Unit 4 · Topic 4.6 · about 20 minutes

Sampling Distributions for the Difference Between Two Sample Means

Describe how the difference between two sample means varies, check that the description applies, and find and interpret probabilities about that difference.

Predict first

Two independent random samples are taken from two different populations. The first sample mean has a standard deviation of 3, and the second has a standard deviation of 4. What is the standard deviation of the difference between the two sample means?

The difference as a statistic

Many questions compare two populations, such as the weights of two apple varieties. The statistic is the difference between the sample means, ${\overline{x}}_{1} - {\overline{x}}_{2}$ , and like any statistic it varies from one pair of samples to the next.

For two independent populations with means $μ_{1}$ and $μ_{2}$ and standard deviations $σ_{1}$ and $σ_{2}$ , when the sampled values are independent, the sampling distribution of ${\overline{x}}_{1} - {\overline{x}}_{2}$ has

μ_{{\overline{x}}_{1} - {\overline{x}}_{2}} = μ_{1} - μ_{2} σ_{{\overline{x}}_{1} - {\overline{x}}_{2}} = \sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}

The center is the true difference in population means, so the statistic does not systematically overshoot or undershoot it. Each term under the square root is the square of a $σ / \sqrt{n}$ from Topic 4.1, so the standard deviation combines the variability of both sample means. Decide the order of subtraction at the start and keep it everywhere: ${\overline{x}}_{2} - {\overline{x}}_{1}$ has the same spread but the opposite center.

Conditions and shape

The standard deviation formula needs the sampled values to be independent. When you sample without replacement, two conditions make that reasonable:

Randomization condition: the data come from two independent random samples.
10% condition: each sample is at most 10% of its own population, $n_{1} \leq 0.10 N_{1}$ and $n_{2} \leq 0.10 N_{2}$ .

If the data come from an experiment instead, the only condition is randomization, and it is met when the treatments are randomly assigned to the experimental units.

The shape follows the same logic as for one mean, applied to both samples. If both populations can be modeled by normal distributions, the sampling distribution of ${\overline{x}}_{1} - {\overline{x}}_{2}$ is normal for any sample sizes. If they cannot, the sampling distribution is approximately normal when $n_{1} \geq 30$ and $n_{2} \geq 30$ . Both, not one.

When can the sampling distribution of ${\overline{x}}_{1} - {\overline{x}}_{2}$ be modeled with a normal distribution?
The two populations are	Sample sizes	Normal model for ${\overline{x}}_{1} - {\overline{x}}_{2}$ ?
Both approximately normal	Any sizes	Yes
Not normal, or unknown	$n_{1} \geq 30$ and $n_{2} \geq 30$	Yes, approximately
Not normal, or unknown	Either sample under 30	Not justified

Worked exampleHow often does the gap look bigger than it is?

At a large orchard, Honeycrisp apple weights are approximately normal with mean 210 grams and standard deviation 30 grams. Gala apple weights are approximately normal with mean 180 grams and standard deviation 20 grams. A buyer picks independent random samples of 12 Honeycrisp and 15 Gala apples. Find the probability that the Honeycrisp sample mean is more than 40 grams larger than the Gala sample mean.

Define the statistic. Let ${\overline{x}}_{H} - {\overline{x}}_{G}$ be the Honeycrisp sample mean minus the Gala sample mean.
Center and spread. $μ_{{\overline{x}}_{H} - {\overline{x}}_{G}} = 210 - 180 = 30$ grams and $σ_{{\overline{x}}_{H} - {\overline{x}}_{G}} = \sqrt{\frac{30^{2}}{12} + \frac{20^{2}}{15}} = \sqrt{75 + 26.67} \approx 10.08$ grams.
Conditions and shape. The samples are independent random samples, and 12 and 15 apples are far less than 10% of each variety at a large orchard. Both populations are approximately normal, so the sampling distribution is normal even though both samples are small.
Calculate. $z = \frac{40 - 30}{10.08} \approx 0.99$ , and the area to the right is about 0.161.
Interpret. In about 16% of pairs of random samples like these, the Honeycrisp sample mean would exceed the Gala sample mean by more than 40 grams, even though the true difference in mean weight is 30 grams.

Answer.

About 0.161.

Sampling distribution of ${\overline{x}}_{H} - {\overline{x}}_{G}$ for samples of 12 and 15 apples

Centered at the true difference of 30 grams. The shaded area, about 0.161, is the chance the sample difference exceeds 40 grams.

Read the picture from the other side too. A difference of 0 is almost 3 standard deviations below the center, $z = (0 - 30) / 10.08 \approx - 2.98$ , so the chance that the Gala sample comes out heavier on average is only about 0.0015. With these sample sizes, a buyer comparing the two samples will nearly always see the right variety come out ahead, but the size of the gap can easily be off by 10 grams or more.

Check your understanding

Scores on a reading assessment have standard deviation 6 points in District 1 and 8 points in District 2, both large districts. Independent random samples of 36 students from District 1 and 32 students from District 2 are selected. Find the standard deviation of the sampling distribution of ${\overline{x}}_{1} - {\overline{x}}_{2}$ . Round to three decimal places.

Your answer

University records show that the mean commute distance is 11.2 miles for all graduate students and 7.5 miles for all undergraduates. Independent random samples of 40 graduate students and 50 undergraduates are selected. What is the mean of the sampling distribution of (undergraduate sample mean minus graduate sample mean)?

Household incomes in two large counties are strongly skewed to the right. An economist takes independent random samples of 45 households from County A and 18 households from County B. Which statement about the sampling distribution of ${\overline{x}}_{A} - {\overline{x}}_{B}$ is correct?

At an orchard, Honeycrisp apples have mean weight 210 grams and Gala apples 180 grams. For independent random samples of 12 Honeycrisp and 15 Gala apples, the sampling distribution of ${\overline{x}}_{H} - {\overline{x}}_{G}$ has standard deviation 10.08 grams. Which is a correct interpretation of 10.08 grams?

A sleep researcher randomly assigns 40 volunteers to wear either blue-light-blocking glasses or regular glasses in the evening and records how long each one takes to fall asleep. Which condition must these data meet for the mean and standard deviation formulas for the sampling distribution of ${\overline{x}}_{1} - {\overline{x}}_{2}$ to apply?

Course alignment, for teachers

AP Statistics topic 4.6, Unit 4: Inference for Quantitative Data: Means.

Skill 3.D: Calculate means, standard deviations, and parameters for probability distributions.
Skill 4.D: Interpret statistical calculations and results to assess meaning or a claim.
Skill 4.E: Justify the use of a chosen statistical inference method by verifying conditions.