A soft-drink manufacturer purchases aluminum cans from an outside vendor. A random sample of 70 cans is selected from a large shipment, and each is tested for strength by applying an increasing load to the side of the can until it punctures. Of the 70 cans, 52 meet the specification for puncture resistance.

a. Find a 95% confidence interval for the proportion of cans in the shipment that meet the specification.

b. Find a 90% confidence interval for the proportion of cans in the shipment that meet the specification.

c. Find the sample size needed for a 95% confidence interval to specify the proportion to within ±0.05.

d. Find the sample size needed for a 90% confidence interval to specify the proportion to within ±0.05.

e. If a 90% confidence interval is computed each day for 300 days, what is the probability that more than 280 of the confidence intervals cover the true proportions?

Solution:

Step 1 of 4:

A random sample of 70 cans is selected from a large shipment, and each is tested for strength by applying an increasing load to the side of the can until it punctures. It is observed that among 70 cans 52 meet the specification for puncture resistance.

We have to find

- 95 % CI for the proportion of cans in the shipment that meet the specification.
- 90 % CI for the proportion of cans in the shipment that meet the specification
- The sample size needed for 95% CI to specify the proportion with in .
- The sample size needed for 90% CI to specify the proportion with in .
- Probability that more than 280 of the confidence intervals cover the true proportion.