AP Statistics 3.5-3.6: Hypothesis Tests for Proportions

Null hypothesis ($H_0$)

A population-level claim representing no difference, no association, or a specified status-quo value. Statistical tests evaluate whether sample evidence is strong enough to reject $H_0$.

Alternative hypothesis ($H_a$ or $H_1$)

The population-level claim that contradicts the null hypothesis and is supported when the null hypothesis is rejected. It commonly expresses the researcher's expected effect or difference.

What are the two possible conclusions of a hypothesis test?

Reject $H_0$ when the sample provides sufficient evidence for $H_a$; otherwise, fail to reject $H_0$. Failing to reject $H_0$ does not prove that $H_0$ is true.

Why must the null and alternative hypotheses be mutually exclusive?

They represent opposing claims about a population parameter, so evidence is evaluated to decide whether it is sufficiently inconsistent with $H_0$ to support $H_a$.

How are equality symbols assigned in null and alternative hypotheses?

$H_0$ includes equality: $=$, $\le$, or $\ge$. $H_a$ uses a strict inequality: $\ne$, $<$, or $>$.

A study tests whether more than 40% of students pass a first attempt. State $H_0$ and $H_a$ using the population proportion $p$.

$H_0: p \le 0.40$ and $H_a: p > 0.40$. The null includes the boundary value because it contains equality.

What does a hypothesis test use as evidence about a population proportion claim?

It uses sample data, summarized by a test statistic or equivalent probability measure, to determine how consistent the observed sample proportion is with $H_0$.

Test statistic for a one-proportion hypothesis test

For testing $H_0:p=p_0$, the one-proportion $z$ statistic is $$z=\frac{\hat p-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}},$$where$\hat p$ is the observed sample proportion.

What is a p-value in a hypothesis test?

The probability, assuming $H_0$ is true, of obtaining a test statistic at least as extreme as the observed value in the direction specified by $H_a$. A sufficiently small p-value provides evidence against $H_0$.

How is a p-value compared with the significance level $\alpha$?

If $p$-value $< \alpha$, reject $H_0$; if $p$-value $\ge \alpha$, fail to reject $H_0$. The significance level is the chosen maximum probability of a Type I error.

Type I error

Rejecting a true null hypothesis: concluding that an effect or difference exists when it does not. Its probability is $\alpha = P(\text{reject }H_0\mid H_0\text{ is true})$.

How is a sample proportion calculated?

The sample proportion is $\hat p = x/n$, where $x$ is the number of successes and $n$ is the number of independent trials.

What conditions support a one-proportion $z$ test?

The data should come from a random sample or randomized process, observations should be independent, and the large-count condition should hold under the null: $np_0\ge 10$ and $n(1-p_0)\ge 10$.

What is the approximate sampling distribution of a sample proportion?

When the normal approximation conditions are met, $\hat p$ is approximately normal with mean $p$ and standard error $\sqrt{p(1-p)/n}$: $$\hat p\sim N\left(p,\sqrt{\frac{p(1-p)}{n}}\right).$$

Why is the null-hypothesis value typically used when calculating a proportion-test standard error?

A hypothesis test models the sampling distribution under the assumption that $H_0$ is true. Therefore, the null value of $p$ is used in $\sqrt{p_0(1-p_0)/n}$ rather than automatically using the observed $\hat p$.

Critical value

A cutoff value for a test statistic that separates the region where $H_0$ is rejected from the region where $H_0$ is not rejected. It is determined by the chosen significance level and whether the test is one-tailed or two-tailed.

Rejection region (critical region)

The set of test-statistic values considered sufficiently unlikely under $H_0$ that the null hypothesis is rejected. Its location depends on the direction of $H_a$.

How does the form of $H_a$ determine the type of hypothesis test?

$H_a: p>p_0$ gives a right-tailed test; $H_a: p<p_0$ gives a left-tailed test; and $H_a: p\ne p_0$ gives a two-tailed test.

What does rejecting $H_0$ mean—and what does it not mean?

It means the sample provides statistically significant evidence against $H_0$ at the chosen significance level. It does not prove $H_a$ with certainty or establish that the probability $H_0$ is false equals the p-value.

How should the conclusion of a proportion hypothesis test be stated?

State the decision about $H_0$, then describe the evidence in terms of the population proportion and the original question. For example: 'Because the p-value is less than $\alpha$, we reject $H_0$ and conclude that there is convincing evidence that the population proportion is greater than $p_0$.'

What is the relationship between significance level and Type I error?

The significance level $\alpha$ is the probability of rejecting $H_0$ when $H_0$ is true, so it is the test's specified Type I error rate, subject to the assumptions of the procedure.