LSRL

1
Chapter 5

• LSRL

2
Bivariate data

• x variable is the independent or explanatory
  variable
• y- variable is the dependent or response
  variable
• Use x to predict y

3
- (y-hat) means the predicted y

• b is the slope
• it is the approximate amount by which y increases
  when x increases by 1 unit
• a is the y-intercept
• it is the approximate height of the line when x
  0
• in some situations, the y-intercept has no meaning

Be sure to put the hat on the y
4
Least Squares Regression LineLSRL

• The line that gives the best fit to the data set
• The line that minimizes the sum of the squares of
  the deviations from the line

5
(0,0)
Sum of the squares 61.25
6
What is the sum of the deviations from the
line? Will it always be zero?
Use a calculator to find the line of best fit
The line that minimizes the sum of the squares of
the deviations from the line is the LSRL.
Sum of the squares 54
7
Interpretations
Slope For each unit increase in x, there is an
approximate increase/decrease of b in y.
Correlation coefficient There is a direction,
strength, linear of association between x and y.
8
The ages (in months) and heights (in inches) of
seven children are given. x 16 24 42 60 75 102 120
y 24 30 35 40 48 56 60 Find the LSRL.
Interpret the slope and correlation coefficient
in the context of the problem.
9
Correlation coefficient There is a strong,
positive, linear association between the age and
height of children.
Slope For an increase in age of one month, there
is an approximate increase of .34 inches in
heights of children.
10
The ages (in months) and heights (in inches) of
seven children are given. x 16 24 42 60 75 102 120
y 24 30 35 40 48 56 60 Predict the height of a
child who is 4.5 years old. Predict the height of
someone who is 20 years old.
11
Extrapolation

• The LSRL should not be used to predict y for
  values of x outside the data set.
• It is unknown whether the pattern observed in the
  scatterplot continues outside this range.

12

• The ages (in months) and heights (in inches) of
  seven children are given.
• The LSRL is
• Can this equation be used to estimate the age of
  a child who is 50 inches tall?
• Calculate LinReg L2,L1

For these data, this is the best equation to
predict y from x.
Do you get the same LSRL?
However, statisticians will always use this
equation to predict x from y
13
The ages (in months) and heights (in inches) of
seven children are given. x 16 24 42 60 75 102 120
y 24 30 35 40 48 56 60 Calculate x y. Plot
the point (x, y) on the scatterplot.
14
The correlation coefficient and the LSRL are both
non-resistant measures.
15
Formulas on chart
16
The following statistics are found for the
variables posted speed limit and the average
number of accidents.
Find the LSRL predict the number of accidents
for a posted speed limit of 50 mph.