Central Limit Theorem Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

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.

Solution

  1. 1
    CLT statement: for any population with mean μ\mu and standard deviation σ\sigma, if the sample size nn is sufficiently large (n≥30n \geq 30 as rule of thumb), then Xˉ∼N(μ,σ/n)\bar{X} \sim N(\mu, \sigma/\sqrt{n}) approximately
  2. 2
    Conditions: (1) random sample, (2) sufficiently large nn (or normally distributed population for any n)
  3. 3
    Conclusion: shape of sampling distribution becomes normal regardless of population shape

Answer

CLT: for large n (≥30), Xˉ\bar{X} is approximately N(μ,σ/n)N(\mu, \sigma/\sqrt{n}) regardless of population shape.
The CLT is the most important theorem in statistics because it justifies using the normal distribution for inference about means, regardless of the population's shape. Without CLT, we would need different inference methods for every population shape.

About Central Limit Theorem

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.

Learn more about Central Limit Theorem →

More Central Limit Theorem Examples

Example 1 medium

A highly skewed population (times between bus arrivals) has [formula] min and [formula] min. For sam

Example 2 hard

A fair die (μ=3.5, σ=1.71) is rolled [formula] times. By CLT, find the approximate probability that

Example 4 hard

Customers arrive at a store with mean [formula] per minute, [formula] per minute (Poisson-like). For

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