# The distribution of the sample means tends to be a

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The mean of all sample means coincides with the population mean.The distribution of the sample means tends to be a normal distribution.The Central Limit Theorem (CLT)Assume thatxis a random variable from a population withmean"μ"andstandard deviation"σ".If a sample of sizenis selected from a such population and eithern30 orxis normally distributed, then the sampling distribution of the mean(ofx)approaches a normal distribution for whichmean:μx=μ,andstandard deviation:σx=σpn.A. Bourhim (Syracuse University)Sampling Distributions and The Central Limit TheoremMAT121, Spring 20156 / 9
When you should use CLTA CLT word problem will most likely contain the phrase "assume the variableis normally distributed", or "the given sample sizenwill be large".You will be given:The population meanμand standard deviationσ.A sample sizen.You may be asked:find the probability thatthe mean of a sample of sizenis greater or lessthan a number.find the probability thatthe mean of a sample of sizenis between twonumbers.Be careful:When working with a sample meanx, its correspondingz-score isz-score=x-μxσx=x-μσpn.