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Carrying Out a Test for a Population Mean or Population Mean Difference

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The one-sample t statistic is $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ with $df = n - 1$, and the p-value is a tail area of that t distribution chosen by the alternative. For the battery data, $t = \frac{124.6 - 128}{1.84} \approx -1.85$ at $df = 24$, giving 0.0385 one-sided and 0.077 two-sided. At $\alpha = 0.05$ the one-sided test rejects, and the conclusion states the comparison, the decision, and the context: convincing evidence that the mean lifetime of all batteries of this type is below 128 hours.

The conclusion drops its comparison, announcing reject or fail to reject with the p-value never set beside $\alpha$, or lands on the null instead of on battery lifetime, or flips which decision a small p-value supports. The p-value gets read from the normal table rather than $t_{24}$, where the area beyond $-1.85$ is about 0.032 instead of 0.0385 and the evidence looks stronger than it is. And robustness gets treated as unlimited: a t test on $n = 8$ with a wild outlier reported without comment, or an outlier quietly deleted to make the procedure apply, when the range of t's robustness depends on the sample size.

25 batteries: x-bar = 124.6 h, s = 9.2 h 1 STATE H0: mu = 128 vs Ha: mu < 128, alpha = 0.05 2 PLAN one-sample t test; random; 25 <= 0.10N; dotplot roughly symmetric, no outliers 3 DO SE = 9.2 / sqrt(25) = 1.84, df = 24 t = -3.4 / 1.84 = -1.85, p = 0.0385 4 CONCLUDE 0.0385 < 0.05, so reject H0. convincing evidence that the mean lifetime of ALL batteries of this type is below 128 hours the normal table would give 0.032: read t at df = 24
The comparison in step 4 is the part a reader can check, so it is written out. Reading the tail from the normal curve instead of the t curve shrinks the p-value by about a sixth here.
robust does not mean shape never matters sample size what the data must look like n < 15 roughly symmetric, NO outliers 15 to 29 mild skew fine; strong skew or outliers not 30 and up almost any shape, short of extreme outliers never delete an outlier to make the procedure apply investigate it: a recording error can be corrected, a real value stays, and results are reported with and without it if it matters
Two opposite errors sit at the ends of this table: reporting a p-value from eight observations with a wild outlier, and refusing a valid analysis at forty-five over mild skew.

The work

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Lesson
Carrying Out a Test for a Population Mean or Population Mean Difference ›

Runs the one-sample t test end to end, reads the p-value at the right degrees of freedom, insists the conclusion show its comparison to alpha and land in context, and maps how far the robustness of t procedures extends by sample size.

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Diagnostic
10-item topic check ›

Ten items on carrying out a t test: decisions announced without comparing p to alpha, p-values read from the normal table, conclusions about the null rather than the population, and outliers deleted to make a procedure apply. Take it cold to find your habit, or after the lesson to check it is gone.

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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.

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