Carrying Out a Test for the Difference Between Two Population Means
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalThe two-sample t statistic is $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$, with nothing further subtracted because the null puts the difference at 0. For the teaching methods, $t = \frac{7.2}{2.341} \approx 3.08$, giving a one-sided p-value of about 0.0024 at the conservative $df = 27$ and about 0.0017 at technology's $df \approx 53$. The conclusion states the comparison to $\alpha$, the decision, the direction, and both populations, and the two-sided test agrees with the interval when both use the same df and standard error.
The standard error gets built by subtracting variances or adding standard errors instead of adding variances, and the samples get pooled as though this were a two-proportion test, though equal means say nothing about equal variances. The conclusion announces a decision with the p-value never set beside $\alpha$, or lands on the null instead of on the two populations, or reports that a difference exists when a one-sided test established a direction. And a large p-value gets converted into a finding that the two methods are equally effective, which is acceptance in other words.
The work
3 ways in · any order
Lesson
Carrying Out a Test for the Difference Between Two Population Means
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Runs the two-sample t test end to end with a standard error that adds the two variances, reads the p-value at the stated degrees of freedom, writes a conclusion carrying the comparison, the direction, and both populations, and explains why means never pool.
Diagnostic
10-item topic check
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Ten items on carrying out a two-sample t test: standard errors combined wrong, samples pooled as if proportions, decisions announced without comparing p to alpha, and equality concluded from a large p-value. Take it cold to find your habit, or after the lesson to check it is gone.
Targeted Practice
Drill a single misconception
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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.