Many diagnostic tests for detecting the presence of disease are based on biomarkers;

broadly speaking, a biomarker is any biological indicator that can be measured, such as

blood pressure or fasting blood glucose level.

Consider a diagnostic test for a hypothetical disease based on measuring the amount of a

certain biomarker present in blood. High levels of the biomarker are often found in individuals

with the disease, but a number of non-disease conditions can also cause high levels

of the biomarker. Individuals without the disease have biomarker levels that are normally

distributed with mean 1.6 ng/mL (nanograms per milliliter of blood), and standard deviation

0.50 ng/mL. Individuals with the disease have biomarker levels that are normally

distributed with mean 5 ng/mL and standard deviation 1.2 ng/mL. Values of 2.5 ng/mL

and higher constitute a positive test result.

iv. Clinicians would like to increase the probability from part ii. by changing the cutoff

value that constitutes a positive test result. Suggest a reasonable new cutoff value x,

such that values of x ng/mL and higher qualify as a positive test result. Explain the

reasoning behind your answer. Use R code for answer