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The question is about testing whether magnets are more effective in reducing pain compared to a sham (placebo) treatment. The null and alternative hypotheses need to be set up for a hypothesis test at a significance level of 0.05.

Here is the interpretation:

1. **Null hypothesis (\( H_0 \))**: The mean reduction in pain for the magnet treatment group is less than or equal to the mean reduction for the sham treatment group. Mathematically: 
   \[
   H_0: \mu_1 \leq \mu_2
   \]
   This represents the assumption that the magnet treatment does not reduce pain more than the sham treatment.

2. **Alternative hypothesis (\( H_1 \))**: The mean reduction in pain for the magnet treatment group is greater than for the sham treatment group. Mathematically: 
   \[
   H_1: \mu_1 > \mu_2
   \]
   This represents the claim that the magnet treatment reduces pain more effectively than the sham.

Thus, the correct hypothesis test setup, based on the question's wording, would be **Option B**:
- \( H_0: \mu_1 = \mu_2 \) (the treatments have the same effect)
- \( H_1: \mu_1 > \mu_2 \) (magnet treatment is more effective than sham).

Would you like more details on how to carry out the hypothesis test, or do you have any other questions?

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Here are 5 related questions to expand your understanding:

1. How do you calculate the test statistic for comparing two independent means?
2. What assumptions must be checked before performing a two-sample t-test?
3. What does a p-value represent in the context of hypothesis testing?
4. How can you interpret the confidence interval in the context of this experiment?
5. What would be the conclusion if the p-value was greater than 0.05?

**Tip:** Always check if the data follows a normal distribution or if sample sizes are large enough for the Central Limit Theorem to apply before conducting a hypothesis test.

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 

(a) Use a 0.05 significance level to test the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment (similar to a placebo).

What are the null and alternative hypotheses?

A. H₀: μ₁ = μ₂ 
    H₁: μ₁ ≠ μ₂

B. H₀: μ₁ = μ₂
    H₁: μ₁ > μ₂

C. H₀: μ₁ < μ₂
    H₁: μ₁ ≥ μ₂

D. H₀: μ₁ ≠ μ₂
    H₁: μ₁ < μ₂

Explore the hypothesis testing process to determine if magnet treatment is more effective than sham treatment in reducing back pain. Learn how to set up null and alternative hypotheses and calculate the test statistic using a two-sample t-test. Ideal for high school students in Grades 11-12.

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