Practice Central Limit Theorem in Statistics

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

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Quick Recap

The Central Limit Theorem (CLT) states that for sufficiently large sample sizes (usually n≥30n \geq 30), the sampling distribution of the sample mean xˉ\bar{x} is approximately normal, regardless of the shape of the original population distribution.

This is statistics' magic trick: no matter how weird your population looks, if you take big enough samples and average them, those averages will form a bell curve. This is why normal distribution methods work so often.

Showing a random 20 of 76 problems.

Example 1

medium
True or false: the CLT applies to the sum of i.i.d. random variables, not just to averages.

Example 2

medium
μ=70\mu=70, σ=14\sigma=14, n=49n=49. Find P(xˉ<68)P(\bar{x}<68).

Example 3

hard
Why does the CLT make xˉ\bar{x} normal but the population shape stays whatever it is?

Example 4

medium
For a sample proportion, n=100n=100, p=0.30p=0.30. What is the approximate distribution of p^\hat{p} by the CLT?

Example 5

hard
A population has σ=12\sigma=12. What is the smallest nn such that the CLT-normal approximation gives an SE of at most 1.51.5?

Example 6

medium
A population has μ=100\mu=100, σ=24\sigma=24, n=16n=16. By the CLT, give the mean and SD of the approximate normal distribution of xˉ\bar{x}.

Example 7

hard
A skewed population has μ=200\mu=200, σ=60\sigma=60. For n=144n=144, find P(195<xˉ<207)P(195 < \bar{x} < 207).

Example 8

challenge
XX is exponential with mean μ=10\mu=10 (so σ=10\sigma=10). For n=100n=100, approximate P(xˉ>11)P(\bar{x}>11) using the CLT.

Example 9

medium
For what sample size does the CLT begin to perform well when the population is symmetric and bell-ish?

Example 10

medium
Distinguish the Central Limit Theorem from the Law of Large Numbers in one sentence each.

Example 11

medium
What is the difference between the CLT and the Law of Large Numbers in one sentence?

Example 12

easy
State the conclusion of the CLT in one phrase.

Example 13

easy
A population has μ=100\mu=100, σ=15\sigma=15. For a sample of size n=225n=225, what is the mean of the sampling distribution of xˉ\bar{x}?

Example 14

easy
As nn increases, the CLT approximation becomes ____.

Example 15

easy
Does the CLT apply equally to skewed and normal populations?

Example 16

hard
A uniform distribution has μ=5\mu = 5 and σ=2.89\sigma = 2.89. For samples of size 36, what are the mean and standard error of the sampling distribution? Is it approximately normal?

Example 17

medium
A population has σ=6\sigma=6. What sample size gives SExˉ=0.5SE_{\bar{x}}=0.5?

Example 18

medium
A population has μ=500\mu=500, σ=80\sigma=80, n=64n=64. Within what symmetric interval around 500 will about 95% of sample means fall?

Example 19

easy
A heavy-tailed population requires what relative to the n≥30n \ge 30 rule for the CLT to look good?

Example 20

medium
n=200n=200, p=0.40p=0.40. By the CLT, approximately what is P(p^>0.45)P(\hat{p}>0.45)?
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