Mistake Master
Student view — seeing the site as a student does
Home› Unit 4 · Inference for Quantitative Data: Means› 4.1·4.2·4.3·4.4·4.5·4.6·4.7·4.8·4.9·4.10

Setting Up a Test for a Population Mean or Population Mean Difference

▶︎  Watch it animatedinteractive step-through · ~3 min · optional

A one-sample t test states $H_0: \mu = \mu_0$ against an alternative whose direction comes from the research question, with the parameter defined in context. For a claimed mean of 128 hours and a consumer group suspecting less: $H_0: \mu = 128$ against $H_a: \mu < 128$, where $\mu$ is the mean lifetime of all batteries of this type. Paired designs subtract first and test $\mu_d$, usually against a null of 0, with $n$ counting pairs and $df = n - 1$. The conditions are random, 10%, and a shape check, and the procedure is t because $\sigma$ is unknown.

Hypotheses get written about $\bar{x}$, which is 124.6 and known, or about the 25 batteries tested rather than the population, or with an inequality in the null where the p-value needs one specific value to compute from. The alternative gets its direction from the sample instead of the question. Paired data gets $n$ counted as measurements rather than pairs, doubling the degrees of freedom. And z gets used where t belongs, which understates the uncertainty by using critical values that are too small at every finite df.

the maker claims 128 h; a consumer group suspects less H0: mu = 128 Ha: mu < 128 mu = mean lifetime of ALL batteries of this type direction from SUSPECTS LESS, not from the sample's 124.6 three that are not hypotheses H0: x-bar = 128 x-bar is known: it is 124.6 H0: mu >= 128 the null needs ONE value H0: the 25 tested average 128 that is the sample, not the target
The null must name one value of the parameter, because the p-value is read from a single sampling distribution centered there. A hypothesis about the sample mean tests a number the data simply report.
two setups that share every formula and differ in what they are about design parameter and null n and df ONE SAMPLE 25 batteries mu = mean lifetime H0: mu = 128 n = 25 df = 24 PAIRED 12 students, twice mu_d = mean change H0: mu_d = 0 n = 12 PAIRS df = 11 paired with n = 24 and df = 23 counts measurements, not pairs and the order of subtraction has to be stated with mu_d
The two designs run the identical formula on different objects. Counting measurements instead of pairs doubles the degrees of freedom and answers a question the study never posed.

The work

3 ways in · any order
Lesson
Setting Up a Test for a Population Mean or Population Mean Difference ›

Sets up the one-sample t test: hypotheses about mu with equality in the null and direction from the research question, the paired version testing a mean difference against zero with n counting pairs, the conditions with a real shape check, and t rather than z because sigma is estimated.

Skill check · 10 scenarios
Open lesson →
Diagnostic
10-item topic check ›

Ten items on mean-test setup: hypotheses about x-bar or about the sampled units, inequalities in the null, alternatives chosen from the data, paired n counted as measurements, and z used where t belongs. Take it cold to find your habit, or after the lesson to check it is gone.

Not started · 10 items · ~15 min
Start diagnostic →
Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.

Take the diagnostic to identify your misconceptions
Diagnostic first