MATH 2400

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MATH 2400

• Chapter 5 Notes

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Regression Line

• Uses data to create a linear equation in the form
  y ax b where
• a is the slope of the line (unit rate of
  change)
• b is the y-intercept (initial value)
• Can be used generalize a set of data, to estimate
  a value, or predict a value.

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Example 1 (Exercise 5.1)

• We expect a cars highway gas mileage to be
  related to its city gas mileage. Data for all
  1040 vehicles in the governments 2010 Fuel
  Economy Guide give the regression line
• HWY MPG 6.554 (1.016 x CITY MPG)
• for predicting highway mileage from city mileage.
• What is the slope of this line? Say in words
  what the numerical value of the slope tells you.
• What is the intercept? Explain why the value of
  the intercept is not statistically meaningful.
• Find the predicted highway mileage for a car that
  gets 16 mpg in the city. Do the same for a car
  with city mileage 28 mpg.

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Example 2 (Exercise 5.2sort of)

• You use the same bottle of body wash every day.
  The volume was initially 355 ml. You estimate
  you use 7 ml of body wash each day. What is the
  equation of the regression line for predicting
  the volume of body wash left in the bottle after
  each day?

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Least-Square Regression Line

•  

 
 
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Example 3

• This table displays the data regarding 8 U.S
  airports and their total number of passengers for
  the year 1992 and 2005. Use the 1992 data as the
  explanatory variable and the 2005 data as the
  response variable. Create a least-squares
  regression line and use that line to estimate how
  many passengers Raleigh-Durham International had
  in 2005 if the airport had 4.9 million passengers
  in1992.

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r and r2

• r tells us if there is a positive or negative
  relationship between the explanatory variable and
  the response variable.
• r also tells us how strong of a relationship the
  variables have.
• r2 tells us what portion of the linear
  relationship between the variables can be
  explained by the explanatory variable.
• 1 r2 tells us what portion of the linear
  relationship between the variables can not be
  explained by the explanatory variable.
• Ex If r 0.6, ? r2 0.36. 36 of the linear
  relationship can be explained by the explanatory
  variable and 64 cannot be explained.
• Ex If r -1, ? r2 1. 100 of the linear
  relationship can be explained by the explanatory
  variable and 0 cannot be explained.

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Example 4
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Example 5
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Residuals

• A residual is the difference between an observed
  value of the response variable and the value
  predicted by the regression line. That is, a
  residual is the prediction error that remains
  after we have chosen the regression line
• Residual observed y predicted y
• y -

 
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Residualscontinued

• A residual plot makes it easier to see unusual
  observations and patterns. The regression line
  is horizontal (think about it).

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Residual Graphing

• Use the following data to create a least-squares
  regression line and plot the residuals on the
  graph provided.

AGE HEIGHT
0 20
1 31
2 36
3 39
4 43
5 46
6 48
7 51
8 54
9 56
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CAUTION!!!

• Correlation and regression lines describe only
  linear relationships.
• Correlation and least-squares regression lines
  are not resistant to influential data (data
  drastically outside the norm). We should always
  plot our data and look for observations that
  might be influential.
• Ecological Correlation is based on averages
  rather than on individuals.
• Ex There is a large positive correlation
  between average income and number of years of
  education. The correlation is smaller if we
  compare the incomes of individuals with number of
  years of education. The correlation based on
  average income ignores the large variation in the
  incomes of individuals having the same amount of
  education.

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CAUTION!!!

• Extrapolation is the use of a regression line for
  prediction far outside the range of values of the
  explanatory variable that you used to obtain the
  line.
• Ex Using the least-squares regression line for
  the height of the child from ages 0-9 to predict
  their height at age 30.
• Lurking Variables should always be thought about
  before drawing conclusions based on correlation
  or regression.

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Correlation ? Causation???

• NO!!!
• A serious study once found that people with two
  cars live longer than people who own only one
  car. Owning three cars is even better, and so
  on. There is a substantial positive correlation
  between number of cars x and length of life y.
• Lurking variables?

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HW 5.17
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HW 5.25
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HW 5.27
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HW 5.29
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HW 5.53