Confidence Intervals

Worksheet #8

STA 291 (022-024)

3/19/08

Confidence Intervals

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.

A confidence interval is based on three elements:

(a) a value of a point estimator (the sample mean, etc.)
(b) the standard error of the point estimator; and
(c) the critical value of Z or t (e.g., the 95% confidence interval or the 99% confidence interval).

Confidence Intervals for a single Mean (μ)

normal population with σ known

σ unknown and n ≥ 30

σ unknown and n < 30

Sample Size

Sample size needed to construct a 100(1-α)% confidence interval for μ with a margin of error of B.

Examples

A survey of 100 retailers revealed that the mean after-tax profit was $90,000 with a standard deviation of $15,000. Determine the 95% confidence interval of the mean after-tax profit for all retailers.

During a water shortage, a water company randomly sampled residential water meters in order to monitor daily water consumption. On a particular day, a sample of 100 meters showed a sample mean of 250 gallons. It is already known that the standard deviation of daily water consumption is 50 gallons. Provide a 90% confidence interval of the mean water consumption for the population.

The number of rooms rented daily during the month of February (28 days) has a mean of 37 rooms per day and a standard deviation of 23 rooms per day. Use this information to estimate the number of rooms rented daily during February with 95% confidence.

A statistician wants to estimate the mean weekly family expenditure on clothes. He believes that the standard deviation of the weekly expenditure is $125. Determine with 95% confidence the number of families that must be sampled to estimate the mean weekly family expenditure on clothes to within $15.