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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

{\bar{x}}_{1} - {\bar{x}}_{2}

Shape

The shape of a sampling distribution of

{\bar{x}}_{1} - {\bar{x}}_{2}

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 ${\bar{x}}_{1} - {\bar{x}}_{2}$ ‍ 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 ${\bar{x}}_{1} - {\bar{x}}_{2}$ ‍ 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:

μ_{{\bar{x}}_{1} - {\bar{x}}_{2}} = μ_{1} - μ_{2}

Variability

The standard deviation of the difference is:

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

(where

n_{1}

and

n_{2}

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 %

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

7.5

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

({\bar{x}}_{A} - {\bar{x}}_{B})

Question 1.1

What are the mean and standard deviation (in minutes) of the sampling distribution of ${\bar{x}}_{A} - {\bar{x}}_{B}$ ‍?

Example 2

A large university has over

30,000

students and over

1,500

professors. Suppose that the ages of students are strongly skewed to the right with a mean and standard deviation of

21

years and

3

years, respectively. The ages of professors are also right-skewed, and their mean and standard deviation are

50

years and

5

years, respectively.

A student conducting a study plans on taking separate random samples of

100

students and

20

professors. They'll look at the difference between the mean age of each sample

({\bar{x}}_{P} - {\bar{x}}_{S})

The student wonders how likely it is that the difference between the two sample means is greater than

35

years.

Question 2.1

Why is it inappropriate to use a normal distribution to calculate this probability?

(Choice A)
We're sampling more than $10 %$ ‍ of each population, so we don't know the standard deviation of the sampling distribution.
(Choice B)
We're sampling less than $10 %$ ‍ of each population, so the sampling distribution isn't approximately normally distributed.
(Choice C)
Neither population is normal, so we need both sample sizes to be at least $30$ ‍, but the sample size of professors is too small.
(Choice D)
Neither population is normal, so we need both sample sizes to be less than $30$ ‍, but the sample size of students is too large.

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