Simple Linear Regression and Correlation: Inferential Methods

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Simple Linear Regression and Correlation
Inferential Methods

• Chapter 13
• AP Statistics
• Peck, Olsen and Devore

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Topic 2 Summary of Bivariate Data

• In Topic 2 we discussed summarizing bivariate
  data
• Specifically we were interested in summarizing
  linear relationships between two measurable
  characteristics
• We summarized these linear relationships by
  performing a linear regression using the method
  of least squares

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Least Squares Regression

• Graphically display the data in a scatterplot
• Form, strength and direction
• Calculate the Pearsons Correlation Coefficient
• The strength of the linear association
• Perform the least squares regression
• Inspect the residual plot
• Determine if the model is appropriate
• No patterns
• Determine the Coefficient of Determination
• How good is the model as a prediction tool
• Use the model as a prediction tool

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Interpretation

• Pearsons correlation coefficient
• Coefficient of Determination
• Variables in
• Standard deviation of the residuals

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Minitab Output
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Simple Linear Regression Model

• Simple because we had only one independent
  variable
• We interpreted as a predicted value of
  y given a specific value of x
• When we can describe this as a
  deterministic model. That is, the value of y is
  completely determined by a given value x
• That wasnt really the case when we used our
  linear regressions. The value of y was equal to
  our predicted value /- some amount. That is,
  
  We call this a probabilistic model.
• So, without e, the (x,y) pairs (observed points)
  would fall on the regression line.

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Now consider this

• How did we calculate the coefficients in our
  linear regression models?
• We were actually estimating a population
  parameter using a sample. That is, the simple
  linear regression is an
  estimate for the population regression line
• We can consider estimates for

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Basic Assumptions for the Simple Linear
Regression Model

• The distribution of e at any particular value of
  x has a mean value of 0. That is,
• The standard deviation of e is the same for any
  value of x. Always denoted by
• The distribution of e at any value of x is normal
• The random deviations are independent.

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Another interpretation of

• Consider , where the
  coefficients are fixed and e is distributed
  normally. Then the sum of a fixed number and a
  normally distributed variable is normally
  distributed (Chapter 7). So y is normally
  distributed.
• Now the mean of y will be equal to
  plus the mean of e which is equal to 0
• So another interpretation is the mean y value for
  a given x value

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Distribution of y

• Where we can now see that
  y is distributed normally with a mean of
• The variance for y is the same as the variance of
  e -- which is
• An estimate for is

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Assumption

• The major assumption to all this is that the
  random deviation e is normally distributed.
• Well talk more about how this assumption is
  reasonable later.

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Inferences about the slope of the population
regression line

• Now we are going to make some inferences about
  the slope of the regression line. Specifically,
  well construct a confidence interval and then
  perform a hypothesis test a model utility test
  for simple linear regression

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Just to repeat

• We said the population regression model is
• The coefficients of this model are fixed but
  unknown (parameters) so using the method of
  least squares, we estimate these parameters using
  a sample of data (statistics) and we get

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Sampling distribution of b

• We use b as an estimate for the population
  coefficient in the simple regression model
• b is therefore a statistic determined by a random
  sample and it has a sampling distribution

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Sampling distribution of b

• When the four assumptions of the linear
  regression model are met
• The mean value of the sampling distribution of b
  is . That is,
• The standard deviation of the statistic b is
• The sampling distribution of b is normally
  distributed.

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Estimates for

• The estimate for the standard deviation of b is
• When we standardize b it has a t distribution
  with n-2 degrees of freedom

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Confidence Interval

• Sample Statistic /- Crit Value Std Dev of Stat

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Hypothesis Test

• Were normally interested in the null
  because if we reject the null, the data
  suggests there is a useful linear relationship
  between our two variables
• We call this Model Utility Test for Simple
  Linear Regression

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Summary of the Test

• Test Statistic
• Assumptions are the same four as those for the
  simple linear regression model.

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Minitab Output