Final answer:
To answer the student's question, a two-way table was constructed based on the given probabilities, and the probability of a positive test result was calculated to be 0.058 or 5.8% when assuming a study of 2000 subjects.
Step-by-step explanation:
The question involves constructing a two-way table and calculating the probability related to the screening of Down syndrome for women who conceive at age 40. Given that the probability of a child having Down syndrome is 0.01 for a woman at age 40, and a first trimester test for Down syndrome has a 5% false positive rate and is 85% accurate in identifying Down syndrome when it is actually present, we can create a two-way table with these probabilities based on a study involving 2000 subjects.
Constructing the Two-Way Table
First, we must calculate the expected numbers:
- Number of children with Down syndrome: 0.01 × 2000 = 20
- Number of children without Down syndrome: (1 - 0.01) × 2000 = 1980
From the 20 children with Down syndrome, the test is expected to correctly identify 85% of them, so:
- True positives: 0.85 × 20 = 17
- False negatives (15% missed): 0.15 × 20 = 3
Out of the 1980 without the condition, 5% will have a false positive:
- False positives: 0.05 × 1980 = 99
- True negatives: 1980 - 99 = 1881
Now we can present this data in a two-way table:
Down SyndromeNo Down SyndromePositive Test1799Negative Test31881
Probability Calculation
To find the probability of a positive test result P(Positive), we add the true positives to the false positives and divide by the total number of subjects:
P(Positive) = (True Positives + False Positives) / Total Subjects
P(Positive) = (17 + 99) / 2000
P(Positive) = 116 / 2000 = 0.058