One difference over one combined standard error

Under $H_0: \mu_1 = \mu_2$, the difference has mean 0, so

$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}.$$

Nothing is subtracted in the numerator beyond the two sample means, because the null puts $\mu_1 - \mu_2$ at 0.

For the teaching methods: $\bar{x}_1 - \bar{x}_2 = 7.2$ and $SE \approx 2.341$, so

$$t = \frac{7.2}{2.341} \approx 3.08.$$

At the conservative $df = 27$, the one-sided p-value is about $0.0024$ and the two-sided about $0.0048$. At technology's $df \approx 53$ they are about $0.0017$ and $0.0033$. Either is acceptable, and which was used has to be stated, because the p-values differ.

Unlike the two-proportion test, there is no pooling here. A two-proportion test pools because its null says the two populations share a single proportion, and a proportion determines its own variance. Two population means being equal says nothing about the two variances, so each group keeps its own $s$.

"Since the one-sided p-value of 0.0024 is less than $\alpha = 0.05$, we reject $H_0$. There is convincing evidence that the mean score for all students taught by method A is higher than for all students taught by method B."

Four things, and a missing one costs credit:

1. The comparison, written out, so a reader can check it.
2. The decision: reject or fail to reject, never accept.
3. The direction: which mean is higher. "There is a difference" is true and weaker than what a one-sided test established.
4. Both populations: a sentence about the 58 students in the study describes groups whose means are already known exactly.

Whether the verb may be causal depends on the design, as always. Random assignment licenses "method A produces higher scores"; two intact classes license only the comparison.

Had the p-value been 0.24, the report would read: there is not convincing evidence of a difference in mean score between the two methods. Not that the methods are equally effective, and not that the means are equal.

With modest group sizes the test may have had little chance of detecting a difference worth caring about, so a non-significant two-sample result is reported alongside both sample sizes. A reader can then see whether the study was capable of finding what it was looking for, which is the power question from Topic 3.8 arriving in a new setting.

A two-sided test at level $\alpha$ and a confidence interval at level $1 - \alpha$ answer the same question two ways. Here the two-sided p-value of 0.0048 is below 0.05, and the 95% interval $(2.40, 12.00)$ excludes 0. Both say that 0 is not a plausible difference.

Unlike the two-proportion case, the correspondence is exact when the same degrees of freedom and the same standard error are used in both, because a two-sample t test and a two-sample t interval share their standard error. Using the conservative df for one and technology's df for the other is what breaks it, so keep the choice consistent across the two procedures.

The two still answer different questions, which is why both are worth reporting. The test says whether a difference is detectable; the interval says how large the plausible differences are, and 2.4 to 12.0 points is information no p-value carries.