Question

1. A scientist is investigating whether a new computer-aided scoring system for x-rays does a better job of identifying cancerous tumors than current practice. The scientist has x-rays from 200 known cancer cases and 200 known controls. Using the new test, 175 of the cases and 44 of the controls test positive.

a) Calculate a 95% confidence interval for the sensitivity of the new test.

b) Calculate a 90% confidence interval for the specificity of the new test.

c) Would a 95% confidence interval for the specificity to be wider or narrower than the 90% interval?

Answer 1

True positives = 175

False positive = 44

False negative = 25

True Negative = 156

sensitivity = True positive/(true positive+false negative) = 175/(175+25) = 0.875

specificity = true negative/(true negative+false positive) = 156/(156+44) = 0.78

a)

z_α/2 = 1.9599639861

Lower Bound = p̂ - z_α/2•√p̂(1 - p̂)/n = 0.875 - (1.9599639861)(0.023385358667337135) = 0.8291655392

Upper Bound = p̂ + z_α/2•√p̂(1 - p̂)/n = 0.875 + (1.9599639861)(0.023385358667337135) = 0.9208344608

Confidence Interval = (0.8291655392, 0.9208344608)

b)

z_α/2 = 1.6448536251

Lower Bound = p̂ - z_α/2•√p̂(1 - p̂)/n = 0.78 - (1.6448536251)(0.029291637031753616) = 0.7318195446

Upper Bound = p̂ + z_α/2•√p̂(1 - p̂)/n = 0.78 + (1.6448536251)(0.029291637031753616) = 0.8281804554

Confidence Interval = (0.7318195446, 0.8281804554)

c)

Wider as confidence increases the margin of error also increases