Central Limit Theorem: Central Limit Theorem for Sums 1.5 2008/06/06 15:05:15 GMT-5 2008/07/18 17:06:31.205 GMT-5 Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Connexions cnx@cnx.org Central Limit Theorem statistics Sums Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution). Suppose: a μ X = the mean of X b σ X = the standard deviation of X If you draw random samples of size n, then as n increases, the random variable ΣX which consists of sums tends to be normally distributed and Σ X ~ N ( n ⋅ μ X , n ⋅ σ X ) The Central Limit Theorem for Sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution. The distribution has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size.The random variable Σ X has the following z-score associated with it:aΣX is one sum. b z = Σ X - n ⋅ μ X n ⋅ σ X a n ⋅ μ X = the mean of ΣX b n ⋅ σ X = standard deviation of ΣX An unknown distribution has a mean of 90 and a standard deviation of 15. A sample of size 80 is drawn randomly from the population. aFind the probability that the sum of the 80 values (or the total of the 80 values) is more than 7500. bFind the sum that is 1.5 standard deviations below the mean of the sums. Let X = one value from the original unknown population. The probability question asks you to find a probability for the sum (or total of) 80 values.ΣX = the sum or total of 80 values. Since μ X = 90 , σ X = 15 , and σ X = 80 , then Σ X ~ N ( 80 ⋅ 90 , 80 ⋅ 15 ) amean of the sums = n ⋅ μ X = ( 80 ) ( 90 ) = 7200 bstandard deviation of the sums = n ⋅ σ X = 80 ⋅ 15 csum of 80 values = Σx = 7500 Find P ( ΣX 7500 ) Draw a graph. P ( ΣX 7500 ) = 0.0127 normalcdf(lower value, upper value, mean of sums, stdev of sums)The parameter list is abbreviated (lower, upper, n ⋅ μ X , n ⋅ σ X )normalcdf(7500,1E99, 80 ⋅ 90 , 80 ⋅ 15 ) = 0.0127 Reminder: 1E99 = 10 99 . Press the EE key for E. Normal Distribution A continuous random variable (RV) with pdf=1σ2πe−(x−μ)2/2σ2 size 12{ ital "pdf"= { {1} over {σ sqrt {2π} } } e rSup { size 8{ - \( x - μ \) rSup { size 6{2} } /2σ rSup { size 6{2} } } } } {}, where μ is the mean of the distribution and σ is its standard deviation. Notation: X ~ N μ σ 2 . If μ=0 and σ=1, the RV is called standard normal distribution, or z-score.