Sampling Distribution Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sampling Distribution.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The probability distribution of a statistic (such as the sample mean) computed from all possible random samples of the same size drawn from a population.

Imagine you survey 50 random people about their height, compute the average, then repeat with a different group of 50, again and again. Each group gives a slightly different average. The pattern of all those averages forms the sampling distribution. It's like taking the temperature of a city by sending out 100 different thermometers—each reads slightly differently, but together they cluster around the truth.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A sampling distribution is the spread of a statistic (like the sample mean) over all possible samples of the same size.

Common stuck point: The procedure for sampling distribution is the easy part; the trap is using the population SD $\sigma$ as the spread of the mean. Asking "Am I describing how a statistic (like $\bar{x}$ ) varies across many samples, rather than how raw values vary?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I describing how a statistic (like $\bar{x}$ ) varies across many samples, rather than how raw values vary?

Worked Examples

Example 1

medium

A population has

\mu=50

and

\sigma=12

. For random samples of size

n=36

, describe the sampling distribution of

\bar{X}

: find its mean, standard error, and shape.

Answer

\bar{X} \sim N(\mu=50,\ SE=2)

. Sample means are normally distributed around 50 with SD=2.

First step

Mean of sampling distribution:

\mu_{\bar{X}} = \mu = 50

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Example 2

hard

For a population with

\mu=100

\sigma=20

, and

n=25

: (a) find

P(\bar{X} > 104)

, (b) find the value

c

such that

P(\bar{X} < c) = 0.90

Example 3

medium

A population has

\mu = 200

\sigma = 40

, and we draw samples of size

n = 64

. Find

P(\bar{X} > 210)

Example 4

medium

A population has

\mu = 80

\sigma = 15

. For samples of size

n = 9

, find

P(78 < \bar{X} < 84)

Example 5

medium

A factory's bolts have

\mu = 10

mm,

\sigma = 0.2

mm. A QC inspector samples

n = 100

bolts. Find

P(\bar{X} > 10.03)

Example 6

medium

A sampling distribution has mean

\mu = 100

and SE

4

. Find the 90th percentile of

\bar{X}

(use

z_{0.90} = 1.28

Example 7

hard

A population has

\mu = 1000

\sigma = 50

. For samples of size

n = 100

, find the value

c

so that

P(\bar{X} > c) = 0.05

(use

z_{0.95} = 1.645

Example 8

hard

A sampling distribution of

\hat{p}

for

p = 0.6

and

n = 150

is approximately normal. Find

P(0.55 < \hat{p} < 0.65)

Example 9

hard

Two independent samples of size

n_1 = 36

from population A (

\mu_A=50

\sigma_A=12

) and

n_2 = 64

from population B (

\mu_B=45

\sigma_B=16

). Find the mean and SE of

\bar{X}_A - \bar{X}_B

Example 10

challenge

A heavily right-skewed population has

\mu = 5

\sigma = 10

. Samples of size

n = 4

are drawn. Why is it incorrect to claim

\bar{X} \sim N(5, 5)

here?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

The standard error of a sample mean is

\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}

. If

\sigma=10

and

n=100

, find the standard error. What happens to the SE if n is quadrupled to 400?

Example 2

hard

A population proportion is

p=0.40

. For samples of size

n=100

, describe the sampling distribution of

\hat{p}

and find

P(\hat{p} > 0.45)

Example 3

easy

Does the sampling distribution describe individual data values or a statistic (like the sample mean) across many samples?

Example 4

easy

As sample size

n

increases, does the sampling distribution of the mean get narrower or wider?

Example 5

easy

A population has

\sigma = 10

. For samples of size

n=25

, what is the standard error of the mean?

Example 6

easy

Must you physically take thousands of samples to have a sampling distribution?

Example 7

easy

The central limit theorem says the sampling distribution of the mean approaches what shape as

n

grows?

