Question

6. Normal cytokine count for non disease carrying patients are found to be distributed normally with a mean and standard deviation of 55 and 10 respectively. If a patient is found to have a cytokine count below two standard deviations from the mean (<35) they are declared disease positive.1

Unbeknownst to anyone (except you), disease carrying patients have a normally distributed cytokine count with mean and standard deviation of 20 and 10 respectively

.i) What proportion of disease carrying patients who are tested will not be declared disease positive? (find the false negative rate).

ii) What proportion of non disease carrying patients tested will falsely be declared disease positive? (find the false positive rate)

Answer 1

i)

Proportion of disease carrying patients who are tested but will not be declared disease positive = P(cytokine count for disease carrying patient is greater than 35)

= P{Z > (35 - 20)/10} = P(Z > 1.5) = 0.0668

The false negative rate = 0.0668 = 6.68%

ii) Proportion of non disease carrying patients tested but falsely declared as disease positive = P(cytokine count for non disease carrying patient is smaller than 35)

= P{Z < (35 - 55)/10}

= P(Z < -2)

= 0.0228 = 2.28%

The false positive rate = 2.28%