The problem requires statistical analysis to test the hypothesis that magnets are effective in reducing pain levels compared to a sham treatment. Let's analyze it step by step based on the provided image:

Part (a): Null and Alternative Hypotheses

The hypothesis test is to determine if magnets lead to greater pain reduction compared to sham treatment.

• Null hypothesis ($H_0$): $\mu_1 = \mu_2$ (No difference in mean pain reduction between magnet and sham treatment)
• Alternative hypothesis ($H_1$): $\mu_1 > \mu_2$ (Magnet treatment results in greater mean pain reduction)

Selected Option: D

Test Statistic and P-value

From the image:

• Test statistic ($t$): $0.23$
• P-value: $0.411$

Since the P-value ($0.411$) is greater than the significance level ($\alpha = 0.01$), we fail to reject the null hypothesis. There is insufficient evidence to support the claim that magnets result in a greater mean reduction in pain than sham treatment.

Part (b): Confidence Interval

You are required to construct a confidence interval for the difference in means ($\mu_1 - \mu_2$) using the data. Let me compute this for you based on the information provided:

1. Sample Sizes:

   • $n_1 = 18$, $n_2 = 18$

2. Means:

   • $\bar{x}_1 = 0.56$, $\bar{x}_2 = 0.47$

3. Standard Deviations:

   • $s_1 = 0.98$, $s_2 = 1.38$

4. Standard Error ($SE$) for the difference in means: $SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

   Substituting values: $SE = \sqrt{\frac{0.98^2}{18} + \frac{1.38^2}{18}} = \sqrt{\frac{0.9604}{18} + \frac{1.9044}{18}} = \sqrt{0.053356 + 0.1058} = \sqrt{0.159156} \approx 0.399$

5. Margin of Error (ME): At a 99% confidence level, the critical $t$-value ($t^*$) for $df = 34$ (approximation for two samples of size 18) is approximately $2.728$.

   $ME = t^* \times SE = 2.728 \times 0.399 \approx 1.089$

6. Confidence Interval: $(\bar{x}_1 - \bar{x}_2) \pm ME = (0.56 - 0.47) \pm 1.089 = 0.09 \pm 1.089$

   This gives: $(-1.00, 1.18)$

The 99% confidence interval for $\mu_1 - \mu_2$ is approximately $(-1.00, 1.18)$.

Conclusion

Since the confidence interval includes 0, it supports the earlier result of failing to reject the null hypothesis. There is insufficient evidence to claim that magnets reduce pain more than the sham treatment.

Next Steps

If you'd like, I can provide:

1. Detailed calculations of the standard error and confidence interval formula.
2. Explanation of how to interpret confidence intervals in hypothesis testing.
3. An overview of when to use one-tailed vs. two-tailed tests.
4. Insights on improving experimental design for such studies.
5. A guide on using software like Excel or statistical tools for such tests.

Tip: Always ensure your sample sizes are large enough to improve the reliability of your statistical conclusions.