Confidence Intervals for a Proportion

You’ll be able to

Construct a one-sample z-interval for a population proportion
Interpret a confidence interval and a confidence level correctly
Explain how confidence level and sample size affect the margin of error

Estimate plus or minus a margin

A confidence interval gives a range of plausible values for a parameter: statistic ± margin of error. For a proportion the statistic is p-hat and the margin of error is a critical value z\ times the standard error sqrt(p-hat(1−p-hat)/n). Before computing, check the conditions: Random sample, 10% condition (n ≤ 0.10N for independence), and Large Counts* (n·p-hat ≥ 10 and n(1−p-hat) ≥ 10).

What "95% confident" really means

The confidence level describes the method, not any single interval. "95% confident" means that if we repeated the sampling process many times, about 95% of the intervals produced would capture the true parameter. It does not mean there is a 95% probability that this one interval contains p — the parameter is fixed, and a given interval either caught it or did not.

One-sample z-interval for a proportion

p-hat ± z\* · sqrt( p-hat(1−p-hat) / n )

z\* is the critical value (1.96 for 95%, 2.576 for 99%, 1.645 for 90%). The margin of error is everything after the ± sign.

Worked example

In an SRS of 400 voters, 240 support a measure. Construct and interpret a 95% confidence interval for the true proportion of supporters.

1.p-hat = 240/400 = 0.60. Conditions: random SRS, population presumably > 4000 (10% ok), np-hat = 240 and n(1−p-hat) = 160 both ≥ 10.
2.Standard error = sqrt(0.60·0.40/400) = sqrt(0.0006) ≈ 0.0245.
3.Margin of error = 1.96 × 0.0245 ≈ 0.048.
4.Interval: 0.60 ± 0.048 = (0.552, 0.648).

Answer: The 95% confidence interval is 0.60 ± 0.048 = (0.552, 0.648). We are 95% confident that the true proportion of all voters who support the measure lies between about 0.552 and 0.648.

Checkpoint

A 95% confidence interval for a proportion is (0.48, 0.56). Which interpretation is correct?

Watch out

A confidence interval estimates a parameter, never an individual or "95% of the population." And confidence is a property of the method over many samples — avoid saying "95% probability this interval contains p." Say "we are 95% confident."

Checkpoint

To cut the margin of error of a proportion interval in half while keeping the same confidence level, you should:

Answer the 2 checkpoints as you read.