ecn10-lecture5-v3

# ecn10-lecture5-v3 - Economics 10 Introduction to...

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Economics 10: Introduction to Statistical Methods Normal Distribution Sampling Distribution, Random Samples

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Recap: Densities In lecture #2 we learned about densities – the continuous analog of histograms, e.g., 0 .005 .01 .015 .02 Density 300 400 500 600 700 reading scale score, K Distribution of Reading Scores, End of K Project Star Participants
“Idealized” Distributions Densities can also be used to describe distributions which do not come from data Mathematical abstractions which can be good approximations for how data are distributed Often used to describe the (unobserved) distribution of data in the entire population Example today : normal distribution Later in class: Chi-squared distribution, t-distribution, F-distribution.

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The Normal Distribution Formula for the normal distribution: Called the “normal pdf*” μ is the mean of variable X σ is the standard deviation of X We often write X ~ N( μ , σ 2 ) “(variable) x is distributed normally with mean μ , variance σ 2 ( 29 2 2 1 2 1 - - = σ μ π x e x f *pdf=probability density function μ , σ : Greek letters for idealized distributions Often represent population parameters
Normal Distribution 0 .1 .2 .3 .4 Density of Normal Distribution -10 -5 0 5 10 x Mean 0, Std Dev 1 Mean 0, Std Dev 2 Mean 5, Std Dev 1 Various Normal Densities

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Characteristics of the Normal Normal distributions are symmetric Normal distributions are completely characterized by μ and σ All other statistics are functions of μ and σ What is the median? What is the variance? The “68-95-99.7 Rule” ≈ 68% of values fall within one σ of μ ≈ 95% of values fall within 2σ of μ ≈ 99.7% of values fall within 3σ of μ
How do you know if your data are normally distributed? Formal statistical tests Too advanced for now Theories say certain types of data are normally distributed You can look at your data! In STATA: kdensity {varname}, normal Adds a normal distribution with the same mean and standard deviation to your density plot.

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Height Data – Normally Distributed? 0 .02 .04 .06 .08 .1 Density 60 65 70 75 80 Height in inches Econ 10 Students Normal PDF kernel = epanechnikov, bandwidth = 1.64 Distribution of Height, Econ 10 Students and Normal Distribution
Height Data – Females Only 0 .05 .1 .15 .2 Density 60 62 64 66 68 Height in inches Econ 10 Females Normal PDF kernel = epanechnikov, bandwidth = .7 Distribution of Height, Econ 10 Females and Normal Distribution

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Height Data – Males Only 0 .05 .1 .15 Density 60 65 70 75 80 height Econ 10 Males Normal PDF kernel = epanechnikov, bandwidth = 1.23 Distribution of Height, Econ 10 Males, and Normal Distribution
0 .02 .04 .06 Density 0 50 100 150 200 250 hourly wage, 1996 Density of Wages Normal PDF kernel = epanechnikov, bandwidth = 1.64 Are Wages Normally Distributed? Observations from the NLSY

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ecn10-lecture5-v3 - Economics 10 Introduction to...

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