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Instructor: Dr. Ozan ERUYGUR e-mail: [email protected]Lecture Notes 14 2( )Var Y. Then, when tis large, the sampling distribution of Yconvergesto 2(,)YYNwhere Yand 22YT. As a result, the sampling distribution of 2YYYwhich is equal to /YTconverges to the standard normalN(0,1)as Tconverges to infinity. The central limit theorem is very important because it holds even when the distribution from which the observations are drawn is not normal. This means that if we make sure that the sample size is large, then we can use the random variable /YTdefined above to answer questions about the population from which the observations are drawn, and we need not know the precise distribution from which the observations are drawn (Ramanathan, 1998, p.37). APPENDIX A. Sampling Distribution of Mean Population: A B C D E 3 1 5 6 2 Population mean, : (3+1+5+6+2)/5=17/5=3.4 Population Size, N: 5 From this population of five, list every possible sample of size 2 (n=2) and calculate the mean for each sample. Repeat this for samples of size 3 (n=3).
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected]Lecture Notes 15 How many samples of size 2? 55410212How many samples of size 3? 5543103123Samples Sample of Size 2 (n=2) Sample Mean Samples Sample of Size 3 (n=3) Sample Mean 1 AB=3,1 (y1=3, y2=1) for Y1and Y2Y=2.0 Y1ABC=3,1,5 (y1=3, y2=1, y3=5) for Y1, Y2 and Y33.00 2 AC=3,5 4.0 Y2ABD=3,5,6 4.67 3 AD=3,6 4.5 Y3ABE=3,1,2 2.00 4 AE=3,2 2.5 Y4ACD=3,5,6 3.67 5 BC=1,5 3.0 Y5ACE=3,5,2 3.33 6 BD=1,6 3.5 Y6ADE=3,6,2 3.67 7 BE=1,2 1.5 Y7BCD=1,5,6 4.00 8 CD=5,6 5.5 Y8BCE=1,5,2 2.67 9 CE=5,2 3.5 Y9BDE=1,6,2 3.00 10 DE=6,2 4.0 Y10CDE=5,6,2 4.33 Average 3.4 3.4 Means of all sample means YY=3.4 For n=2 Y=3.4 For n=3 ConclusionThe mean of all the sample means is the same as the population mean: Y=B. Process Going Into the Sampling Distribution Model oWe start with a population model, which can have any shape. It can even be bimodal or skewed (as this one is). We label the mean of this model and its standard deviation, . oWe draw one real sample (solid line) of size n and show its histogram and summary statistics. We imagine (or simulate) drawing many other samples (dotted lines), which have their own histograms and summary statistics. oWe (imagine) gathering all the means into a histogram