Statistics Review, Lecture 2

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Statistics Review, Lecture 2

• Econ 326
• Eric Bettinger
• (Notes created in part from Woolridge (2000))

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Class Announcement

• Have you opened Weatherhead accounts?
• Have you been able to access the lecture notes on
  the course website?
• First assignment will be due on 1/24/03 in my box.

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Review

• Summation Operator
• Mean
• Median the middle data point
• Slope
• Intercept
• Marginal and Partial Effects
• Linear in Coefficients

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Review (more)

• How to use Excel and Stata
• Example from Colombia

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Probability

• Random Variable
• Has some probability (pj) of taking on a specific
  numeric value (xj) each time it is drawn
• Has a support or possible numbers it can take
  on.
• Examples flipping a coin, ages in our classroom,
  level in school
• Notation

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Probability Continued

• Discrete or Continuous?
• Common Distributions Uniform, Normal
• Probability Density Function
• Sums to one and shows probability that each value
  occurs.
• f(xj) pj , j1,2,,k

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Probability Continued

• P(a
• Draw what it looks like for a normal curve with
  arbitrary points a and b
• Cumulative Distribution Function
• P(X
• Bounded between 0 and 1
• Draw it for Binomial, Uniform, and Normal
  distributions

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More Stats(I dont expect you to be experts in
this part)

• Joint Distributions
• fX,Y(x,y)P(Xx,Yy)
• Examples Test scores and Ability, Height and
  Weight, Test Scores and Gender
• What does it mean for distributions to be
  independent?
• fX,Y(x,y)fX(x) fY(y)
• Examples Suppose that I give good lectures 20
  of the time. What is the probability of having
  two consecutive good lectures? Is this
  calculation valid if after giving a good lecture,
  I relax more and spend less time preparing for
  the next.

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More Stats(Again, I dont expect you to be
experts in this part)

• Conditional Distribution Functions
• fXY(xy)P(XxYy)
• The probability of X given Y
• Examples Given that you are female, what is
  your probability of getting a test score over 10?

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Last Example for Condl Distributions My Wifes
Mood
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Last Example for Condl Distributions

• What is the joint distribution?
• What is the marginal distribution?
• What is the probability she is in a bad mood
  given that she said, OK?
• What is the probability that she will tell me
  OK given that she is in a good mood?

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Expected Value

• The expected value is the value we expect the
  variable to be if we drew one realization of it.
  It is the weighted average.
• For example, f(x) .50 when x-1 .25
  when x0 .25 when x1
• E(X)?

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Properties of Expectations

• E(aXb)aE(X)b
• What is E(X) if X is normally distributed?

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Variance

• Let EXµ
• The Variance of X is defined as
• VXE(X- µ)2
• Can we simplify VX using expectations?
• Properties of Variance
• VaXba 2 VX
• Practice
• E.g. EXbar
• E.g. VXbar

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Covariance

• Cov(X,Y)E(X- µx)(Y- µy)
• If X and Y are independent, then Cov(X,Y)0
• VaraXbY
• a 2 VXb2 VY2abCov(X,Y)
• What is V(X-Y) if X and Y are independent?

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More Expectations

• Conditional Expectations
• One major note
• If X and Y are independent,

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Why do we Care?

• Suppose we want to measure the effect of the
  private schools on students.
• Is that the same as
• EYXin Private School-EYXno PS?
• Can we measure EYXno PS?
• What if we were to look at vouchers?

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Z-scores

• How do we compare different distributions?
• For example, Average Age in Econ 326
• Show Graph of Ages

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Ages
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Z-scores (cont.)

• Suppose you found someone in the quad who was 18
  years old.
• How likely is it that the person could be an
  undergrad at CWRU?
• Suppose you found someone in the quad who was 15
  years old.
• How likely is it that the person could be an
  undergrad at CWRU?

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Hours Worked by an Asst Prof
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Z-scores (cont.)

• Are you more likely to find an assistant
  professor working less than 30 hours a week or a
  CWRU student younger than 15?
• How do we figure it out precisely?
• We have to make the distributions comparable.
• Z-scores

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Z-scores (cont.)

• Suppose that we have a number of variables with
  similar histograms.
• We can Standardize these variables to make them
  comparable
• Suppose X is a variable with mean µ and standard
  deviation s
• Then standardize X by

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Z-scores (cont.)

• What is the expectation of Z?
• What is the variance of Z?
• What is the intuition?

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1. Z-scores (cont.)

• Central Limit Theorem
• Let Z be a standardized sum of n independent,
  identically distributed random variables with a
  finite, nonzero deviation
• Then the PDF of Z approaches a Normal
  Distribution as n increases
• What does this mean?
• Most random events can be modified to appear like
  a normal distribution
• Using Standardization, we can make most things
  look like a Normal with mean0 and standard
  deviation1

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1. Z-scores (cont.)

• T-distribution
• In samples smaller than 120, we may not reach a
  Normal Distribution
• We use the sample mean and std. Deviation to get
  a t-distribution with n degrees of freedom