The two questions are in green and my answers are read. I hope that helps in reading. I just want make sure that I'm doing this all right. Thanks

Breast cancer is considered largely a hormonal disease. An important hormone in breast-cancer research is estradiol. The data in Table 3.19 on serum estradiol were obtained from 213 breast-cancer cases and 432 age-matched controls. All women were age 50-59 years.


Code:
Table 3.19 Serum -estradiol data
Serum estradiol (pg/mL)                Cases(N=213)     Controls(N=432)
1-4                                       28                      72
5-9                                       96                     233
10-14                                     53                      86
15-19                                     17                      26
20-24                                     10                       6
25-29                                      3                       5
30+                                        6                       4
Suppose a serum-estradiol level of 20+ pg/mL is proposed as a screening criterion for indentifying breast-cancer cases.

3.138 What is the sensitivity of this test?

(19 breast cancer cases with 20+pg/mL serum-estradiol levels)/(213 total breast cancer cases)
=19/213= 0.0892 ≈9%


3.139 What is the specificity of this test?

(417 controls with less than 20 pg/mL serum-estradiol levels)/(432 total controls)
417/432= 0.9653 ≈97%


The preceding sample was selected to oversample cases. In the general population, the prevalence of breast cancer is about 2% among women 50 to 59 years old.

3.140 What is the probability of breast cancer among 50 to 59 year old women in the general population who have a serum-estradiol level of ≥20 pg/mL? What is another name for this quantity?


B= has cancer,A = 20+ serum level
P(B) = 0.02
P(A) = 0.035
P(A|B) = sensitivity = 0.09
P(A ̅|B ̅) = specificity = 0.97

=(Pr⁡(A│B) Pr⁡(B))/(Pr⁡(A│B) Pr⁡(B)+Pr⁡(A│B ̅ ) Pr⁡(B ̅))
(0.09*0.02)/((0.09*0.02)+(0.03*0.98))= 0.0577 ≈6%