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Introductory Statistics Unit 6: Inference for proportions

Confidence intervals estimate a population proportion; significance tests check whether data give convincing evidence against a claimed value.

Covers: Confidence intervals & significance tests for one and two proportions, errors & power

Study guide

Inference uses sample data to draw conclusions about a population. A confidence interval estimates a parameter: statistic ± margin of error. For one proportion, the interval is p̂ ± z*·√(p̂(1 − p̂)/n), where z* comes from the confidence level (1.96 for 95%). Interpret it as: we are 95% confident the interval captures the true proportion. The confidence level describes the method: about 95% of intervals made this way capture the parameter. Larger samples and lower confidence levels give narrower intervals.

A significance test checks a claim. State hypotheses H₀ (no effect, p = p₀) and Hₐ (the claim you're looking for evidence of). Check conditions: random sample, 10% condition, and large counts (np₀ ≥ 10 and n(1 − p₀) ≥ 10). Compute z = (p̂ − p₀)/√(p₀(1 − p₀)/n) and the p-value, the probability of a result at least as extreme as the one observed if H₀ were true. If the p-value is below α (often 0.05), reject H₀; otherwise you fail to reject it. Never "accept" H₀.

For two proportions, compare groups with (p̂₁ − p̂₂) ± z*·√(p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂), and use a pooled proportion for the test.

Errors: a Type I error rejects a true H₀ (probability α); a Type II error fails to reject a false H₀. Power, 1 − P(Type II), is the chance of detecting a real effect; it increases with larger samples, larger α, and bigger true effects.

To keep the margin of error at most ME, choose n ≥ (z*/ME)²·p̂(1 − p̂), using p̂ = 0.5 if no estimate is available, and round up. Interpret a p-value as follows: assuming H₀ is true, it is the probability of getting a statistic as extreme as or more extreme than the one observed, purely by chance.

For a two-proportion test, check randomness and the 10% condition in each group. Then check large counts with the pooled proportion p̂_c: n₁p̂_c, n₁(1 − p̂_c), n₂p̂_c and n₂(1 − p̂_c) should all be at least 10. To describe an error in context, state both what you concluded and what is actually true. For example, a Type I error here means concluding the proportion changed when it really did not.

Key ideas

• A confidence interval is point estimate ± margin of error.
• Interpretation: "We are C% confident the interval from __ to __ captures the true proportion of [context]."
• Tests: state H₀ and Hₐ about p, check conditions (random, 10%, large counts), compute the test statistic and p-value, then conclude in context.
• The p-value is the probability of a result at least as extreme as the sample's, assuming H₀ is true.
• Type I error: rejecting a true H₀. Type II error: failing to reject a false H₀. Power = 1 − P(Type II).

Formulas and rules

Common mistakes

• Saying "accept H₀". You fail to reject it.
• Interpreting 95% confidence as a 95% chance the parameter is in this particular interval.

Words to know

One-proportion confidence interval

p̂ ± z*·√(p̂(1 − p̂)/n)

One-proportion z test statistic

z = (p̂ − p₀) / √(p₀(1 − p₀)/n)

P-value

The probability of a result at least as extreme as the observed one, assuming H₀ is true.

Type I error

Rejecting H₀ when it's actually true.

Type II error

Failing to reject H₀ when it's actually false.

Power

The probability of correctly rejecting a false H₀ (1 − P(Type II)).

Sample size and confidence intervals

Larger n → smaller margin of error → narrower interval.

Two-proportion confidence interval

(p̂₁ − p̂₂) ± z*·√(p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂)

Confidence interval

Statistic ± margin of error, a range of plausible parameter values.

Margin of error

Critical value × standard error.

Confidence level

The long-run percent of intervals that capture the parameter.

Null hypothesis (H₀)

The claim of no effect or no difference, like p = p₀.

Alternative hypothesis (Hₐ)

The claim you're seeking evidence for.

Significance level (α)

The p-value cutoff for rejecting H₀, often 0.05.

Pooled proportion

Combined p̂ from both samples, used in two-proportion tests.

Critical value (z*)

1.96 for 95% confidence.

Sample practice questions

37 more questions like these, from level 1 to 5, are in the practice set.