8Sampling Distribution of the Mean

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8.1 Sampling Distributions

Population mean and standard deviation, and unknownMaximal Likelihood Estimator of X It is crucial that the sample drawn be random Different samples, different sample means x1, ….. xn

Sampling distribution of estimator of mean

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8.2 Central Limit Theorem

• Considered the following set of measurements for a given population: 55.20, 18.06, 28.16, 44.14, 61.61, 4.88, 180.29, 399.11, 97.47, 56.89, 271.95, 365.29, 807.80, 9.98, 82.73. The population mean is 165.570.

• Now, considered two samples from this population.

• These two different samples could have means very different from each other and also very different from the true population mean.

• What happen if we considered, not only two samples, but all possible samples of the same size ?

• The answer to this question is one of the most fascinating facts in statistics – Central limit theorem.

• It turns out that if we calculate the mean of each sample, those mean values tend to be distributed as a normal distribution, independently on the original distribution. The mean of this new distribution of the means is exactly the mean of the original population and the variance of the new distribution is reduced by a factor equal to the sample size n.

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8.2 Central Limit Theorem

• When sampling from a population with mean and variance , the distribution of the sample mean (or the sampling distribution X) will have the following properties:

• The distribution of distribution X will be approximately normal. The larger the sample is , the more will the sampling distribution resemble the normal distribution.

• The mean x of the distribution of X will be equal to , the mean of the population from which the samples were drawn.

• The variance s2 of distribution X will be equal to 2/n, the variance of the original population of X divided by the sample size. The quantity s is called the standard error of the mean.

http://cnx.org/content/m11131/latest/http://www.riskglossary.com/link/central_limit_theorem.htmhttp://www.indiana.edu/~jkkteach/P553/goals.html

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8.2 Central Limit Theorem

Distribution of sample means for samples of size n has three important properties:

(1) The mean of the sampling distribution is identical to the population mean

(2) The standard deviation of the distribution of sample means is equal to

(3) Provided that n is large enough, the shape of the sampling distribution is approximately normal.

n

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