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Construct a confidence interval appropriate for the hypothesis test in part (a). according to the picture

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!

Construct a confidence interval appropriate for the hypothesis test in part (a).

Learn how to construct a confidence interval for the difference in pain reduction between magnet and sham treatments using a two-sample t-test. This problem involves calculating the confidence interval based on sample sizes, means, and standard deviations, with detailed formulas and explanation. Suitable for students in Grades 11-12 or college level.

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