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Statistics Notes for 6.3
Central Limit Theorem for Means
Given a population with mean µ and standard deviation σ, the sampling distribution of the
sample mean becomes approximately normal with the mean staying the same and the standard
deviation being divided by the square root of n as the sample size gets larger, regardless of the
shape of the population.
N ≥ 30 is large enough samples size to apply the Central Limit Theorem for Means for any
population.
Three Cases:
1. If the population is normal, then the sampling distribution of the means is normal.
2. If the population is either non-normal or of unknown distribution and the sample size is at
least 30, then the sampling distribution of the means is approximately normal by the
Central Limit Theorem.
3. If the population is either non-normal or of unknown distribution and the samples size is
less than 30, then there is insufficient information to conclude anything about the
distribution shape.
Let ẍ be the mean of a simple random sample of size n, drawn from a population with mean µ
and standard deviation σ
1. The mean of the sampling distribution is µẍ = µ
𝜎
2. The standard deviation of the sampling distribution is σẍ = 𝑛
√
3. The standard deviation of the sampling distribution is sometimes called the standard error of
the mean.
Assignment: p. 256 1-8(all), 10-22(evens)