321_F11_slides3

# 321_F11_slides3 - Review of Econ 221 Part II Econ 321 Click...

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Click to edit Master subtitle style Econ 321-Stéphanie Review of Econ 221 Part II Econ 321 Introduction to Econometrics 11

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Econ 321-Stéphanie Main Steps in Statistical Analysis Central Limit Theorem Other basic distributions Hypothesis testing 22
Econ 321-Stéphanie Central Limit Theorem We don’t know the mean and variance of a population, nor do we know the distribution. with the CLT, we can say something about the average from a random sample for any population We know its asymptotic distribution around the “true value” for large samples 33

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Econ 321-Stéphanie 44 The Central Limit Theorem (CLT) If ( Y 1 ,…, Y n ) are i.i.d. and 0 < 2 Y σ < , then when n is large the distribution of Y is well approximated by a normal distribution. Y is approximately distributed N ( μ Y , 2 Y n ) (“normal distribution with mean Y and variance 2 Y / n ”) n ( Y Y )/ Y is approximately distributed N (0,1) (standard normal) That is, “standardized” ( ) var( ) Y E Y Y - = / Y Y Y n - is approximately distributed as N (0,1) The larger is n , the better is the approximation.
Econ 321-Stéphanie 55 Remember: Sampling distribution of when Y is Y

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Econ 321-Stéphanie 66 Same example : sampling distribution of : ( ) var( ) Y E Y Y -
Econ 321-Stéphanie Example Let’s suppose we have an examination in which the mean score is 550 and the variance is 81. What is the probability that we draw a single exam with a score over 560? For a sample with single observation : 77

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Econ 321-Stéphanie Example Now suppose we take a sample of 25 exams 88 => The sample is much less likely to be “far” from the  mean (over 560) than a single observation
Econ 321-Stéphanie 99 Summary: The Sampling Distribution of Y For Y 1 ,…, Y n i.i.d. with 0 < 2 Y σ < , The exact (finite sample) sampling distribution of Y has mean μ Y and variance 2 Y / n Other than its mean and variance, the exact distribution of Y is complicated and depends on the distribution of Y (the population distribution) When n is large, the sampling distribution simplifies: ( ) var( ) Y E Y Y - is approximately N (0,1) (CLT)

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Econ 321-Stéphanie Other Distributions Chi-Squared Distribution is the distribution of the sum of m squared independent standard
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321_F11_slides3 - Review of Econ 221 Part II Econ 321 Click...

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