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.

#### Solution

Let $X$ denote the number of positive tests.

The probability of positive test result is $p=0.004$. And a clinic tests 1000 people who are all free of HIV antibodies. So $n = 1000$.

There are only two possible outcomes, namely positive test result and no positive test results. Also the tests are independent.

(a) The distribution of number of positive tests is Binomial with $n = 1000$ and $p=0.004$. That is, $X\sim B(1000, 0.004)$.

(b) The mean number of positive tests

` $$ \begin{aligned} E(X) &= n*p \\ &= 1000* 0.004\\ &= 4. \end{aligned} $$ `

(c) Here $n*p =4$ and $n*(1-p) = 996$. We cannot use normal approximation for this distribution, because $n*p$ must be $\geq 10$ and $n*(1-p)$ also must be $\geq 10$.

As a rule of thumb, we can use the Normal Approximation when $n$ is so large that $np\geq 10$ and $n(1-p)\geq 10$.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators