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(2) Sampling Distribution & Central Limit Theorem Prof. Mohammad Almahmeed 2 Sampling Distribution And Central Limit Theorem Sampling distribution of the sample mean If we sample a number of samples (say k samples where k is very large number) each of size n, from a normally distributed population with mean and standard deviation . And compute the mean of each of these samples. We will have different sample mean for each sample: x1 , x 2 , . . . , xk All of these means estimate the same unknown population mean . These means are values of a random variable 2 QMIS 220 1 (2) Sampling Distribution & Central Limit Theorem Prof. Mohammad Almahmeed Sampling distribution of the sample mean From mathematical statistics one could prove that this random variable follows the normal distribution with mean equals to 2 (the population mean) and variance equals to . Population n x1 =? N(, σ) n x2 n sample # 1 sample # 2 n xk sample # k X is a random variable σ X ~ N μ, n 3 Sampling distribution of the sample mean In other words, if x is the mean of a sample of size n taken from a normally distributed population with mean () and standard deviation (σ) [i.e. X~N(,σ)] . Then X is a random variable that follows the normal distribution with mean and standard deviation σ . i.e. : n σ X~N μ , n 4 QMIS 220 2 (2) Sampling Distribution & Central Limit Theorem Prof. Mohammad Almahmeed Central Limit Theorem If X is a random variable that follows any distribution (known or unknown) but with mean (μ) and Standard deviation (σ). If X is the mean of a sample of size n (n large i.e n > 30). Then the distribution of X will approach the normal distribution with mean μ and Standard deviation n . ( is known as the standard error of the mean) n That is X . N ( , X ~ Approach N , n n ) N , i.e. Distribution of ( X ) n n 5 Central Limit Theorem Population n large: Sample (n>30) X ~ ? (μ,σ) X X n X .~ N(, ) n 6 QMIS 220 3 (2) Sampling Distribution & Central Limit Theorem Prof. Mohammad Almahmeed Central Limit Theorem Distribution of ( X ) : (1) If X ~ N ( , ) sample n (large / small) X ~ N , n (2) IF X ~ ? ( , ) small sample X ~ ? u nknown ( 3) IF X ~ ? ( , ) large sample . X ~ N , n (C . L.T .) 7 Sampling distribution of the sample mean and Central Limit Theorem Example - 1 X is a random variable that follows the normal distribution with mean 80 and standard deviation equal to 10. X is the mean of a random sample of size 15 taken from that population. Find: (1) P(75 < X < 92) = (2) P(78 < X <83) = 75 - 80 92 80 Z ) 10 10 P(-0.5 Z 1.2) P(0 Z 1.2) P(0 Z 0.5) 0.3849 0.1915 0.5764 (1) P(75 X 92) P( 78 - 80 83 80 Z ) 10/ 15 10/ 15 P(-0.77 Z 1.16) (2) P(78 X 83) P( P(0 Z 1.16) P(0 Z 0.77) 0.3770 0.2794 0.6534 QMIS 220 8 4 (2) Sampling Distribution & Central Limit Theorem Prof. Mohammad Almahmeed Sampling distribution of the sample mean and Central Limit Theorem Example - 2 For the same previous example if the distribution of X is unknown. And we took a sample of size 40. find: (1) P(75 < X < 92) = (2) P(78 < X <83) = (1) we can not find the required probability since the distribution of X is unknown. (2) Since the sample size is large enough we will apply the central limit theorem to have: 78 - 80 83 80 Z P(78 X 83) P( ) 10/ 40 10/ 40 P(-1.26 Z 1.90) P(0 Z 1.90) P(0 Z 1.26) 0.4713 0.3962 0.8675 9 Sampling distribution of the sample mean and Central Limit Theorem Example - 3 If the standard error of the mean for a sample of 36 is 15. In order to decrease the standard error of the mean to 5 what should be the value of n (the sample size)? the standard error of the mean is x 15 5 36 90 n n = 90 90 n = 5 2 n = 324 10 QMIS 220 5 (2) Sampling Distribution & Central Limit Theorem Prof. Mohammad Almahmeed Sampling distribution of the Population Proportion (p) The proportion of any incident in a population is the number of elements in the population that belong to that incident divided by the total number of elements in the population. “P” usually denotes this proportion. P= The number of elements that has certain character in the population the population size This proportion can be estimated from a sample by P̂ where: pˆ = The number of elements with that certain character in the sample sample size 11 Sampling distribution of the Population Proportion (p) The population of sample proportion has a Binomial distribution. But when the sample size is large the population of all possible sample proportions has approximately normal distribution, with mean ( p̂ ) equals P, and standard P(1 P ) . For this approximation to be good, the n following conditions should be met: deviation ( p̂ ) equals to The sample size (n) is large: (1) (n * p) > 5 (2) (n * q) > 5 , where q=1-p 12 QMIS 220 6 (2) Sampling Distribution & Central Limit Theorem Prof. Mohammad Almahmeed The sample size (n) is large when: (1) (n * p) > 5 (2) (n * q) > 5 , where q=1-p Sampling distribution of the Population Proportion (p) When can we consider n as sufficiently large enough? n 10 15 45 55 90 105 p .4 .4 .1 .1 .05 .05 q .6 .6 .9 .9 .95 .95 np 4 6 4.5 5.5 4.5 5.25 nq 6 9 41.5 49.5 85.5 99.75 Not large enough Large enough Not large enough Large enough Not large enough Large enough 13 Application of the sampling distribution of p̂ Example: Ahmad is a broker in Kuwait stock market, if we know from his record that 65% of his deals are profitable. Let p̂ be the proportion in a random sample of 20 of his latest deals. Find the probability that the value of p̂ will be greater than 70%? p = 0.65 q = 0.35 P( p̂ > .70) = ? Conditions: n=20 n * p = 20 * 0.65 = 13 >5 n * q = 20 * 0.35 = 7 >5 14 QMIS 220 7 (2) Sampling Distribution & Central Limit Theorem Prof. Mohammad Almahmeed Application of the sampling distribution of p̂ Example: cont. p̂ = 0.65 p̂ = and pq 0.65* 0.35 0.114 0.1067 n 20 pˆ pˆ p(pˆ 0.70) p pˆ 0.70 - 0.65 0.1067 P( Z > 0.469 ) = 0.5 – P( 0 < Z < 0.469) = 0.5 – 0.1808 = 0.3192 = 31.92% 15 Estimator and Estimate A sample Statistics used to estimate a population parameter is called an Estimator The value(s) assigned to a population parameter based on the value of a sample statistics is called an Estimate To estimate the population mean we took a random sample of size 22. The computed value of the sample mean x is 43.7 Here x is the estimator for and 43.7 is the estimate for it. 16 QMIS 220 8 (2) Sampling Distribution & Central Limit Theorem Prof. Mohammad Almahmeed Biased and Unbiased Estimator An Estimator of the population parameter is said to be Unbiased estimator when the expected value (or the mean) of this estimator is equal to the value of the corresponding population parameter x (i.e., for the case of the sample mean, if E( ) = x If not, (i.e. E( ) ≠ the Estimator is said to be Biased . 17 QMIS 220 9