Can someone please check my working out - 2nd year stat/probability question

[laura_a]

Homework Statement

Can someone please check my answers for this probability question about disease testing (negative/positives)?
I will write the question on about below will write my answers, I would appreciate anyone letting me know if I'm right/wrong. Thanks
Q. A test for a disease has a false negative rate of 5%, false positive rate of 15%. The prevalance of the disease is 1.8%
b) if the popluation were screened, what proportion will test negative to disease
c) what proportion of those testing negative will actually HAVE the disease
d) explain why this proportion is so small

The Attempt at a Solution

b) P(not disease) U P(false neg)

P(Disease) = 18/1000
P(not disease) = 982/1000
p(false neg) = 5/100

P (not D) U (false neg) = 982/1000 + 5/100 - (982/1000 * 5/100)
= 9829/10,000

This means 9829/10000 or 98.29% will test negative
c) how many of these actually have the disease... well I assume it is just 5% of that number because the given details say 5% test false neg.. so 5% * 98.29 = 4.91%

d) the proportion is 5% of just under 100 so that is why it is lower than the given 5% ?? not sure about d)

 

[HallsofIvy]

Homework Statement

Can someone please check my answers for this probability question about disease testing (negative/positives)?
I will write the question on about below will write my answers, I would appreciate anyone letting me know if I'm right/wrong. Thanks
Q. A test for a disease has a false negative rate of 5%, false positive rate of 15%. The prevalance of the disease is 1.8%
b) if the popluation were screened, what proportion will test negative to disease
c) what proportion of those testing negative will actually HAVE the disease
d) explain why this proportion is so small

The Attempt at a Solution

b) P(not disease) U P(false neg)

P(Disease) = 18/1000
P(not disease) = 982/1000
p(false neg) = 5/100

P (not D) U (false neg) = 982/1000 + 5/100 - (982/1000 * 5/100)
= 9829/10,000

This means 9829/10000 or 98.29% will test negative

I don't understand why you did that calculation. In a population of 1000 people (to avoid the fractions), 18 will have the disease and 982 will not. Of the 982 who do not, 5%= .05(982()= 49.1 will have a false negative. Of the 18 who have the disease, 15%= .15(18)= 2.7 will have a false positive so the other 15.3 will report negative. There will be a total of 49.1+ 15.3= 64.4 negatives out of 1000 or 6.44%

c) how many of these actually have the disease... well I assume it is just 5% of that number because the given details say 5% test false neg.. so 5% * 98.29 = 4.91%

5% of the people who actually have the disease will get a false negative, not "5% of the negatives will be false". 5% of 18= 2.7 out of 1000 or 0.5% will have a false negative as in the caculation above.

[/quote]d) the proportion is 5% of just under 100 so that is why it is lower than the given 5% ?? not sure about d)[/QUOTE]
It is so small, because the number of people who actually have the disease is very small compared with the number who do not. The number of false positives is much larger than the number of false negatives because it is based on a larger population.

 

[HallsofIvy]

Homework Statement

Can someone please check my answers for this probability question about disease testing (negative/positives)?
I will write the question on about below will write my answers, I would appreciate anyone letting me know if I'm right/wrong. Thanks
Q. A test for a disease has a false negative rate of 5%, false positive rate of 15%. The prevalance of the disease is 1.8%
b) if the popluation were screened, what proportion will test negative to disease
c) what proportion of those testing negative will actually HAVE the disease
d) explain why this proportion is so small

The Attempt at a Solution

b) P(not disease) U P(false neg)

P(Disease) = 18/1000
P(not disease) = 982/1000
p(false neg) = 5/100

P (not D) U (false neg) = 982/1000 + 5/100 - (982/1000 * 5/100)
= 9829/10,000

This means 9829/10000 or 98.29% will test negative

I don't understand why you did that calculation. In a population of 1000 people (to avoid the fractions), 18 will have the disease and 982 will not. Of the 982 who do not, 5%= .05(982()= 49.1 will have a false positive and the other 932.9 will test negative. Of the 18 who have the disease, 15%= .15(18)= 2.7 will have a false negative. There will be a total of 932.9+ 2.7= 936.6 negatives out of 1000 or 93.66%

c) how many of these actually have the disease... well I assume it is just 5% of that number because the given details say 5% test false neg.. so 5% * 98.29 = 4.91%

5% of the people who actually have the disease will get a false negative, not "5% of the negatives will be false". 5% of 18= 2.7 out of 1000 or 0.5% will have a false negative as in the caculation above.

[/quote]d) the proportion is 5% of just under 100 so that is why it is lower than the given 5% ?? not sure about d)[/QUOTE]
It is so small, because the number of people who actually have the disease is very small compared with the number who do not. The number of true negatives is much larger than the number of false negatives because it is based on a larger population.