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# Central Limit Thereom

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Author: Bob Kibler
##### Description:
• Determine the probability of a sample randomly obtained from an approximately distributed occurring using the standard normal curve
• Compare the probabilities of an event occurring as the sample size increases
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Tutorial

## Notes

The Central Limit Theorem

Not always practical or possible to investigate an entire population, so we must investigate samples.  How will the means of samples differ from the mean of the population?

Distribution of Sample Means:

• A distribution using means of all possible random samples of a specific size taken from a population.
• Sampling Error is difference between sample measure and corresponding population measure due to the fact that the sample is not a perfect representation of the population.

Properties of Distribution of Sample Means:

When ALL possible samples of a specific size are selected with replacement from a population, then:

• Mean of sample EQUALS mean of population.
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• Standard deviation of sample means is

AS SAMPLE SIZE N INCREASES WITHOUT LIMIT, THE SHAPE OF THE DISTRIBUTION OF THE SAMPLE APPROACHES A NORMAL LIMIT.

When sample means are involved, must use:   or   where = sample mean.

The average number of pounds of sugar that a person living in Sugarland consumes each year is 218.4 pounds.  Assume that the standard deviation is 25 pounds and the distribution is approximately normal.

a) Find the probability that a person selected at random consumes less than 224 pounds per year.

b) If a sample of 40 individuals is selected, find the probability that the mean of the sample will be less than 224 pounds per year.