Find the Least Squares Regression Line and interpret its slope, y-intercept, and the coefficients of correlation and determination
Justify the regression model using the scatterplot and residual plot
AP Statistics Objectives Ch8
Model Residuals Slope Regression to the meanInterceptR2
VocabularyLinear model
Predicted valueRegression line
Residual Plot Vocabulary
Chapter 7 Answers
Linear Regression Practice
Regression Line Notes
Chapter 8 Assignments
Chp 8 Part I Day 2 Example
Lurking Variable
Lurking Variable
Chapter 8 #1r
a) 10 2 20 3 0.5b) 2 0.06 7.2 1.2 -0.4c) 12 6 -0.8 200-4xd) 2.5 12 100 -100+50x
π1=π π π¦π π₯
οΏ½ΜοΏ½=ππ+ππ π
οΏ½ΜοΏ½=ππ .π+π .ππ π
Chapter 8 #1r
a) 10 2 20 3 0.5b) 2 0.06 7.2 1.2 -0.4c) 12 6 -0.8 200-4xd) 2.5 12 100 -100+50xπ1=
π π π¦π π₯
οΏ½ΜοΏ½=ππ+ππ π
οΏ½ΜοΏ½=ππ .π+π .ππ π
Chapter 8 #1r
a) 10 2 20 3 0.5b) 2 0.06 7.2 1.2 -0.4c) 12 6 -0.8 200-4xd) 2.5 12 100 -100+50x
π1=π π π¦π π₯200-4x
οΏ½ΜοΏ½=ππ .π+π .ππ ποΏ½ΜοΏ½=ππ .πβππ
Chapter 8 #1r
a) 10 2 20 3 0.5b) 2 0.06 7.2 1.2 -0.4c) 12 6 -0.8 200-4xd) 2.5 1.2 100 -100+50x
π1=π π π¦π π₯
-100+50x
οΏ½ΜοΏ½=ππ .π+π .ππ ποΏ½ΜοΏ½=ππ .πβππ
πππππ
Standardized Foot Length vs Height 2011
NOTE: (0,0) represents the mean of x and the mean of y.
π§ hπ»πππ π‘=0.84 π§πΉπππ‘πππ§π
Slope is the correlation
is part of all regression
lines
Regression Line for Standardized Values
=
is the predicted z-score for the response variable
is the z-score for the explanatory variable
π ππ hπ‘ πππππππππ‘ππππππππππππππ‘
Stand. Regres. Line will always pass through (.
Regression Line for
= +
is the predicted response variable
is the y-intercept
=
is the slope
=
Regression Line will always pass through (.
Explanatory or Response
Now interpret the R2. R2 = .697
According to the linear model, 69.7% of the variability in height is accounted for by variation in foot size.
Explanatory or Response 2011 data resulted in the following linear equation:
CAREFUL! The equations are not the same when you switch
explanatory and response variables.
Explanatory or Response 2011 data resulted in the following linear equation:
CAREFUL! The equations are not the same when you switch
explanatory and response variables.
Residual Plot Example
Residual Plot Example
REMEMBER: POSITIVE RESIDUALS are UNDERESTIMATES
e = y -
Residual Plot Example
NEGATIVE RESIDUALS are OVERESTIMATES
e = y -
Assignment
CHAPTER 8 Part I: pp. 189-190 #2,4,8&10,12&14Part II: pp. 190-192 #16,18,20,28&30
Chapter 7 Answers
a) #1 shows little or no associationb) #4 shows a negative associationc) #2 & #4 each show a linear
associationd) #3 shows a moderately strong,
curved associatione) #2 shows a very strong association
Chapter 7 Answers
a) -0.977b) 0.736c) 0.951d) -0.021
Chapter 7 Answers
The researcher should have plotted the data first. A strong, curved relationship may have a very low correlation. In fact, correlation is only a useful measure of the strength of a linear relationship.
Chapter 7 Answers
If the association between GDP and infant mortality is linear, a correlation of -0.772 shows a moderate, negative association.
Chapter 7 Answers
Continent is a categorical variable. Correlation measures the strength of linear associations between quantitative variables.
Chapter 7 Answers
Correlation must be between -1 and 1, inclusive. Correlation can never be 1.22.
Chapter 7 Answers
A correlation, no matter how strong, cannot prove a cause-and-effect relationship.
Chapter 8 Vocabulary1) Regression to the mean β each predicted response variable (y) tends to be closer to the mean (in standard deviations) than its corresponding explanatory variable (x)
Chapter 8 Vocabulary2) β predicted response variable
3) Residual β the difference between the actual response value and the predicted response value
e = y - 4) Overestimate β produces a negative residual
5) Underestimate β produces a positive residual
Chapter 8 Vocabulary6) Slope β rate of change given in units of the response variable (y) per unit of the explanatory variable (x)
7) intercept β response value when the explanatory value is zero
8) R2 β Must also be interpreted when describing a regression model (aka Coefficient of Determination)
Chapter 8 Vocabulary8) R2 β Must also be interpreted when describing a regression model
βAccording to the linear model, _____% of the variability in _______ (response variable) is accounted for by variation in ________ (explanatory variable)β
The remaining variation is due to the residuals
Chapter 8 VocabularyCONDITIONS FOR USING A LINEAR REGRESSION
1) Quantitative Variables β Check the variables2) Straight Enough β Check the scatterplot 1st
(should be nearly linear) - Check the residual plot next
(should be random scatter)3) Outlier Condition-
- Any outliers need to be investigated
Chapter 8 Vocabulary9. Residual Plot - a scatterplot of the residuals and either x or
If you find a pattern in the Residual Plot, that means the residuals (errors) are predictable. If the residuals are predictable, then a better model exists. ---- LINEAR MODEL IS NOT APPROPRIATE. A residual plot is done with the RESIDUALS on the y-axis. On the x-axis, put the explanatory variable.
