Central Limit Theorem Statistics Example 1

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

Example 1

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

Solution

  1. 1
    Step 1: The Central Limit Theorem states that for sufficiently large nn (typically nโ‰ฅ30n \geq 30), the sampling distribution of xห‰\bar{x} is approximately normal.
  2. 2
    Step 2: Here n=50โ‰ฅ30n = 50 \geq 30, so the CLT applies even though the population is skewed.
  3. 3
    Step 3: The sampling distribution of xห‰\bar{x} is approximately normal with mean 40 and SE = 1050โ‰ˆ1.41\frac{10}{\sqrt{50}} \approx 1.41.

Answer

Approximately normal with mean 40 and SE โ‰ˆ 1.41, by the Central Limit Theorem.
The CLT is one of the most important results in statistics. It explains why normal-based inference works even for non-normal populations, provided the sample size is large enough.

About Central Limit Theorem

The Central Limit Theorem (CLT) states that for sufficiently large sample sizes (usually nโ‰ฅ30n \geq 30), the sampling distribution of the sample mean xห‰\bar{x} is approximately normal, regardless of the shape of the original population distribution.

Learn more about Central Limit Theorem โ†’

More Central Limit Theorem Examples

Example 2 hard

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

Example 3 hard

A uniform distribution has [formula] and [formula]. For samples of size 36, what are the mean and st

Example 4 hard

A population is strongly right-skewed. If samples of size 100 are taken repeatedly, what does the Ce

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