Practice Central Limit Theorem in Statistics

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

Quick Recap

The Central Limit Theorem (CLT) states that for sufficiently large sample sizes (usually n \geq 30), the sampling distribution of the sample mean \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.

Example 1

hard
A population has a right-skewed distribution with \mu = 40 and \sigma = 10. If we take samples of size 50, describe the shape of the sampling distribution of \bar{x}.

Example 2

hard
Explain why the Central Limit Theorem is important for making confidence intervals.

Example 3

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

Example 4

hard
A population is strongly right-skewed. If samples of size 100 are taken repeatedly, what does the Central Limit Theorem say about the sampling distribution of the sample mean?
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