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 ($H_0$)

The default/status quo claim about a parameter; typically includes an equals sign and a specific value (e.g., $H_0: p = 0.40$).

Alternative hypothesis (Hₐ)

The competing claim the test seeks evidence for; can be two-sided (
e), right-tailed (\gt), or left-tailed (\lt).

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 $H_0$ 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 H_0 (common values: 0.05 or 0.01).

Reject $H_0$

Decision made when p-value $\boldsymbol{≤} \boldsymbol{α}$; the data provide evidence supporting $H_a$.

Fail to reject $H_0$

Decision made when p-value $> \boldsymbol{α}$; the data do not provide enough evidence to support $H_a$ (this is not the same as accepting $H_0$).

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 p_0: np_0 \ge 10 and n(1-p_0) \ge 10.

Test statistic (one-proportion z)

z = \frac{(\hat{p} - p_0)}{\sqrt{p_0(1-p_0)/n}}; measures how many standard deviations \hat{p} is from p_0 under H_0.

Standard error under the null (one-proportion)

\sqrt{p_0(1-p_0)/n}; the standard deviation of \hat{p} assuming H_0 is true.

Right-tailed test

A test with H_a: p \gt p_0; the p-value is the area to the right of the observed z.

Left-tailed test

A test with H_a: p \lt p_0; the p-value is the area to the left of the observed z.

Two-sided test

A test with H_a: p
e p_0; 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: \hat{p} \pm z* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.

z* (critical value)

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

Two-proportion z interval (for $p_1 - p_2$)

A confidence interval estimating $\boldsymbol{\text{p}}_1 - \boldsymbol{\text{p}}_2$ using an unpooled standard error: (\boldsymbol{\text{p}}_1 - \boldsymbol{\text{p}}_2) \text{ } \boldsymbol{±} z^* \text{ } \boldsymbol{\sqrt}(\frac{\boldsymbol{\text{p}}_1(1-\boldsymbol{\text{p}}_1)}{n_1} + \frac{\boldsymbol{\text{p}}_2(1-\boldsymbol{\text{p}}_2)}{n_2}).

Two-proportion z test

A significance test comparing two population proportions, often with H_0: p_1 = p_2 (equivalently p_1 - p_2 = 0), using a pooled estimate for the standard error under H_0.

Pooled proportion (p̂)

For testing $H_0: p_1 = p_2$, the combined estimate $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$, used to compute the pooled standard error in a two-proportion z test.