Question

Consider the next 1000 98% CIs for μ that a statistical consultant will obtain for various clients. Suppose the data sets on which the intervals are based are selected independently of one another. How many of these 1000 intervals do you expect to capture the corresponding value of μ?

Answer 1

980 intervals.

Explanation:

For each interval, there are only two possible outcomes. Either it captures the population mean, or it does not. One interval is independent of other intervals. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected value of the binomial distribution is:

`E(X) = np`

98% confidence interval

Has a 98% probability of capturing the population mean, so
`p = 0.98`

1000 intervals

This means that
`n = 1000`

How many of these 1000 intervals do you expect to capture the corresponding value of μ?

`E(X) = np = 1000*0.98 = 980`

980 intervals.