Studies have found that women diagnosed with cancer in one breast also sometimes have cancer in the other breast that was not initially detected by mammogram or physical examination (“MRI Evaluation of the Contralateral Breast in Women with Recently Diagnosed Breast Cancer,” The New England Journal of Medicine [ 2007]: 1295– 1303). To determine if magnetic resonance imaging (MRI) could detect missed tumors in the other breast, 969 women diagnosed with cancer in one breast had an MRI exam. The MRI detected tumors in the other breast in 30 of these women.
a. Use p = 30/969 = 0.031 as an estimate of the probability that a woman diagnosed with cancer in one breast has an undetected tumor in the other breast. Consider a random sample of 500 women diagnosed with cancer in one breast. Explain why it is reasonable to think that the random variable x = number in the sample who have an undetected tumor in the other breast has a binomial distribution with n = 500 and p = 0.031.
b. Is it reasonable to use the normal distribution to approximate probabilities for the random variable x defined in Part (a)? Explain why or why not.
c. Approximate the following probabilities: i. P(x , 10) ii. P(10 # x # 25) iii. P(x . 20)
d. For each of the probabilities computed in Part (c), write a sentence interpreting the probability.