Inference for Proportions: Hypothesis Tests, Errors, and Two-Proportion Inference

Statistical inference

Using data from a sample to draw conclusions about a population parameter.

Population proportion (p)

The true (usually unknown) proportion of individuals in a population with a specified characteristic.

Sample proportion (p̂)

The proportion of individuals in a sample with the characteristic; computed as p̂ = x/n.

Parameter

A fixed (but often unknown) numerical value that describes a population (e.g., p).

Statistic

A numerical value computed from sample data that estimates a parameter (e.g., p̂).

Significance test (hypothesis test)

A formal procedure that uses sample data to evaluate a claim about a population parameter by assessing how surprising the sample result would be if a “status quo” claim were true.

Null hypothesis (H₀)

The default/status quo claim about a parameter; typically includes an equals sign and a specific value (e.g., H₀: p = 0.40).

Alternative hypothesis (Hₐ)

The competing claim the test seeks evidence for; can be two-sided (≠), right-tailed (>), or left-tailed (<).

Null value (p₀)

The hypothesized value of the population proportion stated in the null hypothesis (used in test calculations and conditions).

p-value

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

Significance level (α)

A chosen cutoff for deciding when evidence is strong enough to reject H₀ (common values: 0.05 or 0.01).

Reject H₀

Decision made when p-value ≤ α; the data provide evidence supporting Hₐ.

Fail to reject H₀

Decision made when p-value > α; the data do not provide enough evidence to support Hₐ (this is not the same as accepting H₀).

One-proportion z test

A significance test for a single population proportion p that uses a Normal approximation to the sampling distribution of p̂ when conditions are met.

Conditions for a one-proportion z test

(1) Random sample or randomized experiment, (2) Independence via the 10% condition if sampling without replacement, (3) Large counts using p₀: np₀ ≥ 10 and n(1−p₀) ≥ 10.

Test statistic (one-proportion z)

z = (p̂ − p₀) / sqrt(p₀(1−p₀)/n); measures how many standard deviations p̂ is from p₀ under H₀.

Standard error under the null (one-proportion)

sqrt(p₀(1−p₀)/n); the standard deviation of p̂ assuming H₀ is true.

Right-tailed test

A test with Hₐ: p > p₀; the p-value is the area to the right of the observed z.

Left-tailed test

A test with Hₐ: p < p₀; the p-value is the area to the left of the observed z.

Two-sided test

A test with Hₐ: p ≠ p₀; the p-value is twice the tail area beyond |z|.

Confidence interval (for p)

An interval estimate giving a range of plausible values for p; for a one-proportion z interval: p̂ ± z* sqrt(p̂(1−p̂)/n).

z* (critical value)

The standard Normal cutoff used in confidence intervals, determined by the desired confidence level (e.g., z* ≈ 1.96 for 95%).

Two-proportion z interval (for p₁ − p₂)

A confidence interval estimating p₁ − p₂ using an unpooled standard error: (p̂₁ − p̂₂) ± z* sqrt[p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂].

Two-proportion z test

A significance test comparing two population proportions, often with H₀: p₁ = p₂ (equivalently p₁ − p₂ = 0), using a pooled estimate for the standard error under H₀.

Pooled proportion (p̂)

For testing H₀: p₁ = p₂, the combined estimate p̂ = (x₁ + x₂)/(n₁ + n₂), used to compute the pooled standard error in a two-proportion z test.