32 Least Squares Regression Line Correlation measures the strength and direction of a linear relationship between two variables How do we summarize the overall pattern of a linear relationship Draw a line ID: 738443
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Slide1
GET OUT p.159 HW!Slide2
Least-Squares Regression
3.2 Least Squares Regression LineSlide3
Correlation measures the strength and direction of a linear relationship between two variables.
How do we summarize the overall pattern of a linear relationship?
Draw a line!
Recall from 3.1:Slide4
Regression Line
A
regression line
is a line that describes how a response variable
y
changes as an explanatory variable
x changes. We often use a regression line to predict the value of y for a given value of x.Slide5
Example p. 165
How much is a truck worth?
Everyone knows that cars and trucks lose value the more they are driven. Can we predict the price of a used Ford F-150
SuperCrew
4 x 4 if we know how many miles it has on the odometer? A random sample of 16 used Ford F-150
SuperCrew
4 x 4s was selected from among those listed for sale at autotrader.com. The number of miles driven and price (in dollars) was recorded for each of the trucks. Here are the data:Miles driven70,583129,48429,932
29,953
24,495
75,678
8359
4447
Price (in
dollars)
21,994
9500
29,875
41,995
41,995
28,986
31,891
37,991
Miles
driven
34,077
58,023
44,447
68,474
144,162
140,776
29,397
131,385
Price (in dollars)
34,995
29,988
22,896
33,961
16,883
20,897
27,495
13,997Slide6
Example p. 165
Miles driven
70,583
129,484
29,932
29,953
24,495
75,678
8359
4447
Price (in
dollars)
21,994950029,87541,99541,99528,98631,89137,991Miles driven34,07758,02344,44768,474144,162140,77629,397131,385Price (in dollars)34,99529,98822,89633,96116,88320,89727,49513,997Slide7
Interpreting a Regression Line
Suppose that
y
is a response variable (plotted on the vertical axis) and
x
is an explanatory variable (plotted on the horizontal axis).
A regression line relating y to x has an equation of the formŷ = a + bxSlide8
Interpreting a Regression Line
ŷ
=
a
+
bx
In this equation,ŷ (read “y hat”) is the predicted value of the response variable y for a given value of the explanatory variable x.b is the slope, the amount by which y is predicted to change when x increases by one unit.a is the
y intercept
, the predicted value of
y
when
x
= 0.Slide9
Example p. 166: Interpreting slope and
y
intercept
The equation of the regression line shown is
PROBLEM
: Identify the slope and y intercept of the regression line.
Interpret each value in context.
SOLUTION
: The slope
b
= -0.1629 tells us that the price of a used Ford F-150 is predicted to go down by 0.1629 dollars (16.29 cents) for each additional mile that the truck has been driven. Slide10
The equation of the regression line shown is
PROBLEM
: Identify the slope and y intercept of the regression line.
Interpret each value in context.
SOLUTION
: The
y
intercept
a
= 38,257 is the predicted price of a Ford F-150 that has been driven 0 miles.
Example p. 166: Interpreting slope and
y
interceptSlide11
Prediction – Example, p. 167
We can use a regression line to predict the response
ŷ
for a specific value of the explanatory variable
x
.
Use the regression line to predict price for a Ford F-150 with 100,000 miles driven.Slide12
Extrapolation – p. 167
Suppose we wanted to predict the price of a vehicle that had 300,000 miles.
According to the regression line, the vehicle would have a negative price. A negative price doesn’t make sense.Slide13
Extrapolation
Extrapolation
is the use of a regression line for prediction far outside the interval of values of the explanatory variable x used to obtain the line. Such predictions are often not accurate
.
Don
’
t make predictions using values of x that are much larger or much smaller than those that actually appear in your data.Slide14
Facts about LSRL:
1.
x
&
y
assignments matter.
LSRL will always go through The slope of the LSRL will always have the same sign as the correlation. Slide15
To plot the line on the scatterplot by hand:
Use the equation for
for two values of
x
, one near each end of the range of
x
in data. Plot each point. Slide16
For Example:
Use the equation:
Smallest x = 15.8,
Largest x = 36.8
Use these two x-values to predict y.
From data set: p. 146Slide17
For Example:
Use the equation:
Smallest x = 15.8,
Largest x = 36.8
(15.8, 3.1572)
(36.8, 12.3384)