Math Core

Many of the most interesting questions in statistics are comparisons. Do more seniors than juniors have part-time jobs? Do plants given a new fertilizer grow taller than plants without it? To answer these, you compare two sample statistics, and you need to know how much their difference varies by chance alone.

Building from what you know

In Unit 4 you learned how to combine two random variables $X$ and $Y$ :

\mu_{X - Y} = \mu_X - \mu_Y, \qquad \sigma^2_{X - Y} = \sigma^2_X + \sigma^2_Y \ \text{ (if } X \text{ and } Y \text{ are independent)}.

Means subtract, but variances add, even for a difference. Subtracting a random quantity adds uncertainty; it never cancels it. The sampling distributions of $\hat{p}_1 - \hat{p}_2$ and $\bar{x}_1 - \bar{x}_2$ come directly from these rules, applied to the formulas from the last few lessons.

Difference of two sample proportions

Take independent random samples of sizes $n_1$ and $n_2$ from two populations with success proportions $p_1$ and $p_2$ .

Sampling distribution of p̂₁ − p̂₂

Center: $\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$ .
Spread: $\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}$ , when the samples are independent and each satisfies the 10% condition.
Shape: approximately normal when all four Large Counts are at least 10: $n_1p_1$ , $n_1(1-p_1)$ , $n_2p_2$ , $n_2(1-p_2)$ .

Difference of two sample means

Take independent random samples of sizes $n_1$ and $n_2$ from two populations with means $\mu_1, \mu_2$ and standard deviations $\sigma_1, \sigma_2$ .

Sampling distribution of x̄₁ − x̄₂

Center: $\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2$ .
Spread: $\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}$ , when the samples are independent and each satisfies the 10% condition.
Shape: normal if both populations are normal; approximately normal if both $n_1 \ge 30$ and $n_2 \ge 30$ (CLT).

Conditions, twice

For differences, every condition is checked for each sample, plus one new requirement: the two samples must be independent of each other. Separate random samples from two populations, or two groups formed by random assignment, are independent. Measurements taken twice on the same people (before and after) are not independent samples; that is paired data, which you will handle in Unit 7.

Common mistake

Never subtract standard deviations. A common wrong answer is $\sigma_1/\sqrt{n_1} - \sigma_2/\sqrt{n_2}$ , or taking the square root of each variance separately and subtracting. Add the variances (the terms under the square root), then take one square root at the end.

Here is a table excerpt with the values used in this lesson.

$z$	$-1.50$	$-1.00$	$-0.80$	$1.25$	$2.00$
Area to the left	$0.0668$	$0.1587$	$0.2119$	$0.8944$	$0.9772$

Worked example: Could the sample reverse the truth?

At a large university, 30% of engineering majors and 25% of business majors say they plan to attend graduate school. A researcher takes independent random samples of 120 engineering majors and 250 business majors. Find the probability that the sample proportion for engineering is smaller than the sample proportion for business, reversing the true order.

Conditions. The samples are random and independent. Each sample is less than 10% of its major at a large university. Large Counts: $120(0.30) = 36$ , $120(0.70) = 84$ , $250(0.25) = 62.5$ , $250(0.75) = 187.5$ , all at least 10. So $\hat{p}_E - \hat{p}_B$ is approximately normal.

Center and spread.

\mu = 0.30 - 0.25 = 0.05,

\sigma = \sqrt{\frac{0.30(0.70)}{120} + \frac{0.25(0.75)}{250}} = \sqrt{0.00175 + 0.00075} = \sqrt{0.0025} = 0.05.

Probability. We want $P(\hat{p}_E - \hat{p}_B < 0)$ :

z = \frac{0 - 0.05}{0.05} = -1.00, \qquad P = 0.1587.

Sampling distribution of the difference in sample proportions: mean 0.05, SD 0.05. The shaded area below 0 is about 0.1587.Open in grapher →

About 16% of the time, the samples would point the wrong way. Sampling variability in a difference can be large even when each sample is reasonably big.

Worked example: Comparing two means

Scores on a science exam are normally distributed at two large schools. At School A, $\mu_A = 78$ and $\sigma_A = 12$ . At School B, $\mu_B = 74$ and $\sigma_B = 16$ . Independent random samples of 16 students are taken from each school. Find the probability that the School B sample mean is higher than the School A sample mean.

Conditions. Independent random samples; each sample of 16 is less than 10% of a large school; both populations are normal, so $\bar{x}_A - \bar{x}_B$ is normal even with small samples.

Center and spread.

\mu_{\bar{x}_A - \bar{x}_B} = 78 - 74 = 4, \qquad \sigma_{\bar{x}_A - \bar{x}_B} = \sqrt{\frac{12^2}{16} + \frac{16^2}{16}} = \sqrt{9 + 16} = 5.

Probability. School B's mean is higher when $\bar{x}_A - \bar{x}_B < 0$ :

z = \frac{0 - 4}{5} = -0.80, \qquad P(\bar{x}_A - \bar{x}_B < 0) = 0.2119.

Tip

Define the order of subtraction once and stick with it. "Group 1 larger than group 2" means the difference $1 - 2$ is greater than 0; "group 2 larger" means it is less than 0. Writing the event in terms of the difference before standardizing prevents sign errors.

Practice

Use the table excerpt above where needed. Give probabilities to four decimal places.

Practice 1

In a large city, 40% of adults under 30 and 30% of adults 30 and over follow a local news account. Independent random samples of 160 younger adults and 100 older adults are selected. What is the mean of the sampling distribution of $\hat{p}_{\text{young}} - \hat{p}_{\text{older}}$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

For the same situation ( $p_1 = 0.40$ , $n_1 = 160$ ; $p_2 = 0.30$ , $n_2 = 100$ ), find the standard deviation of the sampling distribution of $\hat{p}_1 - \hat{p}_2$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

For the same situation, find the probability that $\hat{p}_1 - \hat{p}_2$ is less than 0.01.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Independent random samples are taken from two populations. The standard deviation of $\bar{x}_1$ is 3 and the standard deviation of $\bar{x}_2$ is 4. What is the standard deviation of $\bar{x}_1 - \bar{x}_2$ ?

Practice 5

A researcher compares the proportion of left-handed people in two large towns. She takes independent random samples of 150 people from Town 1, where $p_1 = 0.10$ , and 80 people from Town 2, where $p_2 = 0.11$ . Which statement about using a normal model for $\hat{p}_1 - \hat{p}_2$ is correct?

Practice 6

Two large populations have standard deviations $\sigma_1 = 15$ and $\sigma_2 = 20$ . Independent random samples of size 25 are taken from each. Find the standard deviation of the sampling distribution of $\bar{x}_1 - \bar{x}_2$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

The weights of adult males of a certain breed of dog are normally distributed with mean 50 pounds and standard deviation 9 pounds. The weights of adult females are normally distributed with mean 45 pounds and standard deviation 8 pounds. A veterinarian selects independent random samples of 9 males and 4 females from a large registry. Find the probability that the male sample mean exceeds the female sample mean by more than 11.25 pounds.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.