Question

14.32 Women diagnosed with breast cancer whose tumors have not spread may be faced with a decision between two surgical treatments-mastectomy (removal of the breast) or lumpectomy (only the tumor is removed). In a long-term study of the effectiveness of these two treatments, 701 women with breast cancer were randomly assigned to one of two treatment groups. One group received mastectomies, and the other group received lumpectomies and radiation. Both groups were followed for 20 years after surgery. It was reported that there was no statistically significant difference in the proportion surviving for 20 years for the two treatments (Associated Press, October 17,2002 ). Suppose that this conclusion was based on a \(90 \%\) confidence interval for the difference in treatment proportions. Which of the following three statements is correct? Explain why you chose this statement. Statement 1: Both endpoints of the confidence interval were negative. Statement 2: The confidence interval included \(0 .\) Statement 3 : Both endpoints of the confidence interval were positive.

Short answer

Statement 2 is correct because a confidence interval that includes 0 indicates that there is no statistically significant difference in the proportions, consistent with the study's reported findings.

Step-by-step solution

Understanding Confidence Intervals

A confidence interval provides a range of values that likely contains the true parameter value in the population. In this case, the parameter is the difference in survival proportions between women receiving different treatments. A 90% confidence interval means that the difference in proportions can lie within this range 90% of the time if the study were to be repeated many times under the same conditions.

Analyzing the Problem Statement

According to the problem, it was reported that there was no statistically significant difference in the proportion surviving for 20 years for the two treatments. This means that the confidence interval for the difference in treatment proportions must include 0. Because if 0 is in the confidence interval, it says that there is a chance that the true difference in proportions is 0, which means there is no difference.

Making the Final Decision

With an understanding of confidence intervals and the problem statement, it is possible to determine which statement is correct. Only Statement 2: 'The confidence interval included 0.' is correct because it indicates that there was no statistically significant difference in the survival proportions.