Question

In: Statistics and Probability

Assume there is a medical test to diagnose a disease. If a person has the disease,...

Assume there is a medical test to diagnose a disease. If a person has the disease, the probability of having positive test result is 98 percent. If a person does not have the disease, the probability of having negative test results is 99.6 percent. The probability that a person has a disease is 1 percent in the population.

Answer the following questions:

a) If a person has a positive test result, what is the probability that he/she has the disease?

b) If a person has a positive test result, what is the probability that s/he doesn’t have the disease?

c) If a person has a negative test result, what is the probability that he/she doesn’t have the disease?

d) If a person has a negative test result, what is the probability that s/he has the disease?

Note: Use the following notation in your answer:

D: Person with disease

ND: Person without disease

+T: Positive test result

- T: Negative test result

Write each question in the form of mathematical notation for conditional probability.

Calculate the answer using two methods:

1. Bayes’ rule and conditional probability equations.

2. Draw a table, assume a population (e.g. 1 million) and provide numerical answers

Expert Solution

Has the disease Does not have the disease Total

Test positive 9800 3960 13760

Test negative 200 986040 986240

Total 10000 990000 1000000

a) P(has the disease | Test positive) = P(has the disease and test positive )/P(Test Positive)

= 9800/13760 = 0.7122

b) P(Doesn't have the disease | Test positive) = P(Doesn't have the disease and test positive)/P(Test positive) = 3960/13760 = 0.2878

c) P(Doesn't have the disease | Test negative) = P(Doesn't have the disease and test negative)/P(Test negative) = 986040/986240 = 0.9998

d) P(has the disease | Test negative) = P(has the disease and test negative)/P(Test negative)

= 200/986240 = 0.0002

orchestra answered 2 years ago

A test to diagnose heart disease is correct 99% of the time if the patient has...

A test to diagnose heart disease is correct 99% of the time if the patient has the disease, and 97% correct in its diagnosis if the patient does not have the disease. Only 2% of the population has this heart disease. a).If a patient is randomly selected from the population to perform the test, what is the probability that the disease will be diagnosed? What do you think of that result? b). If the disease is diagnostic, what is the...

Assume that there is a test for a person having antibodies for COVID-19 disease. Let A...

Assume that there is a test for a person having antibodies for COVID-19 disease. Let A denote a test result that shows that a person has antibodies for COVID-19 disease. Let B denote that a person truly has antibodies for COVID-19. Such a person is thought to be immune to again contracting and spreading COVID-19 because that person already had the disease. Assume that the sensitivity of a test for antibodies is .938 and that the specificity of the test...

Sensitivity is the probability that a test will identify a person who has the disease you...

Sensitivity is the probability that a test will identify a person who has the disease you are testing for. Question 9 options: True False Which of the following statements is NOT true when describing Specificity? Question 10 options: A) It measures the effectiveness of a test in returning a negative test result to people who do not have the disease. B) It is the ability of a test to correctly identify people who have the disease you are testing for....

Suppose that a medical test for a certain disease has a sensitivity and specificity of 93%....

Suppose that a medical test for a certain disease has a sensitivity and specificity of 93%. The test is applied to a population of which 11% are actually infected by the disease. 1. Calculate the NPV and the PPV. 2. What percent of the total population will test positive who are disease-free? 3. What percent of the total population will test negative who have the disease?

Assume that on a standardized test of 100 questions, a person has a probability of 80%...

Assume that on a standardized test of 100 questions, a person has a probability of 80% of answering any particular question correctly. Find the probability of answering between 70 and 80 questions, inclusive. (Assume independence, and round your answer to four decimal places.) P(70 ≤ X ≤ 80) =

A new medical test has been designed to detect the presence of the mysterious Brainlesserian disease....

A new medical test has been designed to detect the presence of the mysterious Brainlesserian disease. Among those who have the disease, the probability that the disease will be detected by the new test is 0.78. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is 0.02. It is estimated that 19 % of the population who take this test have the disease. If the test administered...

A medical test has been designed to detect the presence of a certain disease. Among people...

A medical test has been designed to detect the presence of a certain disease. Among people who have the disease, the probability that the disease will be detected by the test is 0.93. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is 0.03. It is estimated that 3% of the population who take this test have the disease. (Round your answers to three decimal places.) (a)...

QUESTION 1 If a person doesn’t have a disease and they test negative for it, it...

QUESTION 1 If a person doesn’t have a disease and they test negative for it, it is an example of: False positive True Positive True negative False negative QUESTION 2 When collecting data with multiple researchers, how a researcher views the data can affect the outcome of the study. To avoid this issue a researcher should assess: Sensitivity Homogeneity Inter-Rater reliability Content validity QUESTION 3 The likelihood of a sick patient testing positive for a disease is: Specificity Reliability Predictive...

Suppose there was no correlation between the Test Screen and the medical assessment of disease (i.e....

Suppose there was no correlation between the Test Screen and the medical assessment of disease (i.e. the test was not able to differentiate between those with or without the disease). Based on the table given above, how many true positives among the sample of 100 do you expect the Test Screen to reveal? Based on this outcome and the observed values given in the initial table above, comment on the association between Test Screen and disease status. Medical...

For a medical test, the probability of testing positive for a healthy person is 5%. Based...

For a medical test, the probability of testing positive for a healthy person is 5%. Based on this, can we find the probability of testing positive for an infected person? If yes, explain how; if no, explain why.