Example 8

easy

If the population mean is

\mu = 50

, what is the mean of the sampling distribution of

\bar{x}

Example 9

easy

To halve the standard error of the mean, by what factor must you increase the sample size?

Example 10

easy

True or false: increasing

n

makes the POPULATION distribution narrower.

Example 11

medium

A population has

\mu = 100

\sigma = 15

. For

n = 9

, give the mean and standard error of

\bar{x}

Example 12

medium

With

\mu=100, \sigma=15, n=9

(so

SE=5

), what is the approximate probability that

\bar{x}

exceeds

110

? (Use the normal model.)

Example 13

medium

A fair coin is flipped

100

times. The sampling distribution of the proportion of heads has what mean and approximate standard error?

Example 14

medium

Population is heavily skewed. For

n=2

the sample mean's distribution is still skewed, but for

n=200

it is nearly normal. What principle explains this?

Example 15

medium

Two surveys estimate a mean: one with

n=100

, one with

n=400

, same population (

\sigma=20

). Compute both standard errors and the ratio.

Example 16

medium

95\%

confidence interval for a mean is

\bar{x} \pm 1.96\,SE

. With

\bar{x}=50

\sigma=10

n=100

, compute the interval.

Example 17

medium

Why can a poll of

1000

people estimate a national proportion within about

\pm3\%

, regardless of the country's population size?

Example 18

medium

A population has

\sigma = 12

. What sample size

n

gives a standard error of the mean equal to

2

Example 19

medium

A sample mean

\bar{x}=52

comes from

n=64

, population

\sigma=16

. How many standard errors is

\bar{x}

above a hypothesized

\mu=50

Example 20

challenge

Derive that the standard error of the sample mean is

\sigma/\sqrt{n}

from

\mathrm{Var}(\bar{x})

for independent observations.

Example 21

challenge

A population is uniform on

\{1,2,3\}

(

\mu=2

). List all

9

samples of size

2

(with replacement), compute each sample mean, and verify the mean of the sampling distribution equals

2

Example 22

challenge

Explain why the sampling distribution of the SAMPLE MAXIMUM does NOT center on the population mean and is not symmetric, contrasting it with the sample mean.

Example 23

easy

A population has

\sigma = 16

. For samples of size

n = 64

, compute the standard error of the sample mean.

Example 24

easy

\sigma = 30

and

n = 9

, find the standard error of

\bar{X}

Example 25

easy

For a population proportion

p = 0.5

and

n = 400

, compute

SE(\hat{p}) = \sqrt{p(1-p)/n}

Example 26

easy

A population has

\mu = 75

and

\sigma = 6

. For samples of size

n = 36

, give the mean and SE of

\bar{X}

Example 27

medium

A population has

\mu = 500

\sigma = 100

. For

n = 25

, find

P(\bar{X} < 480)

Example 28

medium

For a population with

\mu = 60

and

\sigma = 12

, what sample size

n

gives a standard error of exactly

1.5

Example 29

medium

A population proportion is

p = 0.30

. For samples of size

n = 100

, find

P(\hat{p} < 0.25)

Example 30

medium

The sampling distribution of

\bar{X}

has mean

50

and SE

2

. Within what interval do the middle 95% of sample means lie (use

z = 1.96

Example 31

medium

To cut the SE of

\bar{X}

to one-third of its current value, by what factor must

n

grow?

Example 32

hard

A teacher's class scores have

\mu = 70

\sigma = 14

. She averages

n = 49

scores. Compute the probability that the average is more than

4

points away from

70

Example 33

hard

A population has

\mu = 25

\sigma = 5

. What is the smallest

n

such that

SE \le 0.5

Example 34

hard

A population is uniform on

[0,12]

\mu = 6

and

\sigma^2 = 12

. For

n = 48

, find the SE of the sample mean.

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Related Concepts

Normal Distribution Mean Standard Deviation Central Limit Theorem Confidence Interval Hypothesis Testing

Background Knowledge

These ideas may be useful before you work through the harder examples.

normal distributionmeanstandard deviation