Problem 7SE

Leakage from underground fuel tanks has been a source of water pollution. In a random sample of 87 gasoline stations, 13 were found to have at least one leaking underground tank.

a. Find a 95% confidence interval for the proportion of gasoline stations with at least one leaking underground tank.

b. How many stations must be sampled so that a 95% confidence interval specifies the proportion to within ±0.03?

Answer:

Step 1 of 2:

(a)

In this question, we are asked to find a 95% confidence interval for the proportion of gasoline stations with at least one leaking underground tank.

In a random sample of 87 gasoline stations, 13 were found to have at least one leaking underground tank.

Let be the number of success in independent Bernoulli trials with success probability , so that .

Define and Then a level confidence interval for p (proportion) is

………..(1)

Let represent the gasoline stations with at least one leaking underground tank.

The number of successes is , and the number of trials is , then we can say follows the Binomial distribution.

We therefore compute,

,

=

For a 95% confidence interval, the value of is,

and

Then for is , hence

The 95% confidence interval is therefore from equation (1),

(0.0886, 0.241)

Hence 95% confidence interval is (0.0886, 0.241).