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Unformatted text preview: The Central Limit Theorem We can rephrase our previous work (approximating the Binomial with the Normal) as follows: If X1, X2, ... are independent and identically distributed Bernoulli Random Variables, then, for large n, the probability distribution of X1+...+Xn is approximately a normal distribution with mean np and variance np(1p) (the same as the binomial with n=n). This leads us to the following result (just using our Z transformation): X 1 + ... + Xn - np lim P x = ( x) n np (1 - p ) If we replace np and np(1 - p ) with n and * n respectively we get a general formula that holds for ANY IID random variables with FINITE VARIANCE. Central Limit Theorem Let X1, X2, ... be IID random variables with common finite mean and finite nonzero variance 2 . Then n X 1 + ... + Xn - n lim P x = ( x) n for all real values of x. This implies that the standardized versions of sums of IID random variables with finite nonzero variance converge in distribution to the standard normal. Note: We do not need to know the exact distribution of the Xi's to compute (approximate) probabilities. All we need to know is the mean and variance. Recall: Proposition 8.11 Obtaining Probabilities for a Normal R.V. If X is normally distributed with parameters and 2 , then b- a- P ( a < X < b) = - We can directly apply this to the central limit theorem. P(a X1+...+Xn b)= a - n b - n . - n n RULE OF THUMB: The Central Limit Theorem is approximately adequate for n>30. However, the more skewed the original distribution, the larger n must be to obtain a good approximation. The Central Limit Theorem can be rephrased in terms of sample means Suppose x1, x2, ... , xn is a random sample from a distribution with finite mean and finite nonzero x1 + ... + xn variance 2 . The sample mean, x n is found by the average of x1, x2, ..., xn, xn = . n We can rewrite our previous equation as: X P(X) Let x represent the mean from a sample of size 60 from the above distribution. Find the expected value and variance of x . Also, name its distribution and parameters. Use this to find P(1.3< x < 2.2). Margin of Error and Confidence Intervals (using this to compute n) n x - lim P n x = ( x) / n
0 .25 1 .32 for all real values x. 2 .26 3 .12 4 .05 ...
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