Hypothesis Tests for Proportions

Always begin by writing the null hypothesis H₀ and the alternative hypothesis Hₐ. The null hypothesis states that the population proportion equals a specific value p₀: H₀: p = p₀. The alternative can be two-sided (p ≠ p₀), left-sided (p < p₀), or right-sided (p > p₀). Choose the direction of Hₐ based on the research question before you look at the data — never after.

Three conditions must hold for the z-test to be valid. (1) Random: the sample must be a simple random sample or a randomized experiment. (2) Normal (Large Counts): both np₀ ≥ 10 and n(1 − p₀) ≥ 10, ensuring the sampling distribution of p̂ is approximately normal. (3) Independence: the sample size n must be at most 10% of the population size (the 10% condition), so that observations are independent. If any condition fails, the z-test may be unreliable.

The test statistic measures how many standard deviations the sample proportion p̂ falls from the null value p₀. It is computed as z = (p̂ − p₀) / √(p₀(1 − p₀)/n). Notice that the standard error uses p₀ (not p̂) because we are assuming H₀ is true. A large absolute value of z means the sample result is far from what H₀ predicts.

The p-value is the probability of obtaining a test statistic at least as extreme as the observed z, assuming H₀ is true. For a right-sided test, p-value = P(Z ≥ z); for a left-sided test, p-value = P(Z ≤ z); for a two-sided test, p-value = 2·P(Z ≥ |z|). Use a standard normal table or technology. A small p-value (typically < 0.05) means the data are inconsistent with H₀.

Compare the p-value to the significance level α. If p-value < α, reject H₀ and conclude there is statistically significant evidence for Hₐ. If p-value ≥ α, fail to reject H₀ — you do not have sufficient evidence against it. Always state your conclusion in the context of the original problem. 'Fail to reject H₀' is not the same as proving H₀ is true; absence of evidence is not evidence of absence.

• \hat{p} — Sample proportion (dimensionless): The proportion of successes observed in the sample, equal to x/n where x is the count of successes.
• p_0 — Null proportion (dimensionless): The claimed population proportion stated in the null hypothesis H₀.
• n — Sample size (count): The total number of individuals in the sample.
• \sqrt{p_0(1-p_0)/n} — Standard error under H₀ (dimensionless): The standard deviation of the sampling distribution of p̂ when H₀ is assumed true.

Problem. A bottled-water company claims that exactly 70% of teenagers prefer their brand over tap water. A consumer group suspects the true proportion is lower. They randomly survey 150 teenagers and find that 96 prefer the bottled water. Conduct a hypothesis test at the α = 0.05 significance level.

Answer. Fail to reject H₀. Insufficient evidence that the proportion is below 0.70 at α = 0.05.

• Sampling Distribution of a Proportion — The z-test statistic is built directly on the sampling distribution of p̂; understanding its shape, center, and spread is essential before performing hypothesis tests.
• Confidence Intervals for Proportions — Confidence intervals and hypothesis tests are two sides of the same inference coin; a value outside a 95% CI corresponds to rejecting H₀ at α = 0.05 in a two-sided test.
• Type I and Type II Errors — Every hypothesis test risks two kinds of mistakes — rejecting a true H₀ (Type I, probability α) or failing to reject a false H₀ (Type II, probability β) — which are critical for interpreting conclusions.
• Significance Level and P-value — The p-value and significance level α are the decision-making tools at the heart of every hypothesis test, and understanding them deeply prevents common misinterpretations.
• Two-Sample Z-Test for Proportions — A natural extension of the one-sample test is comparing two population proportions, such as testing whether a treatment group differs from a control group.