3.2 Estimating a Population Proportion

Confidence Interval for a Proportion

A confidence interval estimates a population proportion using a sample statistic plus a margin of error: p̂ ± z\*·SE, where SE = √(p̂(1−p̂)/n). Before calculating, check conditions: random sample, independence (10% condition: n ≤ 10% of population), and large counts (np̂ ≥ 10 and n(1−p̂) ≥ 10) to ensure the sampling distribution is approximately normal.

Interpreting Confidence Level and Interval

Confidence level describes the long-run capture rate of the *method*: 'If we repeated this process many times, about C% of intervals would capture the true proportion.' The interval interpretation states: 'We are C% confident the true population proportion is between (lower, upper) based on this sample.' Never say a specific interval has a C% chance of containing the parameter.

Margin of Error and Sample Size

Margin of error (ME) = z\*·SE shrinks as sample size increases or confidence level decreases. To find the minimum sample size needed for a desired ME, solve n ≥ (z\*/ME)²·p̂(1−p̂); use p̂ = 0.5 as a conservative estimate when no prior estimate exists, since it maximizes p̂(1−p̂).

Key terms

Confidence interval formula for a proportion

p̂ ± z*·√(p�̂(1−p̂)/n)

Large counts condition

np̂ ≥ 10 and n(1−p̂) ≥ 10, ensures normal approximation is valid.

Correct interpretation of confidence level

In repeated sampling, about C% of intervals constructed this way would capture the true population proportion.

Margin of error

z*·SE; the amount added/subtracted from p̂ to create the interval; increases with confidence level, decreases with sample size.

Conservative p̂ for sample size calculations

Use p̂ = 0.5 when no prior estimate exists, since it maximizes p̂(1−p̂) and gives the largest required n.

Try an AP-style question

In a sample of 150 students, 45 walk to school. What is p̂?

1. 0.45
2. 45
3. 0.15
4. 0.3