ECON301_Handout_02_1213_02

This means that if we make sure that the sample size

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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 54 10 2 12     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
ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 16 Figure 1 Process Going Into the Sampling Distribution Model C. The Central Limit Theorem (CLT) o The sampling distribution of any mean becomes more nearly Normal as the sample size grows. o All we need is for the observations to be independent and collected with randomization. o We don’t even care about the shape of the population distribution! In fact, this is a fundamental theorem of statistics and it is called the Central Limit Theorem (CLT) . The CLT is surprising: o Not only does the histogram of the sample means get closer and closer to the Normal model as the sample size grows, but this is true regardless of the shape of the population distribution. The CLT works better (and faster) the closer the population model is to a Normal itself.

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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 17 It also works better for larger samples.
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