In Exercises 5–20, 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. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of n1-1 and n2-1)


Magnet Treatment of Pain People spend around \$5 billion annually for the purchase of magnets used to treat a wide variety of pains. 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 given below are among the results obtained in the study (based on data from “Bipolar Permanent Magnets for the Treatment of Chronic Lower Back Pain: A Pilot Study,” by Collacott, Zimmerman, White, and Rindone, Journal of the American Medical Association, Vol. 283, No. 10). Higher scores correspond to greater pain levels.


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).

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). H₀: μ₁ = μ₂ (the mean reduction in pain for the magnet treatment group is equal to the sham treatment group). H₁: μ₁ > μ₂ (the mean reduction in pain for the magnet treatment group is greater than the sham treatment group). This is a one-tailed test.
2. Step 2: Identify the given data. For the magnet treatment group: n₁ = 20, x̄₁ = 0.49, s₁ = 0.96. For the sham treatment group: n₂ = 20, x̄₂ = 0.44, s₂ = 1.4. The significance level is α = 0.05.
3. Step 3: Calculate the test statistic using the formula for a two-sample t-test for independent samples: t = (x̄₁ - x̄₂) / √((s₁²/n₁) + (s₂²/n₂)). Substitute the given values into the formula, but do not compute the final value.
4. Step 4: Determine the degrees of freedom (df) using the smaller of n₁ - 1 and n₂ - 1. In this case, df = min(20 - 1, 20 - 1) = 19. Use this df to find the critical t-value from the t-distribution table for a one-tailed test at α = 0.05.
5. Step 5: Compare the calculated t-value to the critical t-value. If the calculated t-value is greater than the critical t-value, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the problem.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0 in favor of H1. In this case, the null hypothesis would state that there is no difference in mean pain reduction between the magnet and sham treatments.

Independent Samples

Independent samples refer to two or more groups that are not related or paired in any way. In this context, the pain reduction scores from the magnet treatment group and the sham treatment group are independent, meaning the outcome of one group does not affect the other. This independence is crucial for applying certain statistical tests, such as the t-test for comparing means.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether a result is statistically significant. In this scenario, a significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none (Type I error). If the p-value obtained from the hypothesis test is less than 0.05, the null hypothesis can be rejected, suggesting that the magnet treatment may be effective.