Question

14.32 false positives in testing for hiv. a rapid test for the presence in the blood of antibodies to hiv, the virus that causes aids, gives a positive result with probability about 0.004 when a person who is free of hiv antibodies is tested. a clinic tests 1000 people who are all free of hiv antibodies. a. what is the distribution of the number of positive tests? b. what is the mean number of positive tests? c. you cannot safely use the normal approximation for this distribution. explain why.

Explanation:

Step1: Identify the distribution

A binomial distribution \(X\sim B(n,p)\) has \(n\) independent trials, each with probability of success \(p\). Here, \(n = 1000\) (number of people tested) and \(p=0.004\) (probability of a positive test for a person without HIV antibodies). So the distribution of the number of positive tests \(X\) is \(X\sim B(1000,0.004)\)

Step2: Calculate the mean of a binomial distribution

The mean of a binomial distribution \(\mu=np\). Substitute \(n = 1000\) and \(p = 0.004\) into the formula: \(\mu=1000\times0.004\)

Step3: Check the Normal approximation conditions

For a binomial \(X\sim B(n,p)\), the Normal approximation \(X\sim N(np,np(1 - p))\) is appropriate when \(np\geq5\) and \(n(1 - p)\geq5\). Calculate \(np=1000\times0.004 = 4<5\)

Answer:

a. The distribution of the number of positive tests is a binomial distribution \(X\sim B(1000,0.004)\)
b. The mean number of positive tests is \(\mu = 4\)
c. The Normal approximation cannot be used because \(np=4<5\) (one of the conditions \(np\geq5\) for using the Normal approximation to the binomial is not met)