Sampling Distribution And Central Limit Theorem

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1 () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size, from a ormally distributed populatio with mea ad stadard deviatio. Ad compute the mea of each of these samples. We will have differet sample mea for each sample: 1,,..., k All of these meas estimate the same ukow populatio mea. These meas are values of a radom variable QMIS 0 1

2 () Samplig Distributio & Cetral Limit Samplig distributio of the sample mea From mathematical statistics oe could prove that this radom variable follows the ormal distributio with mea equals to (the. populatio mea) ad variace equals to Populatio =? N(, σ) 1 sample # 1 sample # k sample # k is a radom variable ~ σ Nμ, 3 Samplig distributio of the sample mea I other words, if is the mea of a sample of size take from a ormally distributed populatio with mea () ad stadard deviatio (σ) [i.e. ~N(,σ)]. The is a radom variable that follows the ormal distributio with mea ad stadard deviatio σ. i.e. : ~N μ, σ 4 QMIS 0

3 () Samplig Distributio & Cetral Limit Cetral Limit If is a radom variable that follows ay distributio (kow or ukow) but with mea (μ) ad Stadard deviatio (σ). If is the mea of a sample of size ( large i.e > 30). The the distributio of will approach the ormal distributio with mea μ ad Stadard deviatio. ( is kow as the stadard error of the mea) That is. N (, ) ~ N, Approach i.e. Distributio of ( ) N, 5 Cetral Limit Populatio ~? (μ,σ) large: (>30) Sample. ~ N(, ) 6 QMIS 0 3

4 () Samplig Distributio & Cetral Limit Cetral Limit Distributio of ( ) : (1) If ~ N(, ) sample ~ N, (large / small) ( ) IF ~? (, ) small sample ~? ukow ( 3) IF ~? (, ) large sample. ~ N, ( C. L. T.) 7 Samplig distributio of the sample mea ad Cetral Limit Eample - 1 is a radom variable that follows the ormal distributio with mea 80 ad stadard deviatio equal to 10. is the mea of a radom sample of size 15 take from that populatio. Fid: (1) P(75 < < 9) = () P(78 < <83) = (1) P(75 9) P( Z ) P(-0.5 Z 1.) P(0 Z 1.) P(0 Z 0.5) () P(78 83) P( Z ) 10/ 15 10/ 15 P(-0.77 Z 1.16) P(0 Z 1.16) P(0 Z 0.77) QMIS 0 4

5 () Samplig Distributio & Cetral Limit Samplig distributio of the sample mea ad Cetral Limit Eample - For the same previous eample if the distributio of is ukow. Ad we took a sample of size 40. fid: (1) P(75 < < 9) = () P(78 < <83) = (1) we ca ot fid the required probability sice the distributio of is ukow. () Sice the sample size is large eough we will apply the cetral limit theorem to have: P(78 83) P( Z ) 10/ 40 10/ 40 P(-1.6 Z 1.90) P(0 Z 1.90) P(0 Z 1.6) Samplig distributio of the sample mea ad Cetral Limit Eample - 3 If the stadard error of the mea for a sample of 36 is 15. I order to decrease the stadard error of the mea to 5 what should be the value of (the sample size)? the stadard error of the mea is 15 = = 5 = QMIS 0 5

6 () Samplig Distributio & Cetral Limit Samplig distributio of the Populatio Proportio (p) The proportio of ay icidet i a populatio is the umber of elemets i the populatio that belog to that icidet divided by the total umber of elemets i the populatio. P usually deotes this proportio. P = The umber of elemets that has certai character the populatio size i the populatio This proportio ca be estimated from a sample by Pˆ where: p ˆ = The umber of elemets with that certai character i the sample sample size 11 Samplig distributio of the Populatio Proportio (p) The populatio of sample proportio has a Biomial distributio. But whe the sample size is large the populatio of all possible sample proportios has approimately ormal distributio, with mea ( ˆp ) equals P, ad stadard P(1 P) deviatio ( ˆp ) equals to. For this approimatio to be good, the followig coditios should be met: The sample size () is large: (1) ( * p) > 5 () ( * q) > 5, where q=1-p 1 QMIS 0 6

7 () Samplig Distributio & Cetral Limit Samplig distributio of the Populatio Proportio (p) Whe ca we cosider as sufficietly large eough? The sample size () is large whe: (1) ( * p) > 5 () ( * q) > 5, where q=1-p p q p q Not large eough Large eough Not large eough Large eough Not large eough Large eough 13 Applicatio of the samplig distributio of ˆp Eample: Ahmad is a broker i Kuwait stock market, if we kow from his record that 65% of his deals are profitable. Let ˆp be the proportio i a radom sample of 0 of his latest deals. Fid the probability that the value of ˆp will be greater tha 70%? p = 0.65 q = 0.35 =0 P( ˆp >.70) =? Coditios: * p = 0 * 0.65 = 13 >5 * q = 0 * 0.35 = 7 >5 14 QMIS 0 7

8 () Samplig Distributio & Cetral Limit Applicatio of the samplig distributio of ˆp Eample: cot. ˆp = 0.65 ad pq 0.65* pˆ pˆ p( pˆ 0.70) p pˆ ˆp = P( Z > ) = 0.5 P( 0 < Z < 0.469) = = = 31.9% 15 Estimator ad Estimate A sample Statistics used to estimate a populatio parameter is called a Estimator The value(s) assiged to a populatio parameter based o the value of a sample statistics is called a Estimate To estimate the populatio mea we took a radom sample of size. The computed value of the sample mea is 43.7 Here is the estimator for ad 43.7 is the estimate for it. 16 QMIS 0 8

9 () Samplig Distributio & Cetral Limit Biased ad Ubiased Estimator A Estimator of the populatio parameter is said to be Ubiased estimator whe the epected value (or the mea) of this estimator is equal to the value of the correspodig populatio parameter (i.e., for the case of the sample mea, if E( ) = If ot, (i.e. E( ) the Estimator is said to be Biased. 17 QMIS 0 9

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