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.


b. Construct the confidence interval appropriate for the hypothesis test in part (a).

Verified step by step guidance

1. Step 1: Identify the given data for both groups. For the Magnet Treatment group: sample size (n₁) = 20, sample mean (x̄₁) = 0.49, and sample standard deviation (s₁) = 0.96. For the Sham Treatment group: sample size (n₂) = 20, sample mean (x̄₂) = 0.44, and sample standard deviation (s₂) = 1.4.
2. Step 2: Use the formula for the confidence interval for the difference between two means when the population standard deviations are not assumed to be equal. The formula is: CI = (x̄₁ - x̄₂) ± t * √((s₁²/n₁) + (s₂²/n₂)), where t is the critical value from the t-distribution with degrees of freedom calculated using the Welch-Satterthwaite equation.
3. Step 3: Calculate the degrees of freedom (df) using the Welch-Satterthwaite equation: df = ((s₁²/n₁ + s₂²/n₂)²) / (((s₁²/n₁)² / (n₁ - 1)) + ((s₂²/n₂)² / (n₂ - 1))). This will determine the appropriate t-value for the confidence interval.
4. Step 4: Find the critical t-value corresponding to the desired confidence level (e.g., 95%) and the calculated degrees of freedom. Use a t-distribution table or statistical software to find this value.
5. Step 5: Substitute the values for x̄₁, x̄₂, s₁, s₂, n₁, n₂, and the critical t-value into the confidence interval formula to compute the interval bounds. This will give the range within which the true difference in means lies with the specified confidence level.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is calculated using the sample mean, standard deviation, and the critical value from the t-distribution, especially when the sample size is small. The width of the interval reflects the level of uncertainty about the estimate, with wider intervals indicating more uncertainty.

Independent Samples

Independent samples refer to two or more groups that are not related or paired in any way. In statistical analysis, this means that the selection of one sample does not influence the selection of another. This concept is crucial when conducting hypothesis tests or constructing confidence intervals, as it allows for the comparison of different treatments or conditions without bias from interdependence.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis (typically stating no effect or no difference) and an alternative hypothesis. The test assesses the evidence against the null hypothesis using a significance level, leading to a decision to either reject or fail to reject the null based on the calculated p-value or confidence interval.