Question

# = 8

1. Suppose next that we have even less knowledge of our patient, and we are only given the accuracy of the blood test and prevalence of the disease in our population. We are told that the blood test is 9# percent reliable, this means that the test will yield an accurate positive result in 9#% of the cases where the disease is actually present. Gestational diabetes affects #+1 percent of the population in our patient’s age group, and that our test has a false positive rate of #+4 percent. Use your knowledge of Bayes’ Theorem and Conditional Probabilities to compute the following quantities based on the information given only in part 2:
   1. If 100,000 people take the blood test, how many people would you expect to test positive and actually have gestational diabetes?
   2. What is the probability of having the disease given that you test positive?
   3. If 100,000 people take the blood test, how many people would you expect to test negative despite actually having gestational diabetes?
   4. What is the probability of having the disease given that you tested negative?
   5. Comment on what you observe in the above computations. How does the prevalence of the disease affect whether the test can be trusted?

Fill in the conditional probability table here, then answer the questions in each part below.

1. Answer part (a) here.

2. Answer part (b) here.

3. Answer part (c) here.

4. Answer part (d) here.

5. Comment on how prevalence of the disease affects your ability to trust the test. Discuss what factors would lead you to trust the blood test, or not trust the blood test.

Answer 1

Suppose, D denotes the event that a person is affected by gestational diabetes whereas + denotes the event that a person is tested positive and - denotes the event that a person is tested negative.

So, from the given data we have as follows.

  

  

(a)

Expected number of persons affected by gestational diabetes = 100000*0.01 = 1000

Expected number of persons not affected by gestational diabetes = 100000*0.99 = 99000

Expected number of persons to test positive who are affected by gestational diabetes = 1000*0.09 = 90

Expected number of persons to test positive who are not affected by gestational diabetes = 99000*0.04 = 3960

So, expected number of persons to test positive = 90+3960 = 4050

Hence, 4050 people are expected to test positive and 1000 people are expected to have gestational diabetes.

(b)

Using Bayes' theorem required probability is given by

Hence, probability of having the disease is 0.02222222 when it is tested positive.

(c)

Expected number of persons affected by gestational diabetes = 100000*0.01 = 1000

Expected number of persons to test negative who are affected by gestational diabetes = 1000*0.91 = 910

Hence, 910 people are expected to test negative despite of actually having gestational diabetes.

(d)

Using Bayes' theorem required probability is given by

Hence, probability of having the disease is 0.009484106 when it is tested negative.

(e)

From the above calculations, we find as follows.

• Whenever a person has gestational diabetes, this test is an absolute failure to detect that.
• Whenever a person does not have gestational diabetes, this test is successful enough to detect that.

Hence, about the test we have trust or not as follows.

• When test shows positive result, we can not trust the test.
• When test shows negative result, we can trust the test.