One-Proportion z-Test

By the end of this lesson, you will be able to conduct a one-proportion z-test, construct confidence intervals for a proportion, and correctly interpret the results.

When to Use the One-Proportion z-Test

Use this test when:

• Testing a claim about a population proportion (p)
• You have a single categorical variable
• Sample size is sufficiently large (conditions below)

Conditions

1. Random: The sample is a random sample from the population.

2. Independent: Observations are independent; n ≤ 10% of population if sampling without replacement.

3. Large counts (Success/Failure condition):

• np₀ ≥ 10 AND n(1-p₀) ≥ 10 where p₀ is the null hypothesis proportion

This ensures the sampling distribution of p̂ is approximately normal.

Hypothesis Test Steps

Hypotheses

H₀: p = p₀ Hₐ: p ≠ p₀ (two-sided), p > p₀ (right), or p < p₀ (left)

Test Statistic

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

Note: We use p₀ (the null value) in the standard error — NOT the sample proportion p̂. This is because under H₀, we assume p = p₀.

P-value

Using the standard normal (z) distribution:

• Two-sided: P = 2 × P(Z > |z|)
• One-sided right: P = P(Z > z)
• One-sided left: P = P(Z < z)

Decision

Reject H₀ if p-value ≤ α.

Example

A cereal company claims that 40% of its customers are children. A consumer group surveys 200 customers and finds 68 are children (p̂ = 68/200 = 0.34). Is there evidence the true proportion differs from 40%? Use α = 0.05.

H₀: p = 0.40 Hₐ: p ≠ 0.40

Conditions:

• Random: stated ✓
• Independent: 200 << customers in population ✓
• Large counts: 200(0.40) = 80 ≥ 10; 200(0.60) = 120 ≥ 10 ✓

Test statistic: $$z = \frac{0.34 - 0.40}{\sqrt{\frac{0.40 \times 0.60}{200}}} = \frac{-0.06}{0.0346} \approx -1.73$$

P(|Z| > 1.73) = 2 × 0.0418 = 0.084