Mathematics
Grade 12
15 min
The Central Limit Theorem
The Central Limit Theorem
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define the Central Limit Theorem and state its conditions for applicability.
Explain how the sampling distribution of the mean approaches a normal distribution as the sample size 'n' approaches infinity, connecting this to the concept of limits.
Calculate the mean and standard error of the sampling distribution.
Recognize that key statistical formulas, like the sample mean and the Z-score, are structured as rational functions.
Apply the Central Limit Theorem to calculate the probability that a sample mean falls within a specific interval.
Distinguish between the standard deviation of a population (σ) and the standard error of the mean (σ/√n).
How can a company test the average lifespan of thousands of light bulbs by only testing a small sample o...
2
Key Concepts & Vocabulary
TermDefinitionExample
Sampling Distribution of the MeanThe theoretical probability distribution of the sample means (x̄) that would be obtained from all possible random samples of a given size 'n' from a population.If you have a population of 10 people and you take every possible sample of size 3, calculate the average height for each sample, the distribution of all those average heights is the sampling distribution.
Central Limit Theorem (CLT)A theorem stating that for a population with any distribution, the sampling distribution of the mean will approach a normal (bell-shaped) distribution as the sample size 'n' gets larger (typically n ≥ 30).Even if the individual waiting times at a bus stop are uniformly distributed, the average waiting time for 50 randomly chosen...
3
Core Formulas
Mean of the Sampling Distribution
\mu_{\bar{x}} = \mu
The mean of the sampling distribution of the mean (μ_x̄) is always equal to the mean of the original population (μ), regardless of sample size.
Standard Error of the Mean
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
The standard deviation of the sampling distribution of the mean (σ_x̄), also called the standard error. It is calculated by dividing the population standard deviation (σ) by the square root of the sample size (n). Notice how as n → ∞, the standard error → 0.
The Central Limit Theorem Z-score Formula
Z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
This formula converts a sample mean (x̄) into a standard normal Z-score. This allows us to use a standard Z-table or...
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
As the sample size 'n' approaches infinity, what is the limit of the standard error of the mean, σ_x̄ = σ/√n, and what does this conceptually imply about the sample mean x̄?
A.The limit is σ, implying x̄ becomes a less precise estimate of μ.
B.The limit is 0, implying x̄ becomes a perfect estimate of μ.
C.The limit is 1, implying x̄ is always within one standard deviation of μ.
D.The limit is infinity, implying x̄ becomes a meaningless estimate of μ.
Challenging
The scores on a standardized test are distributed with μ=500 and σ=80. What is the minimum sample size 'n' required to ensure that the probability of the sample mean being within 20 points of the population mean (i.e., P(480 < x̄ < 520)) is at least 0.99? (Use Z=2.576 for 99% confidence).
A.43
B.67
C.107
D.166
Challenging
The height of adult women is normally distributed with μ=65 inches and σ=2.5 inches. Which of the following is more likely, and why? (I) A single randomly selected woman is shorter than 62 inches. (II) The mean height of a random sample of 25 women is less than 64 inches.
A.(I) is more likely, as extreme values are more common for individuals than for sample means.
B.(II) is more likely, as sample means are always closer to the population mean.
C.They are equally likely, as both Z-scores are negative.
D.(II) is more likely because the sample size is large.
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free