chap06PRN econ 325

# Consider the sample mean n 1 i i x n 1 x earlier

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Consider the sample mean: = = n 1 i i X n 1 X Earlier lecture notes stated the standard error of X as: n ) X ( se σ = Define the standardized random variable: n X ) X ( se X Z σ μ - = μ - = The Central Limit Theorem states that as n becomes ‘large’ the distribution of Z approaches the standard normal distribution. That is, ) 1 , 0 ( N ~ Z . Therefore, a random variable that can be viewed as the sum of a ‘large’ number of independently and identically distributed random variables will tend to have a normal distribution.

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Econ 325 – Chapter 6 13 A computer simulation can be used to demonstrate the Central Limit Theorem. Consider a random variable that follows the uniform distribution over the interval [0, 10] . A graph of the probability density function is below. 1/10 10 5 0 Econ 325 – Chapter 6 14 A statistical result is that the average of two ( n =2) independent uniform random variables has a triangular shape for the probability density function. To show this result with a computer simulation, select outcomes from two uniform random variables on the interval [0, 10] and calculate the average of the two values. Repeat this 1000 times. A histogram gives a graph of the frequency distribution of the sample means generated by the computer simulation. This gives a rough picture of the probability density function of the sample mean. The histogram generated by the computer simulation is shown below. The triangular shape of the theoretical probability density function has also been sketched. 10 5 0
Econ 325 – Chapter 6 15 Now consider a sample size of n =25. By the Central Limit Theorem it may be reasonable to assume that the sample mean follows a normal distribution. To start the computer simulation, select outcomes from 25 uniform random variables on the interval [0, 10] and calculate the average of the values. Repeat this 1000 times. The histogram generated by the computer simulation is shown below. A normal probability density function is also sketched to reveal that, with n =25, the distribution of the sample mean is closely approximated by the normal distribution. 6 5 4
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Consider the sample mean n 1 i i X n 1 X Earlier lecture...

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