Practice Central Limit Theorem in Math

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

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Quick Recap

For sufficiently large sample size (n≥30n \geq 30 as a rule of thumb), the sampling distribution of the sample mean is approximately normal with mean μ\mu and standard deviation σn\frac{\sigma}{\sqrt{n}}, regardless of the shape of the population distribution.

Roll a single die and the outcomes are flat (uniform). But average the rolls of 30 dice and the result looks like a bell curve every time. No matter how weird the original data looks—skewed, bimodal, flat—the averages of large enough samples always settle into a normal shape. It's one of the most surprising facts in all of mathematics.

Showing a random 20 of 50 problems.

Example 1

hard
An exponential population (e.g., wait times) has μ=5\mu = 5, σ=5\sigma = 5. For n=100n = 100, find P(Xˉ>5.5)P(\bar{X} > 5.5).

Example 2

medium
A truck-weight population has μ=5000\mu = 5000 lb, σ=800\sigma = 800 lb. For n=64n = 64 trucks, find P(Xˉ<4900)P(\bar{X} < 4900).

Example 3

easy
A population has σ=18\sigma = 18. For samples of size n=81n = 81, what is the SE of Xˉ\bar{X}?

Example 4

medium
A population has μ=120\mu = 120, σ=30\sigma = 30. For n=36n = 36, what is the probability Xˉ\bar{X} falls within 55 of μ\mu?

Example 5

medium
A population has μ=70\mu = 70, σ=12\sigma = 12. For n=36n = 36, find P(Xˉ>73)P(\bar{X} > 73).

Example 6

medium
A population has σ=25\sigma = 25. To cut the standard error of the mean from 55 to 2.52.5, by what factor must nn increase?

Example 7

medium
A population has σ=10\sigma = 10. By what factor does the SE of Xˉ\bar{X} change when nn goes from 2525 to 400400?

Example 8

easy
If the sample size increases from n=25n=25 to n=100n=100 (a factor of 44), the standard error of the mean changes by what factor?

Example 9

medium
A coin is flipped n=100n = 100 times. Let Xˉ\bar{X} be the sample proportion of heads. Use CLT to find P(Xˉ>0.55)P(\bar{X} > 0.55).

Example 10

medium
A casino's roulette bet has expected loss μ=−0.05\mu = -0.05 per dollar and σ=1\sigma = 1. After n=10000n = 10000 bets, find P(Xˉ>0)P(\bar{X} > 0) (player profits on average).

Example 11

medium
A highly skewed population (times between bus arrivals) has μ=15\mu=15 min and σ=8\sigma=8 min. For samples of n=64n=64, describe the shape, mean, and SD of the sampling distribution of Xˉ\bar{X}, and find P(Xˉ<14)P(\bar{X} < 14).

Example 12

easy
According to the common rule of thumb, what minimum sample size makes the CLT a reasonable approximation for the sample mean?

Example 13

easy
True or false: the CLT applies to the sampling distribution of the sample mean, not to individual observations.

Example 14

hard
A population proportion is p=0.20p = 0.20. Verify CLT applies for n=50n = 50 and find P(p^>0.25)P(\hat{p} > 0.25).

Example 15

challenge
A population is strongly right-skewed with μ=5\mu = 5, σ=5\sigma = 5. Explain why n=30n = 30 may be insufficient for the normal approximation of xˉ\bar{x}, and state what determines the needed nn.

Example 16

easy
For σ=6\sigma = 6 and n=36n = 36, find the SE of Xˉ\bar{X}.

Example 17

easy
State the Central Limit Theorem in your own words, including what conditions must be met and what it tells us about the shape of the sampling distribution.

Example 18

medium
A population has μ=40\mu = 40, σ=9\sigma = 9. For n=81n = 81, find P(xˉ>41)P(\bar{x} > 41). Use P(Z>1)≈0.1587P(Z > 1) \approx 0.1587.

Example 19

medium
True or false: as nn grows, the variance of Xˉ\bar{X} shrinks toward 00.

Example 20

medium
For μ=60\mu = 60, σ=8\sigma = 8, n=16n = 16, find the SE.
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