The Central Limit Theorem

The Central Limit TheoremSuppose that a sample of sizenis selected from a population that has mean and standarddeviation . LetX1,X2, ,Xnbe thenobservations that are independent and identicallydistributed ( ). Define now the sample mean and the total of thesenobservations asfollows: X= ni=1 XinT=n i=1 XiThecentral Limit theoremstates that the sample mean Xfollows approximately the normaldistribution with mean and standard deviation n, where and are the mean and stan-dard deviation of the population from where the sample was selected. The sample sizenhasto be large (usuallyn 30) if the population from where the sample is taken is the population follows the normal distribution then the sample sizencan be either smallor summarize: X N( , n).To transform Xintozwe use:z= x nExample: LetXbe a random variable with = 10 and = 4. A sample of size 100 is takenfrom this population. Find the probability that the sample mean of these 100 observations isless than 9.

with mean 50000 gallons and standard deviation 10000 gallons. The starting supply of gasoline is 74000 gallons, and there is a scheduled weekly delivery of 47000 gallons. a. Find the probability that, after 11 weeks, the supply of gasoline will be below 20000 gallons. b.

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  Central, Limits, Theorem, Gallon, The central limit theorem

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