To construct the confidence interval for the hypothesis test, let's summarize the information given:

• Magnet Treatment ($\mu_1$):

  • Sample size ($n_1$) = 17
  • Mean ($\bar{x}_1$) = 0.53
  • Standard deviation ($s_1$) = 1.02

• Sham Treatment ($\mu_2$):

  • Sample size ($n_2$) = 17
  • Mean ($\bar{x}_2$) = 0.45
  • Standard deviation ($s_2$) = 1.38

Given this, we can calculate the confidence interval for $\mu_1 - \mu_2$ using the formula for the confidence interval of the difference in means:

$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \times \sqrt{\left( \frac{s_1^2}{n_1} \right) + \left( \frac{s_2^2}{n_2} \right)}$

From the image, it seems the test statistic $t$ is calculated as $0.19$ and the P-value is 0.425. However, we need to calculate the critical value $t_{\alpha/2}$ for constructing the confidence interval. For a typical 95% confidence level, $\alpha = 0.05$, and the degrees of freedom for this two-sample t-test can be approximated using the following formula:

$df \approx \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}$

Let's calculate this step-by-step.The 95% confidence interval for the difference in means ($\mu_1 - \mu_2$) is:

$-0.77 < \mu_1 - \mu_2 < 0.93$

This means we are 95% confident that the true difference in the reduction of pain levels between the magnet treatment and sham treatment lies between $-0.77$ and $0.93$.

Do you want further details or have any questions?

Here are 5 related questions to expand this topic:

1. What does it mean if 0 is included in the confidence interval?
2. How does the choice of confidence level affect the width of the confidence interval?
3. What are the assumptions of the two-sample t-test used here?
4. How would a larger sample size affect the confidence interval?
5. Can a non-overlapping confidence interval with zero suggest a significant difference between treatments?

Tip: Always check whether the confidence interval includes 0 to infer if there is a significant difference between groups!