AP Statistics Unit 6 (Proportions): Learning to Estimate a Population Proportion with Confidence

Population parameter

A fixed (usually unknown) numerical value that describes an entire population (e.g., the true population proportion p).

Population proportion (p)

The true proportion of individuals in a population who have a characteristic of interest; a fixed but typically unknown parameter.

Random sample

A sample selected using a chance process so that inference about the population is less biased and is valid under the CI method.

Statistic

A numerical value computed from sample data (e.g., p-hat) used to estimate a population parameter.

Sample proportion (p-hat)

The proportion of “successes” in the sample; computed as p-hat = x/n.

Point estimate

A single-number estimate of a parameter (e.g., p-hat as an estimate of p).

Sampling variability

The natural sample-to-sample variation in a statistic caused by random sampling.

Confidence interval (CI)

An interval calculated from sample data that gives plausible values for a population parameter, using a method that succeeds at a stated long-run rate (confidence level).

Confidence level

The long-run percentage of intervals (from repeated random samples using the same method) that would contain the true parameter (e.g., 95%).

Correct interpretation of “95% confident”

If we repeatedly took random samples of the same size and built intervals the same way, about 95% of those intervals would contain the true parameter p.

Statistical inference

Using sample data to draw conclusions about a population (e.g., estimating p with a confidence interval).

Sampling distribution (of p-hat)

The distribution of sample proportions p-hat you would get from many repeated samples of the same size from the same population.

Standard error (SE of p-hat)

An estimate of the standard deviation of p-hat: SE = sqrt([p-hat(1 − p-hat)]/n).

Critical value (z*)

A standard Normal (z) cutoff chosen to match the confidence level for a two-sided interval (e.g., 1.96 for 95%).

Margin of error (ME)

The “±” amount in a confidence interval: ME = z* × SE; equals half the width of the CI.

One-proportion z interval

A confidence interval method for estimating a single population proportion p using p-hat ± z*sqrt([p-hat(1−p-hat)]/n), assuming conditions are met.

Random condition

The data come from a random sample (or random assignment in experiments), supporting unbiased inference/generalization.

Independence (10% condition)

When sampling without replacement, n should be no more than 10% of the population to treat observations as approximately independent.

Large counts condition

For Normal approximation in a one-proportion z interval: n(p-hat) ≥ 10 and n(1 − p-hat) ≥ 10.

Normal approximation for p-hat

For a large enough sample, p-hat is approximately Normal with center p and SD about sqrt(p(1−p)/n).

Common z* values (90%, 95%, 99%)

Approximate critical values: 90% → 1.645; 95% → 1.96; 99% → 2.576.

Effect of confidence level on CI width

Higher confidence level → larger z* → larger margin of error → wider (less precise) interval.

Effect of sample size on margin of error

Larger n decreases SE and ME (at a diminishing rate because SE involves 1/sqrt(n)), producing a narrower interval.

Sample size planning formula for a proportion

To achieve margin of error m at confidence level z* using an anticipated proportion p: n = (z/m)^2 p(1−p), then round up.

Conservative choice p* = 0.5

When no prior estimate is available, use p* = 0.5 because p(1−p) is maximized at 0.5, giving the largest (safest) required sample size.