A test for the presence of a certain diseasehas probability .03 of giving a fals

Question

A test for the presence of a certain diseasehas probability .03 of giving a false-positive reading (apositive test for a healthy person) and probability .02 of giving afalse-negative result. Suppose that four individuals are tested,two of whom have the disease and two of whom do not. Let X = thenumber of positive readings. A) Does X have a binomial distribution? Explain yourreasoning. B) What is the probability that at least two of the four testresults are positive? C) Find the expected number of positive tests. A test for the presence of a certain diseasehas probability .03 of giving a false-positive reading (apositive test for a healthy person) and probability .02 of giving afalse-negative result. Suppose that four individuals are tested,two of whom have the disease and two of whom do not. Let X = thenumber of positive readings. A) Does X have a binomial distribution? Explain yourreasoning. B) What is the probability that at least two of the four testresults are positive? C) Find the expected number of positive tests.

Explanation / Answer

a) Let X is the number of positive readings. Since the number ofindividuals are finite and the number of trails are independent.So, we can say that X has a binomial distribution. The probability of success is given by,p=1/2=0.5,q=1-p=1-0.5=0.5. b) P(X=x)=ncx(0.5)x(0.5)n-x P(X>=2)=P(X=2)+P(X=3)+P(X=4) =0.375+0.25+0.0625 =0.6875. c) The expected number of positive tests is np=4(0.5)=2.

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