NOTE: Some software packages will put on the x-axis. This does not change the presence of (or lack of) of a pattern.
Chapter 8 Vocabulary9. Residual Plot - a scatterplot of the residuals and either x or
If you find a pattern in the Residual Plot, that means the residuals (errors) are predictable. If the residuals are predictable, then a better model exists. ---- LINEAR MODEL IS NOT APPROPRIATE. A residual plot is done with the RESIDUALS on the y-axis. On the x-axis, put the explanatory variable.
NOTE: Some software packages will put on the x-axis. This does not change the presence of (or lack of) of a pattern.
What is the ?
Did you say 2? Wrong. Try again.
It is actually because both (2)2 and (-2)2 is 4.
So what?
Important Note: The correlation is not given directly in this software package. You need to look in two places for it. Taking the square root of the βR squaredβ (coefficient of determination) is not enough. You must look at the sign of the slope too. Positive slope is a positive r-value. Negative slope is a negative r-value.
So here you should note that the slope is negative. The correlation will be negative too. Since R2 is 0.482, r will be -0.694.
S/F Ratio
Grad Rate
-0.07861
Coefficient of Determination =
(0.694)2 = 0.4816
0.4816
With the linear regression model, 48.2% of the variability in airline fares is accounted for by the variation in distance of the flight.
π1=ππ π¦π π₯
ΒΏ0.694ππ .ππ497.8
ΒΏ0.0786
There is an increase of 7.86 cents for every additional mile.
#10. Interpret the slope.
There is an increase of $7.86 for every additional 100 miles.
π1=ππ π¦π π₯
ΒΏ0.694ππ .ππ497.8
There is an increase of 7.86 cents for every additional mile.
#10. Interpret the slope.
There is an increase of $7.86 for every additional 100 miles.
244.33 = + (0.0786)(853.7)
π1=0.0786
π¦=π0+π1π₯
244.33 β (0.0786)(853.7) =
#9. Interpret the y-intercept.
The model predicts a flight of zero miles will cost $177.23. The airline may have built in an initial cost to pay for some of its expenses.
177.2292=
π1=0.0786
177.2292 + 0.0786Distance
π1=0.0786
177.2292 + 0.0786Distance
177.2292 + 0.0786(200)
$192.95
177.2292 + 0.0786Distance
177.2292 + 0.0786(200)
$192.95
177.2292 + 0.0786(2000)
$334.43
8. Using those estimates, draw the line on the scatterplot.
177.2292 + 0.0786(200) = $192.95
177.2292 + 0.0786(2000) = $334.43
177.2292 + 0.0786Distance
177.2292 + 0.0786(1719)
$312.34
y β
212 β
-$100.34
12. In general, a positive residual means
13. In general, a negative residual means
The model underestimatedthe actual value.
The model overestimatedthe actual value.
A linear model should be appropriate, because
1) the scatterplot shows a nearly linear form and
2) the residual plot shows random scatter.
The coefficient of determination is .482, so
the coefficient of correlation is = .694. This shows a moderate strength in association for the model.
$150 for a flight of about 700 miles seems low compared to the other fares.
βfareβ is the response variable. Not all software will call it the dependent variable.Always look for βConstantβ and what is listed beside it. Here above it shows the column is for the βvariableβ and below βdistβ is the explanatory variable.
Recall:For y = 3x + 1 the coefficient of x is β3β.For computer printouts this is the key column for your regression model.
Recall:For y = 3x + 1 the coefficient of x is β3β.For computer printouts this is the key column for your regression model.
The βCoefficientβ of the βConstantβ is the y-intercept for your linear regression.
Recall:For y = 3x + 1 the coefficient of x is β3β.For computer printouts this is the key column for your regression model.
The βCoefficientβ of the βConstantβ is the y-intercept for your linear regression.
The βCoefficientβ of the variable βdistβ is the slope for your linear regression.
177.215 + 0.078619distance
Recall:For y = 3x + 1 the coefficient of x is β3β.For computer printouts this is the key column for your regression model.
The βCoefficientβ of the βConstantβ is the y-intercept for the linear regression.
The βCoefficientβ of the variable βdistβ is the slope for the linear regression.
177.215 + 0.078619distance
177.215 + 0.078619(1000)
5. Predict the airfare for a 1000-mile flight.
ΒΏ $πππ .ππ
Note: Even when we switchthe response and explanatory
variables, the linear modelis still appropriate.
-644.287 + 6.13101fare
R2 doesnβt change, but the equation does.
-644.287 + 6.13101fare
-644.287 + 6.13101
= 924.2 miles
-644.287 + 6.13101fare
-644.287 + 6.13101
= 924.2 miles
8. Residual? e = y - = 924.2 β 1000 = -75.8
Chp 8 #17R squared = 92.4%
17a. What is the correlation between tar and nicotine? (NOTE: scatterplot shows a strong positive linear association.)
+ =
Chp 8 #17R squared = 92.4%
17b. What would you predict about the average nicotine content of cigarettes that are 2 standard deviations below average in tar content.
= r
r=
= 0 = -1.922
I would predict that the nicotine content would be 1.922 standard deviations below the average.
Chp 8 #17R squared = 92.4%
17c. If a cigarette is 1 standard deviation above average in nicotine content, what do you suspect is true about its tar content?
= r
r=
= 0 = 0.961
I would predict that the tar content would be 0.961 standard deviations above the average.
