Pooled or not on a TI-84 two-sample t-test? Choose No

Pooled or not on a two-sample t-test: press No

Your calculator's 2-SampTTest screen ends with a line that reads Pooled: No Yes. Highlight No and press enter. The 2-SampTInt screen asks the same question for a confidence interval, and the answer there is No as well.

That holds for every two-sample means problem in AP Statistics, and the reason is short. The formula sheet gives exactly one standard error for the difference between two means:

That is the Pooled: No formula. No pooled two-sample formula for means appears on the sheet at all, so pressing Yes produces a t statistic, a degrees of freedom, and a p-value that do not come from the procedure the course is built on.

Pooled is not a display option or a rounding preference. It swaps in a different standard error, a different df, and usually a different p-value. Worked example 1 below runs one data set both ways and reaches opposite decisions at $\alpha = 0.05$.

What the Pooled setting actually changes

Pooled: No lets each group keep its own spread. Each group contributes its own variance divided by its own sample size, and the two pieces add under the square root:

Pooled: Yes assumes the two populations share one standard deviation, so it merges both samples into a single estimate of that shared spread, written $s_p$:

Read $s_p^2$ as a weighted average of the two sample variances, each weighted by its own degrees of freedom. The bigger sample gets more say in the result.

Both versions share the same numerator, $\bar{x}_1 - \bar{x}_2$. Everything that differs sits in the denominator and in the df.

Why No is the safe setting

Pooled: No is valid whether or not the two populations have the same spread. Pooled: Yes is valid only when they do. That asymmetry settles the question. One setting works in both worlds, the other works in one, and two samples give you no reliable way to tell which world you are in.

Students try to justify Yes by looking at $s_1$ and $s_2$ and deciding they seem close. Sample standard deviations bounce around a lot at homework sample sizes, so two samples drawn from populations with identical spreads regularly produce $s$ values that differ by a factor of two. A close-looking pair is weak evidence of equal population spreads, and a far-apart pair does not prove they differ either.

The cost of the safe setting is small. When the equal-spread assumption really does hold, pooling buys a few extra degrees of freedom and a slightly smaller p-value. The cost of the unsafe setting can be large, and the dangerous case is unequal sample sizes together with unequal spreads:

• If the smaller sample comes from the population with the larger spread, pooling understates the standard error. Your t is too big, your p-value too small, and you reject a true null hypothesis more often than your significance level claims.
• If the smaller sample comes from the less variable population, pooling overstates the standard error and the test turns too cautious instead.

Neither error announces itself on the screen. That is why R and Minitab treat the unpooled version as their default, and why the AP course teaches only that one.

How to tell whether you pressed Yes

Two lines on the output screen give the setting away, and one of them settles it outright.

The decisive tell is $\text{Sxp}$, the pooled standard deviation. Pooled output carries that extra line and unpooled output has no such line at all, so if $\text{Sxp}$ is on your screen you pooled.

The fast tell is the degrees of freedom.

• A df that is a decimal, such as 4.325 or 12.422, came from the Welch-Satterthwaite formula. You ran Pooled: No.
• A df that is a **whole number equal to $n_1 + n_2 - 2$** almost always came from pooling. With $n_1 = 5$ and $n_2 = 20$, a df reported as exactly 23 is your cue to look again.

Almost, not always. The Welch df is never larger than $n_1 + n_2 - 2$ and never smaller than the smaller of $n_1 - 1$ and $n_2 - 1$, and it reaches that ceiling on its own exactly when $\frac{s_1^2}{n_1(n_1 - 1)} = \frac{s_2^2}{n_2(n_2 - 1)}$. For two samples of the same size that condition is just $s_1 = s_2$, which turns up often enough to trip you: two groups of 10 with $s_1 = s_2 = 5$ report $df = 18$ under either setting. For unequal sample sizes it takes a coincidence in the $s$ values you will not stumble into. So a df parked at the top of the range means you pooled, unless the two groups are the same size with matching spreads, and that is the case where $\text{Sxp}$ earns its keep.

Calculators generally remember the setting from one run to the next, and the same choice carries over to 2-SampTInt, so one stray Yes can follow you through a whole problem set. Check the df every time. For the rest of the fields on that screen, see reading calculator output.

Pooling is right for the two-proportion z-test, not for means

Here is where the confusion starts. Pooling really is the AP method for a two-proportion z-test. Under the null hypothesis $p_1 = p_2$, both samples estimate the same proportion, so you combine the counts of successes $X_1$ and $X_2$ into one estimate:

and build the standard error from that single combined value. That is genuine pooling, it is on the formula sheet, and the calculator's 2-PropZTest does it for you without asking.

The test is the only place it happens. A confidence interval for $p_1 - p_2$ does not pool, because an interval never assumes the two proportions are equal, it estimates how far apart they are. The formula sheet gives the interval its own standard error, $\sqrt{\hat{p}_1(1 - \hat{p}_1)/n_1 + \hat{p}_2(1 - \hat{p}_2)/n_2}$, and 2-PropZInt uses each sample's own $\hat{p}$ without asking either.

Means work differently. The null hypothesis for two means says $\mu_1 = \mu_2$. It says nothing whatsoever about $\sigma_1$ and $\sigma_2$. So the null gives you no license to merge the two spreads, and each group's variability stays its own. That is the whole reason the two-sample t standard error keeps $s_1$ and $s_2$ apart, while the two-proportion z test statistic, working under a null that does claim $p_1 = p_2$, stops keeping $\hat{p}_1$ and $\hat{p}_2$ apart.

Store it as a pair: pool for the two-proportion z-test, never for means. The proportion side is worked out in full in one-proportion vs two-proportion z-test.

What to write on the AP exam

The setting changes your numbers, so it changes your write-up. A clean two-sample t answer names the procedure and reports the unpooled results.

1. State. Define $\mu_1$ and $\mu_2$ in context, then write $H_0: \mu_1 - \mu_2 = 0$ and an alternative that matches the question.
2. Plan. Name it: a two-sample t-test for a difference between two population means. Check random, the 10 percent condition when sampling without replacement, and normal-or-large for each group.
3. Do. Report the t statistic, the df, and the p-value from the Pooled: No run. A decimal df is expected and correct, so do not round it off to look tidier.
4. Conclude. Compare the p-value to $\alpha$ and answer in context.

Working without a calculator, use the conservative df: the smaller of $n_1 - 1$ and $n_2 - 1$. It sits below the true Welch df, so the p-value you read from the t-table comes out a little too large, which never overstates your evidence.

Two-sample means work lives in Unit 4, topics 4.7 through 4.10. Check your numbers with the two-sample t-test calculator, which uses the Welch version and so matches Pooled: No. If the two measurements come from the same subjects, you need a different procedure entirely: see paired vs two-sample t-test.