Question

In: Statistics and Probability

A scientist designed a medical test for a certain disease. Among 100 patients who have the...

A scientist designed a medical test for a certain disease. Among 100 patients who have the disease, the test will show the presence of the disease in 96 cases out of 100, and will fail to show the presence of the disease in the remaining 4 cases out of 100. Among those who do not have the disease, the test will erroneously show the presence of the disease in 3 cases out of 100, and will show there is no disease in the remaining 97 cases out of 100.

What is the probability that a patient who tested positive on this test actually has the disease, if it is estimated that 20% of the population has the disease?

A.

0.9074 or 90.74%

B.

0.04 or 4%

C.

0.96 or 96%

D.

0.9407 or 94.07%

Solutions

Expert Solution

P[ A person has disease ] = 20% = 0.2

P[ A person does not have disease ] = 1 - P[ A person has disease ] = 1 - 0.2 = 0.8

P[ the test is positive | Patient actually has the disease ] = 96/100 = 0.96

P[   the test is positive | Patient does not have the disease ] = 3/100 = 0.03

P[ Patient actually has the disease | the test is positive ] = P[ Patient actually has the disease and the test is positive ] / P[ the test is positive ]

P[ the test is positive ] = P[ Patient actually has the disease and the test is positive ] + P[ Patient does not the disease and the test is positive ]

P[ Patient actually has the disease and the test is positive ] = P[ the test is positive | Patient actually has the disease ]*P[ A person has disease ]

P[ Patient actually has the disease and the test is positive ] = 0.96*0.2

P[ Patient actually has the disease and the test is positive ] = 0.192

P[ Patient does not the disease and the test is positive ] = P[   the test is positive | Patient does not have the disease ] *P[ A person does not have disease ]

P[ Patient does not the disease and the test is positive ] = 0.03*0.8

P[ Patient does not the disease and the test is positive ] = 0.024

P[ the test is positive ] = 0.024 + 0.192

P[ the test is positive ] = 0.216

P[ Patient actually has the disease | the test is positive ] = 0.192/0.216

P[ Patient actually has the disease | the test is positive ] = 0.8889


Related Solutions

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)...
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?
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...
Medical Case Study: Testing for Kidney Disease Patients with kidney disease often have protein in their...
Medical Case Study: Testing for Kidney Disease Patients with kidney disease often have protein in their urine. While small amounts of protein are not very worrisome, more than 1 gram of protein excreted in 24 hours warrants active treatment. The most accurate method for measuring urine protein is to have the patient collect all his or her urine in a container for a full 24-hour period. The total mass of protein can then be found by measuring the volume and...
A certain medical test is known to detect 72% of the people who are afflicted with...
A certain medical test is known to detect 72% of the people who are afflicted with the disease Y. If 10 people with the disease are administered the test, what is the probability that the test will show that: All 10 have the disease, rounded to four decimal places? At least 8 have the disease, rounded to four decimal places? At most 4 have the disease, rounded to four decimal places?
A certain medical test is known to detect 73% of the people who are afflicted with...
A certain medical test is known to detect 73% of the people who are afflicted with the disease Y. If 10 people with the disease are administered the test, what is the probability that the test will show that: All 10 have the disease, rounded to four decimal places? At least 8 have the disease, rounded to four decimal places? At most 4 have the disease, rounded to four decimal places? Assume that 39% of people are left-handed. If we...
A certain medical test is known to detect 50% of the people who are afflicted with...
A certain medical test is known to detect 50% of the people who are afflicted with the disease Y. If 10 people with the disease are administered the test, what is the probability that the test will show that: All 10 have the disease, rounded to four decimal places? At least 8 have the disease, rounded to four decimal places? At most 4 have the disease, rounded to four decimal places?
A certain medical test is known to detect 49% of the people who are afflicted with...
A certain medical test is known to detect 49% of the people who are afflicted with the disease Y. If 10 people with the disease are administered the test, what is the probability that the test will show that: All 10 have the disease, rounded to four decimal places? At least 8 have the disease, rounded to four decimal places? At most 4 have the disease, rounded to four decimal places?
A certain medical test is known to detect 38% of the people who are afflicted with...
A certain medical test is known to detect 38% of the people who are afflicted with the disease Y. If 10 people with the disease are administered the test, what is the probability that the test will show that: All 10 have the disease, rounded to four decimal places? At least 8 have the disease, rounded to four decimal places? At most 4 have the disease, rounded to four decimal places?
A turfgrass scientist is looking for an optimal approach to control a certain plant disease in...
A turfgrass scientist is looking for an optimal approach to control a certain plant disease in Kentucky blue grass. He compares three different management strategies designed so that they would reduce spread of the disease. He has set up a field study with a total of 20 experimental plots and 4 treatments (3 disease prevention treatments and a control treatment. Each treatment has been assigned to 5 randomly selected experimental plots. Plant biomass is then measured from each pot at...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT