Question

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 longterm 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 ). What hypotheses do you think the researchers tested in order to reach the given conclusion? Did the researchers reject or fail to reject the null hypothesis?

Short answer

The researchers tested the null hypothesis \(H_0: p_m = p_l\), where \(p_m\) and \(p_l\) represent the proportions of women surviving for 20 years after mastectomy and lumpectomy (with radiation), respectively. They failed to reject the null hypothesis, pointing to no statistically significant difference in the survival rate between the two treatment methods based on their study.

Step-by-step solution

Identify Null Hypothesis

The null hypothesis would state that there is no statistically significant difference in the survival rate for 20 years between the two treatment methods. It can be formulated as: \(H_0: p_m = p_l\) where \(p_m\) is the proportion of women surviving for 20 years after mastectomy and \(p_l\) is the proportion of women surviving for 20 years after lumpectomy with radiation.

Identify Alternative Hypothesis

The alternative hypothesis would assert that there is a difference in the survival rate. It could state either method is more effective than the other, but based on the information given, no directionality is indicated. So it can be stated as: \(H_1: p_m ≠ p_l\) where \(H_1\) represents the alternate hypothesis.

Conclusion about Null Hypothesis

From the report it's known that there is no 'statistically significant difference' in the survival rate of the two treatment methods. When there's no statistically significant difference, we 'fail to reject the null hypothesis'. Therefore, the researchers failed to reject the null hypothesis, meaning they did not find any significant evidence to support that one treatment method is more effective than the other in terms of 20-year survival rate.