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handout3_1 - Econ 139: Introduction to Econometrics Andrew...

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Econ 139: Introduction to Econometrics Andrew Sweeting 1 Department of Economics Duke University Spring 2011 Econ 139 Handout 3 (Duke) Statistics Review Spring 2011 1 / 36
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Sample: Subset of a population Population: Total collection of objectives or people to be studied (or set of all information of interest to the decision maker) Statistical Inference: Making statements about a population on the basis of a sample Population Parameter Sample Parameter μ X mean ¯ X σ 2 X variance s 2 X N size n usually we do not observe the whole population, but only a sample Econ 139 Handout 3 (Duke) Statistics Review Spring 2011 2 / 36
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Random sampling To be useful, a sample must be representative of the population. Simple Random Sample (SRS) A sample is a simple random sample (SRS) if each individual in the population is equally likely to be included in the sample Example: class (heights) as a sample of current Duke undergrads Is this a SRS? If not, how would you construct one? An iid (independent and identically distributed) sample A sample is an iid random sample if it is a SRS and 1 each of the n observations in the sample, X 1 , X 2 , ..., X n , are independent (e.g. no relatives), and 2 each observation is drawn from the same overall population distribution. Econ 139 Handout 3 (Duke) Statistics Review Spring 2011 3 / 36
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Point estimates We use samples to get information on population &parameters± . Sample mean, elasticity of female labor supply, e/ectiveness of a cancer treatment. A point estimate is a single number used to estimate an unknown population parameter. We will focus on point estimates in this course Examples: the sample mean ¯ X can be used to estimate the population mean μ X . An estimator is the function of the sample of data that produces the number for the point estimate. Econ 139 Handout 3 (Duke) Statistics Review Spring 2011 4 ² 36
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Point estimators Formulas for some commonly used point estimators: Sample Mean: ¯ X = 1 n n i = 1 X i = 1 n [ X 1 + X 2 + ... + X n ] ; Sample Variance: s 2 X = 1 n & 1 n i = 1 ( X i & ¯ X ) 2 . Why ( n & 1 ) ? It comes from using ¯ X to estimate μ , which introduces a downward bias in ( X i & ¯ X ) 2 which we correct by multiplying by n n & 1 (which is > 1) Sample Covariance: s XY = 1 n & 1 n i = 1 ( X i & ¯ X )( Y i & Y ) Sample Correlation: r = s XY s X s Y Econ 139 Handout 3 (Duke) Statistics Review Spring 2011 5 / 36
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Sample mean as a random variable The sample mean is a random variable because it is a function of the sample draws which are themselves random variables. Even if we knew μ , we do not know the value of the sample mean before we draw the sample Sample mean will therefore have a distribution. The distribution of any point estimator is called the sampling distribution. The sample mean will therefore also have an expected value (mean) and a variance/standard deviation.
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This note was uploaded on 08/02/2011 for the course ECON 139 taught by Professor Alessandrotarozzi during the Spring '08 term at Duke.

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handout3_1 - Econ 139: Introduction to Econometrics Andrew...

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