https://file.aiquickdraw.com/mathbot/file/1727496999178kua97g32.jpg

The question displayed on the screen involves a hypothesis test for the effectiveness of magnets in treating back pain based on a reduction in pain levels. The research compares two treatments:

1. **Magnet Treatment** (\(\mu_1\)): \(n = 21\), \(\bar{x} = 0.56\), \(s = 0.91\)
2. **Sham Treatment** (\(\mu_2\)): \(n = 21\), \(\bar{x} = 0.52\), \(s = 1.51\)

### Part (a) asks about the null and alternative hypotheses:
- The question claims to test whether those treated with magnets have a greater mean reduction in pain compared to the sham treatment.

The correct hypotheses are:
- Null Hypothesis (\(H_0\)): \(\mu_1 = \mu_2\) (the mean reduction in pain is the same for both treatments).
- Alternative Hypothesis (\(H_1\)): \(\mu_1 > \mu_2\) (those treated with magnets have a greater mean reduction in pain).

This is represented as option **B**, which has been correctly selected.

### Test statistic and P-value:
- Test statistic \(t = 0.10\)
- P-value = 0.460

### Conclusion:
- Since the P-value (0.460) is greater than the significance level (\(\alpha = 0.05\)), we **fail to reject the null hypothesis**. This means that there is not enough evidence to support the claim that magnet treatment leads to a greater mean reduction in pain compared to the sham treatment.

### Part (b):
You are asked to construct a confidence interval appropriate for this hypothesis test. However, this part seems to be incomplete in the image, but generally, this would involve finding a confidence interval for the difference in means \(\mu_1 - \mu_2\) based on the data provided.

Would you like me to help with calculating the confidence interval, or do you have any other questions?

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Here are five related questions to extend the learning:

1. How is the test statistic calculated for a two-sample t-test?
2. What assumptions need to be met for the validity of a two-sample t-test?
3. How does the significance level (\(\alpha\)) affect the hypothesis test?
4. What does a confidence interval represent in the context of this study?
5. What are Type I and Type II errors in hypothesis testing?

**Tip:** The P-value indicates the probability of obtaining the observed data (or something more extreme) assuming the null hypothesis is true.

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.

Reduction in Pain Level After Magnet Treatment (μ₁): n = 21, x̄ = 0.56, s = 0.91
Reduction in Pain Level After Sham Treatment (μ₂): n = 21, x̄ = 0.52, s = 1.51

This problem involves testing whether magnets have a greater effect on reducing back pain compared to a sham treatment. Using a two-sample t-test and a significance level of 0.05, we test the claim that those treated with magnets experience a greater reduction in pain. The test statistic is t = 0.10 and the P-value is 0.460, leading to a failure to reject the null hypothesis.

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