Practice Sampling Distribution in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

hard
True or false: the standard error of Xˉ\bar{X} depends on the population size NN (assuming NN is much larger than nn).

Example 2

easy
Fill in the blank: the spread of the sampling distribution of Xˉ\bar{X} is ____ than the spread of the population.

Example 3

easy
The standard error of a sample mean is σXˉ=σn\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}. If σ=10\sigma=10 and n=100n=100, find the standard error. What happens to the SE if n is quadrupled to 400?

Example 4

easy
A population has μ=75\mu = 75 and σ=6\sigma = 6. For samples of size n=36n = 36, give the mean and SE of Xˉ\bar{X}.

Example 5

easy
A population has σ=16\sigma = 16. For samples of size n=64n = 64, compute the standard error of the sample mean.

Example 6

medium
A population proportion is p=0.30p = 0.30. For samples of size n=100n = 100, find P(p^<0.25)P(\hat{p} < 0.25).

Example 7

challenge
A population is uniform on {1,2,3}\{1,2,3\} (μ=2\mu=2). List all 99 samples of size 22 (with replacement), compute each sample mean, and verify the mean of the sampling distribution equals 22.

Example 8

medium
A population has μ=80\mu = 80, σ=15\sigma = 15. For samples of size n=9n = 9, find P(78<Xˉ<84)P(78 < \bar{X} < 84).

Example 9

easy
As sample size nn increases, does the sampling distribution of the mean get narrower or wider?

Example 10

medium
A population has μ=500\mu = 500, σ=100\sigma = 100. For n=25n = 25, find P(Xˉ<480)P(\bar{X} < 480).

Example 11

medium
For a population with μ=60\mu = 60 and σ=12\sigma = 12, what sample size nn gives a standard error of exactly 1.51.5?

Example 12

medium
A sampling distribution has mean μ=100\mu = 100 and SE 44. Find the 90th percentile of Xˉ\bar{X} (use z0.90=1.28z_{0.90} = 1.28).

Example 13

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 14

hard
A population has μ=25\mu = 25, σ=5\sigma = 5. What is the smallest nn such that SE≤0.5SE \le 0.5?

Example 15

easy
If σ=30\sigma = 30 and n=9n = 9, find the standard error of Xˉ\bar{X}.

Example 16

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

Example 17

medium
A sampling distribution of Xˉ\bar{X} has mean 00 and SE 11. What is its name?

Example 18

medium
A sample mean xˉ=52\bar{x}=52 comes from n=64n=64, population σ=16\sigma=16. How many standard errors is xˉ\bar{x} above a hypothesized μ=50\mu=50?

Example 19

medium
A population has μ=50\mu=50 and σ=12\sigma=12. For random samples of size n=36n=36, describe the sampling distribution of Xˉ\bar{X}: find its mean, standard error, and shape.

Example 20

hard
A population proportion is p=0.40p=0.40. For samples of size n=100n=100, describe the sampling distribution of p^\hat{p} and find P(p^>0.45)P(\hat{p} > 0.45).
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