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.