Topic 5.8 Sampling Distributions for Differences in Sample Means: describe the mean, standard deviation, and shape of the sampling distribution of the difference between two independent sample means, adding variances and checking the conditions for normality.

What this topic is asking

The College Board (Topic 5.8) wants you to describe the mean, standard deviation, and shape of the sampling distribution of the difference between two independent sample means $\bar{x}_1 - \bar{x}_2$, adding the variances and checking the conditions for normality.

Center and spread of the difference

The mean is the difference of the two population means. The standard deviation adds the two sample-mean variances $\dfrac{\sigma_1^2}{n_1}$ and $\dfrac{\sigma_2^2}{n_2}$ and square-roots the sum. Each term is the variance of one sample mean (the square of $\dfrac{\sigma}{\sqrt{n}}$ from Topic 5.7), so this is the same combining rule from Topic 4.9 applied to two independent sample means.

Why variances add (again)

This is the identical principle to Topic 5.6 for proportions: variances of independent quantities add under both addition and subtraction. The persistent error is to subtract the variances or to combine the standard deviations directly. The correct procedure is always: square each sample mean's standard deviation to get its variance, add the two variances, then take the square root.

The conditions for normality

The shape is approximately normal when each sample mean is approximately normal, so the shape conditions of Topic 5.7 must hold for both samples. Each sample mean is normal if its population is normal (any $n$) or its sample is large (CLT, commonly $n \ge 30$). In a typical two-sample problem you justify normality by stating that both populations are normal, or that both sample sizes are large, or a mix (one normal population, one large sample). In addition, the two samples must be independent of each other, and each sample should be at most $10\%$ of its population for the standard deviation formula to be valid. A complete answer checks all of this: the normality justification for each sample, the between-sample independence, and the $10\%$ condition for each. This is the two-mean analogue of the doubled conditions seen for two proportions, and it is the main way the two-sample topic extends the one-sample Topic 5.7.

Why this matters for inference

Topic 5.8 is the sampling-distribution foundation for comparing two means, the basis of the two-sample $t$ procedures in Unit 7. A confidence interval for $\mu_1 - \mu_2$ is centered at $\bar{x}_1 - \bar{x}_2$ with a width built from this added-variances standard deviation (estimated as a standard error using $s_1$ and $s_2$, which is why a $t$-distribution is used in practice), and a two-sample significance test computes a standardized statistic with it. Being able to state the center $\mu_1 - \mu_2$, compute the added-variances spread, justify the normal shape for both samples, and find a probability is exactly the preparation for those procedures. As with proportions, an illuminating question asks for the probability that the observed difference is large or even has the opposite sign to $\mu_1 - \mu_2$, which underscores that a single observed difference of sample means is one draw from a distribution of possible differences, not the true difference itself. Completing the full template here, center, added-variances spread, shape justification, standardize, interpret, rounds out Unit 5 and feeds straight into two-sample mean inference.

Try this

Q1. Write the standard deviation formula for $\bar{x}_1 - \bar{x}_2$ and state why variances are added. [2 points]

• Cue. $\sigma = \sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}$; variances add because the samples are independent (and add even for a difference).

Q2. What must be true for $\bar{x}_1 - \bar{x}_2$ to be approximately normal? [1 point]

• Cue. Each sample mean must be approximately normal (both populations normal, or both samples large by the CLT), with the samples independent and each at most $10\%$ of its population.