A standard test for cholesterol is based on the total serum cholesterol level. Y

Question

A standard test for cholesterol is based on the total serum cholesterol level. Your total serum cholesterol consists of high-density lipoprotein, or HDL cholesterol; low-density lipoprotein, or LDL cholesterol; and triglycerides. Levels are reported in milligrams per deciliter of blood, or mg/dL. The total serum choles- terol level, in mg/DL, is approximately Normally distributed: with mean 175 and standard deviation 40 among healthy individuals; and with mean 250 and standard deviation 38 among individuals with high cholesterol. (a). Suppose a test is set up in such a way that people are diagnosed as having high cholesterol if their total serum cholesterol is greater than 235 mg/DL. i Find the sensitivity of this test. ii Find the specificity of this test. iii If 1000 randomly selected 18-24 year old Australians were subjected to this test, how many would you expect to be incorrectly diagnosed? You may assume the prevalence of high total cholesterol is 10% in this population. Show full working and reasoning. iv For this test, what seems to be a priority getting it right for people with healthy cholesterol levels or getting it right for people with high cholesterol levels? Support your answer by referring to answers to previous part/s of this question.

Explanation / Answer

Solution:

(i) Sensitivity = True positives/(True positive + False negative) = number of true positives/ number of positive samples

True positives in this case wll be the persons who's cholestrol level is being detected above 235 mg/DL in the persons with high cholestral problems.

And false negative in this case will be the persons who's cholestrol level is being detected below 235 mg/DL even if they are the persons with high cholestral problems.

So Here sensitivity ss= Pr ( X >= 235 mg/Dl ; 250; 38) by normal distribution

Z = (235 - 250)/38 = - 0.3947

so Pr ( X >= 235 mg/Dl ; 250; 38) = 1 - (-0.395) = 1 - 0.3446 = 0.655

(ii) Specificity = True negatives/ True negatives + False positives

True negatives in this case wll be the persons who's cholestrol level is being detected below 235 mg/DL in the healthy persons.

And false positives in this case will be the persons who's cholestrol level is being detected sbove 235 mg/DL even they are healthy persons.

Specificity sp= Pr ( X <= 235 mg/Dl ; 175; 40) by normal distribution

Z = (235 -175)/40 = 1.5

so Pr ( X <= 235 mg/Dl ; 250; 38) = (1.5) = 0.9332

(iii) If we select 1000 Australians , Expected they are incorrectly diagnosed. Here 10% that means 100 are having high cholestrol and 900 are healthy.

Incorrectly diagonesed = P( detected healthy/ Having High cholestrol) + P( detected high cholestrol/ is healthy)

                                    = 100 × ( 1- ss) + 900 × (1 - sp) = 100 * ( 1 - 0.655) + 900 × (1- 0.9332) = 34.5 + 60.12 = 94.62 or say 95 people are incorrectly diagnosed.

(iv) It is obvious that priority would be getting it right for people with high cholestrol level because sensitivity of this teest is only 65% so only 65% of diseased peopleget their test done right. Second thing is diseased people must be given priority so they can have treatment. They must correctly know that they have the disease. Thirs the health person have a good rate of non - detection of 94% so there are only 6% chance that the test will fail in their case.

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