Module 5 · Inference · Topic 5

CI for Proportions

Conditions. Requires: random sample, np̂ ≥ 10 and n(1 − p̂) ≥ 10 (success/failure condition), and sample is less than 10% of the population.

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Key formulas

Proportion CI Formula

p̂ ± z* × √(p̂(1 − p̂) / n)

From the Inference formula sheet

Confidence Interval for a Mean: xbar +/- critical value * standard error — Estimate plus or minus a margin of error.
One-Sample Z Test: z = (xbar - mu_0) / (sigma / sqrt(n)) — Tests a sample mean against a null mean when population sigma is known.
Power: Power = 1 - beta — Probability of correctly rejecting a false null hypothesis.

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Worked example

Worked Example

In a survey of 400 people, 260 support a policy. p̂ = 260/400 = 0.65. For 95% CI: ME = 1.96 × √(0.65 × 0.35 / 400) = 1.96 × 0.0239 = 0.047. CI = (0.603, 0.697).

When to use it

Interpretation

We are 95% confident that the true proportion of all people who support this policy is between 60.3% and 69.7%.

Related glossary terms

P-value: The probability of results this extreme or more extreme, assuming the null hypothesis is true. Example: A p-value of 0.03 is evidence against H0 at alpha = 0.05.
Confidence Level: The long-run success rate of a confidence interval procedure. Example: A 95% method captures the true parameter in about 95% of repeated samples.
Type I Error: Rejecting the null hypothesis when it is actually true. Example: The significance level alpha is the Type I error rate.
Type II Error: Failing to reject the null hypothesis when the alternative hypothesis is true. Example: A false negative in a hypothesis test is a Type II error.
Power: The probability of rejecting the null hypothesis when it is false. Example: Power equals 1 - beta, where beta is the Type II error probability.

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