Central Limit Theorem PDF

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CENTRAL LIMIT THEOREM: STATEMENT Consider a large population: Population mean= μY Population variance = σ 2 Y Take all possible samples of size n from this population and compute the sample means Y . The Central Limit Theorem states the following: 1. The mean of the sampling distribution of the mean is equal to the population mean (i.e. E( Y ) = μ Y 2. The variance of the sampling distribution of the mean is equal to the population variance divided by the sample size (n) i.e. σ 2 Y = σ 2 Y /n (Therefore the standard deviation (or standard error) of the mean is the population standard deviation divided by the square root of the number of observations i.e. SE Y ) = σ Y / n ) ( 3. The sampling distribution of the mean will be (approximately) normally distributed. In summary: Y ≈ N μ Y ,

σ2 n

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(

Y

Applications of Central Limit Theorem In a population μY = 100 and σ 2Y = 43 . If a random sample of size n=100 is taken, use the Central Limit Theorem to find P (Y ≤ 101) . Solution Y − μY Y − μY 101− 100 = = 1.52 = If Y = 101 , then Z = SE (Y ) σ Y / n ( 43 / 100 ) Therefore, ( P≤ Y 101) = P (Z ≤ 1.52 ) = φ (1.52 ) = 0.9357 Distribution Table (see Table 1).

from the Standard Normal...