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March 11th, 2011, 09:15 PM   #1
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Challenging Probability and Statistics Questions!

It is known that 5% of the members of a population have disease A, which can be discovered by a blood test.

Suppose that N ( a large no.) people are to be tested
This can be done in 2 ways :
1. Each person is tested separately , or
2. the blood samples of k people are pooled together and analyzed.
( Assume that N =nk , with n an integer)
If the test is -ve , all of them are healthy
( that is , just this 1 test is neeeded)
if the test is +ve , each of the k persons must be tested separately
( that is , a total of k+1 tests are needed)

Questions:
a) For the fixed k, what is the expected no. the tests needed in option 2.
b) Find the k that will minimize the expected no. of tests in option 2.
c) If k is selected as in part b, on the average how many tests does option 2 save in comparison with option1?
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March 12th, 2011, 02:23 PM   #2
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Re: Challenging Probability and Statistics Questions!

Quote:
Originally Posted by GreenBamboo
It is known that 5% of the members of a population have disease A, which can be discovered by a blood test.

Suppose that N ( a large no.) people are to be tested
This can be done in 2 ways :
1. Each person is tested separately , or
2. the blood samples of k people are pooled together and analyzed.
( Assume that N =nk , with n an integer)
If the test is -ve , all of them are healthy
( that is , just this 1 test is neeeded)
if the test is +ve , each of the k persons must be tested separately
( that is , a total of k+1 tests are needed)

Questions:
a) For the fixed k, what is the expected no. the tests needed in option 2.
b) Find the k that will minimize the expected no. of tests in option 2.
c) If k is selected as in part b, on the average how many tests does option 2 save in comparison with option1?
I'll show you how to do a. See if you can do the rest. If you test k people, the probability that none are infected is .95^k (=P). Therefore the expected number of tests is P + (k+1)(1-P).

There is something wrong with the problem statement for b, since k=1 is the obvious answer, but it seems wrong for the underlying problem.
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March 13th, 2011, 07:44 AM   #3
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Re: Challenging Probability and Statistics Questions!

for question (a), the answer given is assume the number of expected no. of test for population which means we need to use N=nk.

for the number of people in sample k have disease A is (k * (1-0.95^k)) is it?

the number of test is k + 1, so is 1+(k * (1-0.95^k))?
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