Researchers conducted a study to determine whether magnets are effective in treating back pain. Pain was measured using the visual analog scale, and the results shown below are among the results obtained in the study. Higher scores correspond to greater pain levels. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. The reduction in pain level after magnet treatment for ų1 is n = 16,  x = 0.47, s = 1.01. The reduction in pain level after sham treatment for ų2 is n = 16. x = 0.38, s = 1.56. Find the test statistic and don’t round answer 

We are tasked with finding the test statistic for comparing the mean reductions in pain level after magnet treatment (\(\mu_1\)) and sham treatment (\(\mu_2\)). Given that we are not assuming equal population standard deviations, we will use **Welch's t-test** for comparing the two means.

The test statistic formula for Welch's t-test is:

\[
t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
\]

### Step-by-Step Breakdown:
- \(\bar{x}_1 = 0.47\): sample mean reduction in pain after magnet treatment
- \(\bar{x}_2 = 0.38\): sample mean reduction in pain after sham treatment
- \(s_1 = 1.01\): sample standard deviation for magnet treatment
- \(s_2 = 1.56\): sample standard deviation for sham treatment
- \(n_1 = 16\): sample size for magnet treatment
- \(n_2 = 16\): sample size for sham treatment

### Calculate the test statistic:
1. Subtract the sample means: \(\bar{x}_1 - \bar{x}_2 = 0.47 - 0.38 = 0.09\).
2. Square the standard deviations:
   - \(s_1^2 = 1.01^2 = 1.0201\)
   - \(s_2^2 = 1.56^2 = 2.4336\)
3. Divide the squared standard deviations by the sample sizes:
   - \(\frac{s_1^2}{n_1} = \frac{1.0201}{16} = 0.06375625\)
   - \(\frac{s_2^2}{n_2} = \frac{2.4336}{16} = 0.1521\)
4. Add the two results together:
   - \(0.06375625 + 0.1521 = 0.21585625\)
5. Take the square root:
   - \(\sqrt{0.21585625} \approx 0.46464\)
6. Calculate the test statistic:
   \[
   t = \frac{0.09}{0.46464} \approx 0.1937
   \]

Thus, the test statistic is approximately **0.1937**.

Would you like details on the next steps for hypothesis testing? 

### Follow-up Questions:
1. What are the null and alternative hypotheses for this test?
2. How do we determine the degrees of freedom for Welch’s t-test?
3. Can you show how to calculate the p-value for this test statistic?
4. How do we interpret the result based on a specific significance level (e.g., 0.05)?
5. How does Welch’s t-test differ from the standard two-sample t-test?

### Tip:
When using Welch's t-test, always ensure that you account for unequal variances, as this test provides better control over Type I errors in such cases.

Researchers conducted a study to determine whether magnets are effective in treating back pain. Pain was measured using the visual analog scale, and the results shown below are among the results obtained in the study. Higher scores correspond to greater pain levels. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. The reduction in pain level after magnet treatment for ų1 is n = 16, x = 0.47, s = 1.01. The reduction in pain level after sham treatment for ų2 is n = 16. x = 0.38, s = 1.56. Find the test statistic and don’t round answer.

Explore how to calculate the test statistic for a study on magnet treatment effectiveness for back pain using Welch's t-test. The problem involves hypothesis testing, comparing sample means, and understanding unequal variances. Detailed steps provided with sample sizes, means, and standard deviations.

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