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
Interpret the slope and the y-intercept from the given least squares regression line in context of
the problem. Determine if the y-intercept 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 - Least-Squares
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:Remember:APAP
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
residual = 69.97 inchesJohn jumped much farther than what was
predicted by our least squares regression line. He jumped almost 70 inches farther, based on his
sprint time.
- 81.03
So why least squared regression
line?
http://bcs.whfreeman.com/tps4e/#628644__666392__
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 residualsDoor - 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 residualsDoor - find the sum of the absolute value of the residuals
Stop notes for todayHomework 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 Plotswith 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:Remember:Correlation does NOT assess linearity, just Correlation does NOT assess linearity, just
strength and direction!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=dGJyMPGssk2xqLJNuePfgeyx44Hy
How to Graph?Take each data point and determine the residual
Plot the residuals versus the explanatory variable
i.e. (explanatory data, residual)
explanatory variable
residual
21.51
0.50
-0.5-1
-1.5-2
use the same numbers as your scatterplot
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?