CentralLimitTheorem(Chapter7)

CentralLimitTheorem( - The distribution of the sample means In this lesson we shall consider the cases where the population is at least 20 times

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Unformatted text preview: The distribution of the sample means. In this lesson, we shall consider the cases where the population is at least 20 times larger than the size of the sample. For example: If we take a simple random sample of size 30 from the employees of a large corporation which has 1100 employees, 1100 30  36. 66666667 , the population is 36.67 times more than the sample size This example meets the at least 20 times larger criterion. On the other hand if we take a simple random sample of 50 students from a small college with 700 students 700 50  14 , the at least 20 times criterion is not met. In such cases, we use what is called ”Finite Population Correction Factor” When the population is at least 20 times larger than the sample, we have the following Theorem: If x (the population) has mean  and standard deviation  then x mean of simple random samples of size n, has mean  and st dev  n If x is normal, x is also normal. Let us consider the following example for an illustration of the above result: Example: Given that the life of a certain make of light bulb shows a normal distribution with 1 mean 1000 hours and st dev  of 250 hours.   1000   250 b  x   1 250 2  e −  x − 1000  2 /  2  250 2  (you may ignore this expression, used only to sketch the graph) The distribution of x is given by b  x  0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 400 600 800 1000 1200 1400 1600 x If we look at the distribution of the average life of 4-pack of bulbs from the above population, that is simple random samples of size 4 from the above population, the mean of those x has mean 1000 hours st dev is 250 4  125 hours and the distribution is normal. a  x   1 125 2  e −  x − 1000  2 /  2  125 2  (you may ignore this expression, I have used it only to sketch the graph) The following picture shows that the distribution of x has less spread than x....
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This note was uploaded on 09/22/2011 for the course STAT 101 taught by Professor Unknown during the Fall '08 term at Alabama State University.

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CentralLimitTheorem( - The distribution of the sample means In this lesson we shall consider the cases where the population is at least 20 times

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