Statistics%20350%20%20Lecture%202

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Statistics 350 Lecture 2
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Today

• Last Day Section 1.1-1.3
• Today Section 1.6
• Homework 1
• Chapter 1 Problems (page 33-38) 2, 5, 6, 7, 22,
  26, 33, 34, 35
• Due January 19
• Read Sections 1.1-1.3 and 1.6

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Simple Linear Regression

• Last day, introduced the simple linear regression
  model
• Yi?0?1Xi?i for i1,2,,n
• In practice, do not know the values of the ?s
  nor ?2
• Use data to estimate model parameters giving
  estimated regression equation
• Want to get the line of best fitwhat does this
  mean?

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Apartment Example
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Least Squares

• Would like to get estimates, bo and b1 to get
  estimated regression function
• Would like the estimated line to come as close as
  possible to the data
• Coming close to one point may make the line
  further from others
• Would like all points to be close on average

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Least Squares

• The single criterion that is commonly used is the
  least squares criterion
• Q
• Want to select values of bo and b1 that minimize
  Q
• How to minimize

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Least Squares

• Partial derivatives
• Normal equations

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Least Squares

• Solving the normal equations
• b0
• b1

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Comments and Properties

• b1 can be re-written as linear combination of the
  Yis
• Therefore
• It is a statistic (function of the data)
• It is a random variable with its own distribution
• It is a linear combination of independent normal
  random variables and thus will have a
• Same is true for bo
• As we shall see, both are unbiased estimators of
  , respectively, so

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Comments and Properties

• The resulting estimated regression line is
• It gives an estimate of E(Y) for a given X
• For the Xis in the sample, can compute the
  predicted (or fitted) values
• The difference between the actual observed data
  and the predicted value is
• They should resemble the ?is (Chapter 3)

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Comments and Properties

• Residuals sum to 0
• The sum of the squared residuals is minimized for
  these data (i.e., property of least squares)

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Comments and Properties

• ? Xiei0
• The predicted response at the mean of the
  observed X is

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Comments and Properties

• The sum of the residuals, weighted by the
  predicted responses, is
• The mean square error is
• It is useful because