To construct a confidence interval for the difference in means $\mu_1 - \mu_2$ as part of the hypothesis test, we follow the following steps using the data provided.

Information extracted from the problem:

• $n_1 = 18$, $\bar{x}_1 = 0.48$, $s_1 = 0.99$ for the magnet treatment.
• $n_2 = 18$, $\bar{x}_2 = 0.39$, $s_2 = 1.57$ for the sham treatment.
• Confidence level: 95% (significance level $\alpha = 0.05$).

Step 1: Find the Standard Error (SE)

We use the formula for the standard error of the difference in means when the two samples are independent: $SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ Where:

• $s_1 = 0.99$, $n_1 = 18$,
• $s_2 = 1.57$, $n_2 = 18$.

Step 2: Find the critical value (t-value)

Since we are using a 95% confidence level and assuming equal variances, we find the $t$-critical value for a two-tailed test with $n_1 + n_2 - 2 = 34$ degrees of freedom.

Step 3: Construct the confidence interval

The confidence interval for $\mu_1 - \mu_2$ is given by: $(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \cdot SE$

Let me calculate this step by step.The confidence interval for the difference in means $\mu_1 - \mu_2$ is approximately:

$-0.799 < \mu_1 - \mu_2 < 0.979$

This interval suggests that there is no statistically significant difference between the mean reduction in pain levels for the magnet treatment and the sham treatment, as the interval contains zero.

Do you have any questions or need further clarification?

Here are five related questions to consider:

1. What does it mean when a confidence interval includes zero?
2. How does the sample size affect the width of the confidence interval?
3. Why is it important to assume that the population standard deviations are not equal?
4. How does changing the confidence level (e.g., from 95% to 99%) affect the interval?
5. What is the impact of using a two-tailed test compared to a one-tailed test?

Tip: The margin of error in a confidence interval reflects the uncertainty around the estimate. A larger margin indicates more uncertainty, often influenced by smaller sample sizes or higher variability.