interval} = 34.6 \pm 1.96 \times 11.9$

Now calculate:

$1.96 \times 11.9 = 23.324$

Thus, the 95% confidence interval is:

$34.6 \pm 23.324$

So the interval is:

$[34.6 - 23.324, 34.6 + 23.324]$ $[11.276, 57.924]$

Therefore, the 95% confidence interval lies between 11.276 and 57.924.

Would you like any more details or have any questions?

Here are 5 numbered questions you might want to ask:

1. How do you interpret a confidence interval in the context of hypothesis testing?
2. What assumptions must be met to use this method for confidence intervals?
3. How would the confidence interval change if we wanted a 99% confidence level?
4. Can you explain the difference between a sample mean and a population mean?
5. How does sample size affect the width of the confidence interval?

**Tip