Transcript middle - Campbell County Schools

3.2 Least Squares
Regression Line
Regression Line
• Describes how a response variable changes
as an explanatory variable changes
• Formula sheet:
• Calculator version:
Slope
• Formula Sheet
• Interpretation: how will the predicted response
variable change for one increase in the
explanatory variable?
Y-Intercept
• Formula Sheet
• Interpretation: what is the predicted response
variable if there is no explanatory variable?
• Mathematically - needed!
• Realistically - might not make sense!
•
Sometimes the explanatory variable
might not make sense being zero
context of the problem.
Determine if the yintercept is realistic for
this problem, explain.
(I will write the equation on the board)
Extrapolation
• When using a regression line to predict a
variable outside the range of the data
gathered
• Unreliable predictions!
Multiple Choice
Problems
Let's do p. 160!
3.2 - LeastSquares
Regression
(Residuals)
Where else have we
seen
“residuals?”
Sx = data point - mean
(observed - predicted)
z-scores = observed - expected
* note: this is just the numerator
of these calculations
Remember:
AP
Below is the LSRL for sprint time (seconds) and the long jump distance (inches)
Find and interpret the residual for John who had a time of 8.09 seconds and a
jump of 151 inches.
predicted long jump distance = 304.56 - 27.63(sprint time)
residual = observed - predicted
151
- 81.03
residual = 69.97 inches
John jumped much farther than what was predicted
by our least squares regression line. He jumped
almost 70 inches farther, based on his sprint time.
So why least squared
regression
line?
Graph (0,0), (0,2), (2,2), and (2,4) and find the least squares
regression line. Then find the residuals.
Windows - find the sum of the square of the residuals
Door - find the sum of the absolute value of the residuals
Now, what if I said the least squares regression line was y = 0.2 + 1.6x? y = x?
Windows find the sum of the square of the residuals
Door - find the sum of the absolute value of the residuals
http://bcs.whfreeman.com/tps4e/#628644__666392__
Stop notes for today
Homework is p193
#43,45,47,53
Activity - "Matching Descriptions to Scatterplots"
Homework hint: you will need to be familiar with the
formulas on your sheet to write the LSRL
Residual Plots
a scatterplot of the residuals
against the explanatory
variable.
used to help assess the
strength of your regression
line
Residual Plots
with Normal Probability Plots we want the
graphs to be linear to support the
Normality of our data.
with Residual Plots we want the residuals
to be very scattered so our data is can be
model with a linear regression.
Remember:
Correlation does NOT assess linearity, just
strength and direction!
What’s a Good
Residual
Plot?
No
obvious pattern
- the LSRL
would be in the middle of the
data, some data above and
some below
Relatively small residuals - the
data points are close to the
LSRL
Do the following residual plots support
or refute a linear model?
http://content.ebscohost.com/pdf23_24/pdf/2009/D8Y/01Sep09/43669525.pdf?T=P&P=AN&K=43669525&S=R&D=aph&EbscoContent=dGJyMNHX8kSeqK84yOvqOLCmr0qep7RSs6%2B4S7aWxWXS&ContentCustomer=
ssk2xqLJNuePfgeyx44Hy
How to Graph?
Take each data point and determine the
residual
Plot the residuals versus the explanatory
variable
i.e. (explanatory data, residual)
residual
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
use the same numbers as your scatterplot
explanatory variable
Calculator
Construction
If you have a lot of data, follow the
instructions on page 178
to construct your residual plot
(you will also have to have done the
technology corner on p. 170)
What is Standard
Deviation?
the average squared distance
a data point is from the mean
Is there a sx? Is there a sy?
So why not s? (standard
deviation of residuals)
Standard Deviation of
Residuals
gives the approximate size of
an “average” or “typical”
prediction error from our LSRL
formula on page 177
Why divide by n-2?