An interval assumes nothing, so it never pools

The two-proportion z-interval keeps the shape of every interval in the course:

$$(\hat{p}_1 - \hat{p}_2) \pm z^{*}\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}.$$

The point estimate is the observed difference; the standard error is the one from the previous topic with $\hat{p}_1$ and $\hat{p}_2$ in place of the unknown $p_1$ and $p_2$; and $z^{*}$ comes from the confidence level as before.

A new checkout system converts 138 of 300 shoppers and the old one converts 90 of 250, so $\hat{p}_1 = 0.46$, $\hat{p}_2 = 0.36$, and the difference is 0.10. Then

$$SE = \sqrt{\frac{0.46(0.54)}{300} + \frac{0.36(0.64)}{250}} \approx 0.0418,$$

and the 95% interval is $0.10 \pm 1.96(0.0418) = 0.10 \pm 0.082$, or about $(0.018, 0.182)$.

The two-proportion test in Topic 3.12 combines the samples into a pooled proportion $\hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2}$. An interval does not, and the difference is not a convention.

Pooling is justified only when the two populations are assumed to share a single proportion. A test makes exactly that assumption, because its null says $p_1 = p_2$, so under the null there is one common proportion and both samples estimate it. An interval assumes nothing about $p_1$ and $p_2$; its whole purpose is to estimate their difference. Combining the samples would build in the assumption that the difference is 0, which is the very thing being estimated.

The practical rule is short: interval, keep the groups separate; test, pool. A standard error containing $\frac{1}{n_1} + \frac{1}{n_2}$ belongs to a test; one containing two separate fractions belongs to an interval.

The checks are the previous topic's, written with the study's numbers:

1. Random: both groups randomly sampled, or subjects randomly assigned to the two conditions.
2. Independence between groups: no shopper appears in both. A before-and-after design on the same people is paired and needs different machinery.
3. 10%: each sample at most a tenth of its own population, checked separately.
4. Large counts: all four at least 10. Here 138 and 162, then 90 and 160.

Since no value is assumed, the counts use the observed $\hat{p}$ values, which makes them simply the successes and failures in each group. That parallels the one-sample interval exactly, and it is the same reason: with nothing hypothesized, the sample's own numbers are all that is available.

The sentence has one more part than the one-sample version, and losing it is the standard deduction:

"We are 95% confident that the interval from 0.018 to 0.182 captures the true difference, new system minus old system, in the proportion of all shoppers who complete a purchase."

Three things must appear: the confidence level, what the interval captures (a difference between two population proportions, not a proportion), and the order of subtraction. Without the order, a reader cannot tell which system the interval favors.

The failed versions are the familiar ones with an extra way to go wrong. "95% of shoppers fall in this interval" describes individuals. "There is a 95% probability the true difference lies here" treats a fixed parameter as random. And "the new system converts between 1.8% and 18.2% of shoppers" drops the word difference, converting an interval about a gap into a claim about one group's rate.

What the interval's position means, and in particular what it means when 0 falls inside it, is the subject of the next topic.