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

Constructing a Confidence Interval for the Difference Between Two Population Means

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

A two-sample t interval is $(\bar{x}_1 - \bar{x}_2) \pm t^{*}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$. For method A ($\bar{x}_1 = 78.4$, $s_1 = 8.1$, $n_1 = 30$) against method B ($\bar{x}_2 = 71.2$, $s_2 = 9.6$, $n_2 = 28$): the difference is 7.2, $SE \approx 2.341$, and with the conservative $df = 27$ and $t^{*} = 2.052$ the interval is $7.2 \pm 4.80$, or about $(2.40, 12.00)$ points. The conditions are random, independence between groups, the 10% condition per group, and a shape check per group.

The standard error gets built by subtracting variances or adding standard errors instead of adding variances. The degrees of freedom get set to $n_1 + n_2 - 2$ from memory rather than to technology's value or the conservative $\min(n_1, n_2) - 1$. The shape condition gets checked once for the combined data or only for the larger group, when a 28-observation group below the guideline needs its own plot. And the interpretation drops the word difference, the order of subtraction, or both populations, leaving a sentence a reader cannot act on.

method A minus method B, 95% confidence 0 7.2 2.40 12.00 margin of error 4.80 7.2 +/- 2.052 x 2.341 2.052 is t* at df = 27 2.341 adds the two variances every endpoint is a GAP in points
The endpoints are plausible gaps between the two population means, not scores. That is why the interpretation has to contain the word difference and the order of subtraction.
how many times each condition is checked condition times for this study random once both samples random independent groups once no student in both 10% condition TWICE 30 and 28, separately shape TWICE A: n = 30 clears it B has n = 28, below the guideline: its plot must be described the boxplot of the 28 method-B scores is roughly symmetric with no outliers: that sentence IS the check
Two of the four conditions are per group, so the smaller group can fail on its own. Naming the condition is not checking it; describing the plot is.

The work

3 ways in · any order
Lesson
Constructing a Confidence Interval for the Difference Between Two Population Means ›

Builds the two-sample t interval with a standard error that adds each group's variance, chooses degrees of freedom between technology's value and the conservative rule, checks the 10% and shape conditions once per group, and writes an interpretation naming both populations and the order of subtraction.

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

Ten items on two-sample intervals for means: standard errors combined wrong, degrees of freedom taken as n1 plus n2 minus 2, shape checked only for the larger group, and interpretations missing the word difference. 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