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.


c. Does it appear that magnets are effective in treating back pain? Is it valid to argue that magnets might appear to be effective if the sample sizes are larger?

Verified step by step guidance

1. Step 1: Identify the problem as a two-sample t-test for independent samples. The goal is to determine if there is a significant difference in the mean reduction in pain levels between the magnet treatment group and the sham treatment group.
2. Step 2: Write down the null hypothesis (H₀) and the alternative hypothesis (Hₐ). H₀: μ₁ = μ₂ (no difference in mean pain reduction between the two groups). Hₐ: μ₁ ≠ μ₂ (there is a difference in mean pain reduction).
3. Step 3: Calculate the test statistic using the formula for a two-sample t-test: t = (x̄₁ - x̄₂) / √((s₁²/n₁) + (s₂²/n₂)), where x̄₁ and x̄₂ are the sample means, s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes.
4. Step 4: Determine the degrees of freedom (df) using the smaller of n₁ - 1 and n₂ - 1, or use a more precise formula for unequal variances if required. In this case, n₁ = n₂ = 20, so df = 19.
5. Step 5: Compare the calculated t-value to the critical t-value from the t-distribution table at the chosen significance level (e.g., α = 0.05) with df = 19. Alternatively, calculate the p-value and compare it to α. If the t-value exceeds the critical value or the p-value is less than α, reject H₀ and conclude that there is a significant difference. Otherwise, fail to reject H₀.

Key Concepts

Independent Samples

Independent samples refer to two or more groups that are not related or paired in any way. In this context, the magnet treatment and sham treatment groups are independent, meaning the results from one group do not influence the other. This is crucial for statistical tests that compare means, as it ensures that the observations in one sample do not affect the observations in the other.

Visual Analog Scale (VAS)

The Visual Analog Scale (VAS) is a measurement instrument used to assess subjective characteristics or attitudes that cannot be directly measured. In this study, it is used to quantify pain levels, where higher scores indicate greater pain. Understanding how VAS works is essential for interpreting the results of the study and determining the effectiveness of the treatments.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, researchers would formulate a null hypothesis (e.g., magnets have no effect on pain) and an alternative hypothesis (e.g., magnets reduce pain). The results from the independent samples will be analyzed to determine if there is enough evidence to reject the null hypothesis, which is critical for concluding the effectiveness of the magnet treatment.