Question

The value of mammography as a screening test for breast cancer has been contro-

versial, particularly among young women. A study was recently performed looking

at the rate of false positives for repeated screening mammograms. The study re-

ported that of a total of 1996 tests given to 40-49 year old women, 156 yielded

false-positive results.

(a) Construct a 95% upper condence bound for the probability of false-positive

result of mammograms.

(b) Some physicians feel a mammogram is not cost-eective unless one can be rea-

sonaly certain that the false-positive rate is less than 10%. Address this is-

sue using the preceding data, using both critical value approach and p-value

method.

Answer 1

a) = 156/1996 = 0.078

For 95% upper confidence interval the critical value is z_0.95 = 1.645

The lower limit of the 95% upper confidence interval is

- z_0.95 * sqrt((1 - )/n)

= 0.078 - 1.645 * sqrt(0.078 * (1 - 0.078)/1996)

= 0.078 - 0.01

= 0.068

So the 95% upper confidence interval is (0.068, )

b) H₀: P = 0.1

    H₁: P < 0.1

The test statistic z = ( - P)/sqrt(P(1 - P)/n)

                              = (0.078 - 0.1)/sqrt(0.1 * 0.9/1996)

                              = -3.28

At = 0.05, the critical value is z_0.05 = -1.645

As the test statistic value is less than the critical value (-3.28 < -1.645), we should reject the null hypothesis.

The P-value = P(Z < -3.28)

                    = 0.0005

Since the P-value is less than (0.0005 < 0.05), we should reject the null hypothesis.