Practice Central Limit Theorem in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
For sufficiently large sample size (n \geq 30 as a rule of thumb), the sampling distribution of the sample mean is approximately normal with mean \mu and standard deviation \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 looksโskewed, bimodal, flatโthe averages of large enough samples always settle into a normal shape. It's one of the most surprising facts in all of mathematics.
Example 1
mediumA highly skewed population (times between bus arrivals) has \mu=15 min and \sigma=8 min. For samples of n=64, describe the shape, mean, and SD of the sampling distribution of \bar{X}, and find P(\bar{X} < 14).
Example 2
hardA fair die (ฮผ=3.5, ฯ=1.71) is rolled n=100 times. By CLT, find the approximate probability that the sample mean is between 3.3 and 3.7.
Example 3
easyState 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 4
hardCustomers arrive at a store with mean \mu=2 per minute, \sigma=1.5 per minute (Poisson-like). For 36-minute observation windows, find P(\text{total arrivals} > 80) using CLT.