Ozan eruygur e mail oeruygurgmailcom lecture notes 14

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Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 14 2 ( ) Var Y . Then, when t is large, the sampling distribution of Y converges to 2 ( , ) Y Y N where Y and 2 2 Y T . As a result, the sampling distribution of 2 Y Y Y which is equal to / Y T converges to the standard normal N(0,1) as T converges 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 / Y T defined 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? 5 5 4 10 2 1 2 How many samples of size 3? 5 5 4 3 10 3 1 2 3 Samples Sample of Size 2 (n=2) Sample Mean Samples Sample of Size 3 (n=3) Sample Mean 1 AB=3,1 (y 1 =3, y 2 =1) for Y 1 and Y 2 Y = 2.0 Y 1 ABC=3,1,5 (y 1 =3, y 2 =1, y 3 =5) for Y 1 , Y 2 and Y 3 3.00 2 AC=3,5 4.0 Y 2 ABD=3,5,6 4.67 3 AD=3,6 4.5 Y 3 ABE=3,1,2 2.00 4 AE=3,2 2.5 Y 4 ACD=3,5,6 3.67 5 BC=1,5 3.0 Y 5 ACE=3,5,2 3.33 6 BD=1,6 3.5 Y 6 ADE=3,6,2 3.67 7 BE=1,2 1.5 Y 7 BCD=1,5,6 4.00 8 CD=5,6 5.5 Y 8 BCE=1,5,2 2.67 9 CE=5,2 3.5 Y 9 BDE=1,6,2 3.00 10 DE=6,2 4.0 Y 10 CDE=5,6,2 4.33 Average 3.4 3.4 Means of all sample means Y Y =3.4 For n=2 Y =3.4 For n=3 Conclusion The mean of all the sample means is the same as the population mean: Y = B. Process Going Into the Sampling Distribution Model o We 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, . o We 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. o We (imagine) gathering all the means into a histogram
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