Blood plasma proteins of patients with malignant breast tumo

Blood plasma proteins of patients with malignant breast tumors differ from proteins of healthy people in their solubility in the presence of various polymers. When the polymers dextran and poly(ethylene glycol) are mixed with water, a two-phase mixture is formed. When plasma proteins of tumor patients are added, the distribution of proteins between the two phases is different from that of plasma proteins of a healthy person. The distribution coefficient (K) for any substance is defined in the equation below.

K = [concentration of the substance in phase A] / [concentration of the substance in phase B]

Proteins of healthy people have a mean distribution coefficient of 0.75 with a standard deviation of 0.07. For the proteins of people with cancer, the mean is 0.92 with a standard deviation of 0.11.

(a) Suppose that K were used as a diagnostic tool and that a positive indication of cancer is taken as K ? 0.92. What fraction of people with tumors would have a false negative indication of cancer because K < 0.92?

(b) What fraction of healthy people would have a false positive indication of cancer? This number is the fraction of healthy people with K ? 0.92, shown by the shaded area in the graph below. Estimate an answer with using this table. (Enter your answer as a percentage.)

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Solution

(a)

For the people with tumours, the mean value of K is equal to 0.92

This means that half of the people with tumours will have a K value greater than 0.92, while the other half will be having values less than 0.92

False negatives means that the test shows K < 0.92 for a person, although that person actually has cancer.

So the proportion of false negatives in the people with tumors = 0.5

(b)

For the healthy people we have:

Mean, m = 0.75

Standard deviation, S = 0.07

For a false positive, the value of K should be greater than 0.92

We first calculate the z-value:

z = (X-m)/S = (0.92-0.75)/0.07 = 2.429

So, proportion of false positives = Proportion of healthy people who have K value greater than 0.92 = P(z > 2.429) = 0.0076