Topic 6.4 Setting Up a Test for a Population Proportion: state null and alternative hypotheses about a population proportion, identify the significance level, and verify the conditions for a one-sample z-test.

What this topic is asking

The College Board (Topic 6.4) wants you to set up a significance test for a population proportion: write the null and alternative hypotheses about $p$, identify the significance level $\alpha$, and check the conditions for the one-sample z-test, using the null value $p_0$ in the large-counts check.

Hypotheses are about the parameter

Hypotheses always concern the population proportion $p$, never the statistic $\hat{p}$; $\hat{p}$ is the evidence we will weigh, not the claim. The null is an equality at the boundary value $p_0$. Choose the alternative from the wording: "more than" gives $>$, "less than" gives $<$, "different from" or "changed" gives $\ne$. Always define $p$ in words first ("let $p$ be the true proportion of ... who ...") so the hypotheses are anchored in context.

Choosing the significance level

You set $\alpha$ in advance to fix how much risk of a false alarm you will accept. A high-stakes decision (a costly or harmful wrong rejection) calls for a small $\alpha$ like $0.01$. The level chosen here is the line a P-value (Topic 6.5) will be compared against to reach a conclusion (Topic 6.6).

Checking the conditions, with the null twist

The distinctive feature of a test (versus an interval) is that the large-counts check and the standard deviation use the null value $p_0$, because the entire test is conducted under the assumption that $H_0$ is true. This contrasts with the interval of Topic 6.2, which used the observed $\hat{p}$. Mixing these up is one of the most common setup errors, so name $p_0$ explicitly. The same $p_0$ will appear in the standard deviation $\sqrt{p_0(1-p_0)/n}$ of the test statistic in Topic 6.6.

Why the setup must come first

The conclusion of a test is only as trustworthy as its setup. Clearly defined hypotheses about $p$ keep you from testing the wrong thing; a stated $\alpha$ prevents you from moving the goalposts after seeing the data; and checked conditions earn the normal model that the P-value depends on. Examiners award setup points independently, and a missing condition or a hypothesis about $\hat{p}$ loses marks even if the later arithmetic is correct.

Try this

Q1. A claim is that exactly $25\%$ of items pass; you suspect the rate has changed. Write the hypotheses. [1 point]

• Cue. $H_0: p = 0.25$ versus $H_a: p \ne 0.25$ (two-sided, because "changed").

Q2. In the large-counts check for a test, do you use $\hat{p}$ or $p_0$, and why? [1 point]

• Cue. Use $p_0$; the test reasons under the assumption that $H_0$ is true, so normality is checked at the null value.