ECON2209 Lecture Notes - Lecture 11: Central Limit Theorem, Sampling Distribution, Standard Error
Feb 22nd:
● Measures of relative standing
○ Percentiles
○ Z scores
■ Population
● x-u/sigma
■ Sample
● x-xbar/s
■ Measures the number of standard deviations that a value is from the mean
■ Xbar is 10
■ Standard deviation is 4
■ Find the Zscore:
● Z = (28-10)/4 = 4.5… so 28 is 4.5 standard deviations away from the
mean
■ If the distribution is symmetrical….
● 68% of all observations have a z score between -1 and 1
● 95% of all observations have a z score between -2 and 2
● 99% of all observations have a z score between -3 and 3
■ 2nd, VARs (DIST), normalcdf, put in z-score
● Measures of Association (aka measures of linear relationships)
○ Cov(x,y) = sigmaXY =
■ Variance is a special case of covariance
■ Problem with covariance? Can't say anything about the strength of the
connection!
○ Correlation:
■ Population:
■ Corr is between -1and 1
■ Ex: Cov = 132.6
● Variance of X = 34.3
● Sampling Distributions
○ Distributions of statistics (measure of a sample)
■ Ex: distribution of xbar
■ Ex: distribution of s2
■ Ex: distributions of p
○ Sampling distributions
■ Need 3 things:
● Mean/central location:
○ E(xbar)= uxbar
●
● Dispersion
○ Standard error is the standard deviation of a sampling
distribution
● Shape: CLT (Central Limit Theorem)
○ CLT1: If X is normally distributed, then xbar is a normally
distributed for any n
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Document Summary
Measures the number of standard deviations that a value is from the mean. Z = (28-10)/4 = 4. 5 so 28 is 4. 5 standard deviations away from the mean. 68% of all observations have a z score between -1 and 1. 95% of all observations have a z score between -2 and 2. 99% of all observations have a z score between -3 and 3. 2nd, vars (dist), normalcdf, put in z-score. Measures of association (aka measures of linear relationships) Variance is a special case of covariance. Can"t say anything about the strength of the connection! Distributions of statistics (measure of a sample) Standard error is the standard deviation of a sampling distribution. Clt1: if x is normally distributed, then xbar is a normally distributed for any n. Clt2: if x is not normally distributed, then xbar will be approx. normally distributed if n is large (n>30) P is at least approximately normal if n is large.