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construct A confidence interval appropriate for the hypothesis test in part (a) with these instructions in pic 2.

To construct the confidence interval for the hypothesis test in part (a), we need to follow the instructions provided in the second image and apply the following formula:

\[
(\bar{x}_1 - \bar{x}_2) - E < \mu_1 - \mu_2 < (\bar{x}_1 - \bar{x}_2) + E
\]

Where:
- \(\bar{x}_1 = 0.53\) (Mean reduction in pain level for the magnet group)
- \(\bar{x}_2 = 0.45\) (Mean reduction in pain level for the sham group)
- \(n_1 = 17\), \(n_2 = 17\) (Sample sizes)
- \(s_1 = 1.02\), \(s_2 = 1.38\) (Sample standard deviations)
- \(t_{\alpha/2}\) is the critical value for a 90% confidence level (since \(\alpha = 0.05\) for a one-tailed test)

### Step 1: Calculate the standard error (SE)

The formula for the standard error \(SE\) is:

\[
SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}
\]

### Step 2: Determine the critical value

For a 90% confidence interval, we need the critical value of the t-distribution, \(t_{\alpha/2}\), corresponding to a 90% confidence level. The degrees of freedom (df) can be approximated by:

\[
df = \min(n_1 - 1, n_2 - 1)
\]

For \(n_1 = n_2 = 17\), \(df = 16\).

### Step 3: Compute the margin of error (E)

The margin of error \(E\) is:

\[
E = t_{\alpha/2} \times SE
\]

### Step 4: Construct the confidence interval

Once \(E\) is calculated, the confidence interval is given by:

\[
(\bar{x}_1 - \bar{x}_2) - E < \mu_1 - \mu_2 < (\bar{x}_1 - \bar{x}_2) + E
\]

I will now calculate the values for you.The calculated values for the confidence interval are:

- Standard Error (SE): 0.4162
- Critical value (t\(_{\alpha/2}\)): 1.746
- Margin of Error (E): 0.7266

Thus, the 90% confidence interval for the difference in means (\(\mu_1 - \mu_2\)) is:

\[
-0.65 < \mu_1 - \mu_2 < 0.81
\]

This means there is no statistically significant difference in pain reduction between the magnet and sham treatments, as the confidence interval includes 0.

Do you need further details or clarifications?

Here are 5 related questions:
1. How does sample size impact the width of the confidence interval?
2. What is the difference between a one-tailed and two-tailed hypothesis test?
3. How would the results change if we used a 95% confidence level instead?
4. Why is it important that the confidence interval contains 0 in this context?
5. How does the critical value \(t_{\alpha/2}\) change with varying degrees of freedom?

**Tip:** A wider confidence interval suggests more uncertainty in estimating the true difference between means.

Researchers conducted a study to determine whether magnets are effective in treating back pain. Construct a confidence interval appropriate for the hypothesis test in part (a) with these instructions in pic 2.

Learn how to construct a confidence interval for determining the effectiveness of magnets in treating back pain using hypothesis testing. This example uses a t-distribution, calculates the standard error, and provides a 90% confidence interval for the difference in means between two independent groups.

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