Confidence Intervals for Proportions

One-Sample z-Interval for $p$

A confidence interval gives a range of plausible values for a population parameter.

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

where:

• $\hat{p}$ = sample proportion
• $z^*$ = critical value from the Standard Normal distribution
• $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ = standard error of $\hat{p}$

Common Critical Values

| Confidence Level | $z^*$ | |-----------------|-------| | 90% | 1.645 | | 95% | 1.960 | | 99% | 2.576 |

Conditions (Check These!)

1. Random: Data comes from a random sample or randomized experiment
2. 10% Condition: $n < 0.10N$ (sample is less than 10% of population)
3. Large Counts: $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$

Interpretation

Correct: "We are [C]% confident that the true proportion of [context] is between [lower bound] and [upper bound]."

Incorrect: "There is a [C]% probability that $p$ is in this interval." (The parameter is fixed; the interval either contains it or it doesn't.)

What Does "95% Confident" Mean?

If we repeated the sampling process many times, approximately 95% of the resulting confidence intervals would contain the true population proportion $p$.

Margin of Error

$ME = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

The margin of error decreases when:

• $n$ increases (more data)
• Confidence level decreases (narrower interval)

Determining Sample Size

To achieve a desired margin of error $ME$ with confidence level $z^*$:

$n = \left(\frac{z^*}{ME}\right)^2 \hat{p}(1-\hat{p})$

If $\hat{p}$ is unknown, use $\hat{p} = 0.5$ for the most conservative (largest) sample size.

Four-Step Process for AP

1. State: Identify the parameter and confidence level
2. Plan: Name the procedure, check conditions
3. Do: Calculate the interval
4. Conclude: Interpret in context

AP Tip: You MUST check all three conditions (Random, 10%, Large Counts) and interpret the interval in context to receive full credit on free-response questions.