Least Squares Regression Line (LSRL)

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Least Squares Regression Line (LSRL)

• Presentation 2-5

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Introduction
Size of Diamond vs. Price

• Many times the scatterplot shows some pattern in
  the data.
• For now, we will look at the analysis of data
  that falls in a straight line pattern.

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Introduction

• When we see a straight line pattern, we want to
  model the data with a linear equation.
• This will allow us to make predictions and
  actually use our data.

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Linear Relations

• We know lines from algebra to come in the form y
  mx b, where m is the slope and b is the
  y-intercept.
• In statistics, we use y a bx for the equation
  of a straight line. Now a is the intercept and b
  is the slope.
• The slope (b) of the line, is the amount by which
  y increases when x increase by 1 unit.
• This interpretation is very important.
• The intercept (a), sometimes called the vertical
  intercept, is the height of the line when x 0.

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Example

• Consider the equation y73x
• The slope is 3.
• For every increase of 1 in the x-variable, there
  will be an increase of 3 in the y-variable.
• The intercept is 7.
• When the x-variable is 0, the y-variable is 7.

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Example

• Consider the equation y17-4x
• The slope is -4.
• For every increase of 1 in the x-variable, there
  will be a decrease of 4 in the y-variable.
• The intercept is 17.
• When the x-variable is 0, the y-variable is 17.

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Least Squares Line

• How can we find the best line to fit the data?
• We would like to minimize the total distance away
  from the line
• This distance is measured vertically from the
  point to the line.
• Go to the following applet and start plotting
  points to see how this process works.
• Make your own regression line

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Least Squares Line

• You first get a line once you plot two points.
• When you plot the third, green bars appear
  representing the error (actually called residual)
  of the line.
• These are how far off your line is for each of
  the points.
• The best line is the one that would minimize the
  total length of the green lines (all put
  together).

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Guess the best fit line

• Go to the following applet to practice your
  skills at estimating an LSRL.
• Plot a bunch of points.
• Then click the draw line button and draw what you
  think is the best fit line
• Then, check the show least squares line
  checkbox
• To the applet

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The details of the LSRL

• The mathematics involved in calculating the LSRL
  is a bit complicated.

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Least Squares Line
The line that gives the best fit to the data is
the one that minimizes this sum it is called
the least squares line or sample regression line.
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Coefficients a and b
S-sub y and s-sub x are the sample standard
deviations of y and x (kinda like rise over run)
The slope is
The intercept is
y-bar and x-bar are the mean y and x respectively
The equation of the least squares regression line
is written as
The little symbol above the y is a hat! The
equation is read as, y-hat equals a plus bx.
The y-hat indicates that this is a regression
line and that the model (equation) is to be used
to make predictions.
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Three Important Questions

• To examine how useful or effective the line
  summarizing the relationship between x and y, we
  consider the following three questions.
• Is a line an appropriate way to summarize the
  relationship between the two variables?
• Are there any unusual aspects of the data set
  that we need to consider before proceeding to use
  the regression line to make predictions?
• If we decide that it is reasonable to use the
  regression line as a basis for prediction, how
  accurate can we expect predictions based on the
  regression line to be?

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Example 1 - Finding the LSRL

• Consider the following data
• With this data, find the LSRL
• Start by entering this data into list 1 and list 2

Shoe Size (mens U.S.) Height (in)
7 64
10 69
12 71
8 68
9.5 71
10.5 70
11 72
12.5 74
13.5 77
10 68
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Example 1 - Finding the LSRL

• You should then see the results of the
  regression.
• a53.24
• b1.65
• r-squared.8422
• r.9177

This is the correlation coefficient for the
scatterplot!!!
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Example 2 Interpreting LSRL

• Interpreting the intercept
• When your shoe size is 0, you should be about
  53.24 inches tall
• Of course this does not make much sense in the
  context of the problem
• Interpreting the slope
• For each increase of 1 in the shoe size, we would
  expect the height to increase by 1.65 inches

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Example 3 Using LSRL

• Making predictions
• How tall might you expect someone to be who has
  a shoe size of 12.5?
• Just plug in 12.5 for the shoe size above, so
• Height 53.241.65 (12.5)73.865 inches
• Of course this is a prediction and is therefore
  not exact.

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Least Squares Regression Line (LSRL)

• This concludes this presentation.