Topic 7.4 Setting Up a Test for a Population Mean: state the null and alternative hypotheses about a population mean, identify the significance level, and verify the conditions for a one-sample t-test.

What this topic is asking

The College Board (Topic 7.4) wants you to set up a significance test for a population mean: write the null and alternative hypotheses about $\mu$, identify the significance level $\alpha$, and check the conditions for the one-sample $t$-test.

Hypotheses are about the mean

Hypotheses always concern the population mean $\mu$, never the sample mean $\bar{x}$; $\bar{x}$ is the evidence, not the claim. The null fixes a specific value $\mu_0$. Pick the alternative from the wording: "greater than / increased" gives $>$, "less than / underfilled" gives $<$, "different / changed / drifted" gives $\ne$. Always define $\mu$ in words first ("let $\mu$ be the true mean ... of ...").

Choosing the significance level

Setting $\alpha$ in advance commits you to a standard of evidence and a false-alarm risk before you can be swayed by the result. Higher-stakes decisions warrant a smaller $\alpha$. This is the line the P-value (Topic 7.5) will be compared against.

Checking the conditions

The shape condition mirrors the interval's. Unlike the proportion test, the mean test's conditions do not change between interval and test: there is no "use $\mu_0$ in the condition" twist, because the t-procedure's standard error uses $s$ (the sample standard deviation) in both the interval and the test. So the same normal/large-sample reasoning applies throughout the unit. State your shape justification explicitly, especially when $n < 30$.

Why the setup earns its own marks

A test's conclusion is only trustworthy if the setup is right. Hypotheses about $\mu$ keep you testing the population parameter, not the sample. A pre-set $\alpha$ stops you adjusting the standard after seeing the data. Checked conditions earn the t-model the P-value relies on. Examiners award these parts independently; hypotheses about $\bar{x}$, or a missing shape check, lose marks even with correct later arithmetic.

Try this

Q1. A claim is that a mean equals $100$; you suspect it has decreased. Write the hypotheses. [1 point]

• Cue. $H_0: \mu = 100$ versus $H_a: \mu < 100$ (one-sided, because "decreased").

Q2. For $n = 12$, how do you justify the normality condition? [1 point]

• Cue. With $n < 30$, examine a graph of the data; if it is roughly symmetric with no outliers, the t-procedure is appropriate.