AP Statistics Topic 4.6: Sampling Distributions for the Difference Between Two Sample Means

01 · Start here

The concept in 60 seconds

Take a random sample from each of $2$ groups and calculate their sample means. Subtract them in a stated order. That difference is an estimate of the difference between the $2$ population means.

If you repeat the sampling process, the $2$ sample means change. Their difference changes too. The distribution of all those possible differences is the sampling distribution of $\bar x_{1}-\bar x_{2}$ .

Three questions guide the model: Where is the distribution centered? How much do the differences vary? Is a normal shape justified?

Its center is $\mu_{1}-\mu_{2}$ . For independent sample means, add their variances to find the variance of the difference, then take a square root to obtain its standard deviation. Subtracting the means does not mean subtracting their variability.

02 · One situation, deeper understanding

Comparing $2$ delivery services

A business compares delivery times from services A and B. Treat each service’s defined set of deliveries as a separate population. For this teaching example, the population summaries are known:

Population information and independent random samples
Group	Population mean and $\operatorname{SD}$	Sample and population sizes
1: Service A	$\mu_{1}=32 \,\mathrm{min}$ ; $\sigma_{1}=8 \,\mathrm{min}$	$n_{1}=40$ ; $N_{1}=2{,}000$
2: Service B	$\mu_{2}=28 \,\mathrm{min}$ ; $\sigma_{2}=6 \,\mathrm{min}$	$n_{2}=36$ ; $N_{2}=1{,}800$

Take the $2$ SRSs independently, without replacement within each population. Individual delivery times are right-skewed. Use the order $\mathrm A-\mathrm B$ , and define $D=\bar x_{A}-\bar x_{B}$ .

The population difference is $32-28=4 \,\mathrm{min}$ . Across repeated pairs of samples, $D$ is centered at $4 \,\mathrm{min}$ . It does not equal $4$ every time.

Pause and predict: Could a particular pair of samples give $D=7 \,\mathrm{min}$ , even though the population difference is $4$ ? Yes. The sampling distribution helps us describe how often results that far above the center would occur.

For these sample sizes, $D$ has a standard deviation of approximately $1.612 \,\mathrm{min}$ . Both samples are large enough for the course’s central-limit-theorem normal approximation, so we can estimate probabilities about $D$ .

03 · Understand the language

Key ideas and notation

Population difference

$\mu_{1}-\mu_{2}$ is the difference between $2$ population means. It is a fixed parameter for the specified populations.

Delivery example: the A-minus-B population difference is $4 \,\mathrm{min}$ .

Sample difference

$\bar x_{1}-\bar x_{2}$ is calculated from $1$ pair of samples. Before sampling, it is a random statistic.

$D$ is a shorter name for this difference; define its order before using it.

$2$ sources of variation

$\sigma_{1}$ and $\sigma_{2}$ describe individual values in the populations. $\frac{\sigma_{1}}{\sqrt{n_{1}}}$ and $\frac{\sigma_{2}}{\sqrt{n_{2}}}$ describe variation of the separate sample means.

Spread of the difference

$\sigma_{D}$ describes variation of $\bar x_{1}-\bar x_{2}$ across repeated samples of the stated sizes.

Its units match the original measurement units.

Independent groups and matched pairs

$2$ separately selected groups can have independent sample means. Repeated measurements on the same people, or deliberately matched observations, are paired data. For pairs, analyze within-pair differences using their own mean and $\operatorname{SD}$ .

Having $2$ samples of equal size does not establish pairing. Having $2$ different group names does not establish independence; check how the data were collected.

04 · Imagine the repetitions

Build the sampling distribution

Keep the populations and sample sizes fixed. Each repetition produces $2$ means and $1$ difference.

2

independent samples →

2

means →

1

difference

1 · Sample separatelyDraw $40$ from A and $36$ from B

Use independent random selections from the same $2$ populations each time.

2 · CalculateRecord $\bar x_{A}-\bar x_{B}$

Calculate the average within each sample, then subtract A minus B.

3 · RepeatPlot the differences

A histogram of many repeated differences approximates the sampling distribution.

Reset to the same populations for each hypothetical repetition. A sampling-distribution axis contains differences of means, not individual delivery times.

Do not mix these distributions: The distribution of individual A delivery times, the distribution of $\bar x_{A}$ , and the distribution of $\bar x_{A}-\bar x_{B}$ describe $3$ different quantities.

$1$ observed pair of sample means supplies $1$ point in this imagined distribution. You do not need to collect thousands of real samples to use the theoretical model; repeated sampling explains what the model represents.

05 · Describe the model

Calculate the center and standard deviation

Center: subtract the population means

$\mu_{D}=\mu_{1}-\mu_{2}$

Each sample mean is centered at its own population mean, so their difference is centered at the population difference. For the deliveries, $\mu_{D}=32-28=4 \,\mathrm{min}$ . This makes $\bar x_{1}-\bar x_{2}$ an unbiased estimator of $\mu_{1}-\mu_{2}$ under the stated sampling design.

Spread: add variances, then take the square root

Under the independent-observation model:

$\operatorname{Var}(D)=\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}$

$\sigma_{D}=\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}$

This variance rule requires independent sample means. For finite samples drawn without replacement, the calculations below use the unadjusted independent-observation approximation justified by the separate sampling checks. The subtraction sign changes the center; it does not cancel independent uncertainty. Squaring the coefficient $-1$ gives $+1$ in the variance calculation.

For A’s sample mean: $\frac{\sigma_{1}^{2}}{n_{1}}=\frac{8^{2}}{40}=1.6\,\mathrm{min}^{2}$ .
For B’s sample mean: $\frac{\sigma_{2}^{2}}{n_{2}}=\frac{6^{2}}{36}=1\,\mathrm{min}^{2}$ .
Add the variances: $\operatorname{Var}(D)=1.6+1=2.6\,\mathrm{min}^{2}$ .
Take the square root: $\sigma_{D}\approx\sqrt{2.6}\approx1.612452 \,\mathrm{min}$ .

Interpretation: Across repeated independent random samples of $40$ A deliveries and $36$ B deliveries, the A-minus-B sample mean difference typically varies from the population difference of $4 \,\mathrm{min}$ by about $1.612 \,\mathrm{min}$ .

What changes when sample sizes grow?

Increasing either sample size decreases its contribution to the variance. Increasing both sample sizes by a factor of $4$ divides the total variance by $4$ and multiplies the standard deviation by $\frac12$ . The center stays at $\mu_{1}-\mu_{2}$ .

Multiplying both sample sizes by

4

halves the spread

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Sampling Distributions for the Difference Between Two Sample Means

By the end of this lesson, you should be able to:

The concept in 60 seconds

Comparing $2$ delivery services