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## AP®︎/College Statistics

### Course: AP®︎/College Statistics>Unit 9

Lesson 7: Sampling distributions for differences in sample means

# Differences of sample means — Probability examples

Practice using shape, center (mean), and variability (standard deviation) to calculate probabilities of various results when we're dealing with sampling distributions for the differences of sample means.

## Intro and review

In this article, we'll practice applying what we've learned about sampling distributions for the differences in sample means to calculate probabilities of various sample results.
Skip ahead if you want to go straight to some examples.
Here's a review of how we can think about the shape, center, and variability in the sampling distribution of the difference between two means x, with, \bar, on top, start subscript, 1, end subscript, minus, x, with, \bar, on top, start subscript, 2, end subscript:

### Shape

The shape of a sampling distribution of x, with, \bar, on top, start subscript, 1, end subscript, minus, x, with, \bar, on top, start subscript, 2, end subscript depends on our sample sizes and the shape of each population distribution from which we sample.
• If both populations are normal, then the sampling distribution of x, with, \bar, on top, start subscript, 1, end subscript, minus, x, with, \bar, on top, start subscript, 2, end subscript is exactly normal regardless of sample sizes.
• If one or both populations are not normal (or their shapes are unknown), then the sampling distribution of x, with, \bar, on top, start subscript, 1, end subscript, minus, x, with, \bar, on top, start subscript, 2, end subscript is approximately normal as long as our sample size is at least 30 from the not-normal population(s).

### Center

The mean difference is the difference between the population means:
mu, start subscript, x, with, \bar, on top, start subscript, 1, end subscript, minus, x, with, \bar, on top, start subscript, 2, end subscript, end subscript, equals, mu, start subscript, 1, end subscript, minus, mu, start subscript, 2, end subscript

### Variability

The standard deviation of the difference is:
sigma, start subscript, x, with, \bar, on top, start subscript, 1, end subscript, minus, x, with, \bar, on top, start subscript, 2, end subscript, end subscript, equals, square root of, start fraction, sigma, start subscript, 1, end subscript, squared, divided by, n, start subscript, 1, end subscript, end fraction, plus, start fraction, sigma, start subscript, 2, end subscript, squared, divided by, n, start subscript, 2, end subscript, end fraction, end square root
(where n, start subscript, 1, end subscript and n, start subscript, 2, end subscript are the sizes of each sample).
This standard deviation formula is exactly correct as long as we have:
• Independent observations between the two samples.
• Independent observations within each sample*.
*If we're sampling without replacement, this formula will actually overestimate the standard deviation, but it's extremely close to correct as long as each sample is less than 10, percent of its population.
Let's try applying these ideas to a few examples and see if we can use them to calculate some probabilities.

## Example 1

Every day, thousands of people at an airport pass through security on one of two levels: level A or level B. Suppose that, on average, it takes people 26 minutes to pass through security on level A with a standard deviation of 7, point, 5 minutes. On level B, the mean and standard deviation are 24 and 4 minutes, respectively.
Each day, the airport looks at separate random samples of 100 people from each level. They calculate the mean time for each sample, then look at the difference between the sample means left parenthesis, x, with, \bar, on top, start subscript, start text, A, end text, end subscript, minus, x, with, \bar, on top, start subscript, start text, B, end text, end subscript, right parenthesis.
Question 1.1
• Current
What are the mean and standard deviation (in minutes) of the sampling distribution of x, with, \bar, on top, start subscript, start text, A, end text, end subscript, minus, x, with, \bar, on top, start subscript, start text, B, end text, end subscript?