Chapter 7, Problem 47P

Suppose a new test is proposed to identify patients with a specific condition.

One specimen is obtained from each of 220 subjects in the population to whom the test will be applied. Fifty-one (51) subjects have the condition and 169 do not, according to a gold standard used to assess the presence of the condition (referred to as a reference standard below). The following results are obtained:

Suppose a quality score is constructed for the test. A true positive identified by the test is given a score of +20. A

true negative identified by the test is given a score of +10. A false positive identified by the test is given a score of -5. A false negative identified by the test is given a score of-20.

The FDA will approve the new test if the lower bound of a 90% Cl for the mean score is $\ge$+5. Should the FDA approve the test? A yes/no answer is not sufficient.

Hint: Assume that the score distribution is approximately normal or that the central limit theorem is applicable.

Hint: Assume that a t distribution with > 200 df is approximately normal.

Hint: To compute the mean and variance from grouped data, use the formulas

$\bar{x}={\displaystyle \underset{i=1}{\overset{k}{\sum }}{f}_i}{x}_i/{\displaystyle \underset{i=1}{\overset{k}{\sum }}{f}_1}$

$s^2=\frac{{\displaystyle {\sum }_{i=1}^k{f}_i{x}_i^2}-\frac{({\displaystyle {\sum }_{i=1}^k{f}_i{x}_i{)}^2}}{N}}{N-1}$

where x_i is the ith score value, f_i is the frequency of the ith score value, k = number of score values, and

$N={\displaystyle {\sum }_{i=1}^k{f}_i=220}$