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# Central limit theorem

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### Central limit theorem

1. 1. Central Limit Theorem Presented By Vijeesh S1-MBA (PT)
2. 2. Introduction  The Central Limit Theorem describes the relationship between the sampling distribution of sample means and the population that the samples are taken from.
3. 3. Normal Populations  Important Fact:  If the population is normally distributed, then the sampling distribution of x is normally distributed for any sample size n.
4. 4. Sampling Distribution of x- normally distributed population n=10 /10  Population distribution: N( , ) Sampling distribution of x: N( ,  /10)
5. 5. Non-normal Populations What can we say about the shape of the sampling distribution of x when the population from which the sample is selected is not normal? 53 490 102 72 35 21 26 17 8 10 2 3 1 0 0 1 0 100 200 300 400 500 600 Frequency Salary (\$1,000's) Baseball Salaries
6. 6. The Central Limit Theorem If a random sample of n observations is selected from a population (any population), then when n is sufficiently large, the sampling distribution of x will be approximately normal. (The larger the sample size, the better will be the normal approximation to the sampling distribution of x.)
7. 7. • Suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. • If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution
8. 8. For eg: Suppose we have a population consisting of the numbers {1,2,3,4,5} and we randomly selected two numbers from the population and calculated their mean. For example, we might select the numbers 1 and 5 whose mean would be 3. Suppose we repeated this experiment (with replacement) many times. We would have a collection of sample means ( millions of them). We could then construct a frequency distribution frequency distribution of these sample means. The resulting distribution of sample means is called the sampling distribution of sample means. Now, having the distribution of sample means we could proceed to calculate the mean of all sample means (grand mean) and their
9. 9. The Central Limit Theorem predicts that regardless of the distribution of the parent population: [1] The mean of the population of means is always equal to the mean of the parent population from which the population samples were drawn. [2] The standard deviation (standard error ) of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (N). SD’= SD/√N
10. 10. If the population from which samples are drawn is normally distributed then the sampling distribution of sample means will be normally distributed regardless of the size of the sample, and the CLT is not needed. But, if the population is not normal, the CLT tells us that the sampling distribution of sample means will be normal provided the sample size is sufficiently large. How large must the sample size be so that the sampling distribution of the mean becomes a normal distribution? If the samples were drawn from a population with a high degree of skewness (not normal), the sample size must be 30 or more before the sampling distribution of the mean becomes a normal distribution. A sample of size 30 or more is called a large sample and as a sample of size less that thirty is called a small sample.