Download - Section 3.3 Linear Regression
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Section 3.3Linear Regression
Statistics
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AP Statistics, Section 3.3, Part 1 2
Linear Regression
It would be great to be able to look at multi-variable data and reduce it to a single equation that might help us make predictions
“What would be the predicted number of wins for a team with a 4.0 ERA?”
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AP Statistics, Section 3.3, Part 1 3
Linear Regression
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AP Statistics, Section 3.3, Part 1 4
The Least-Square Regression
Finds the best fit line by trying to minimize the areas formed by the difference of the real data from the predicted data.
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AP Statistics, Section 3.3, Part 1 5
The Least-Square Regression
Finds the best fit line by trying to minimize the areas formed by the difference of the real data from the values predicted by the model.
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AP Statistics, Section 3.3, Part 1 6
The Least-Square Regression
Statistician use a slightly different version of “slope-intercept” form. y
x
y a bxs
b rs
a y bx
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AP Statistics, Section 3.3, Part 1 7
Predicting Model To put the regression line
on the graph use the Statistics:Eq:RegEQ from the Vars menu to put the Y1 equation.
Then you can use Trace or Table or Y1 to find response values that correspond to particular experimental values.
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AP Statistics, Section 3.3, Part 1 8
Fact about least-square regression
Make sure you know which is the explanatory (x) variable and which is the response (y) variable.Switching them gets a different regression
line.
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AP Statistics, Section 3.3, Part 1 9
Fact about least-square regression
Regression line always goes through the point (x-bar, y-bar)
The coefficient of correlation (r) explains the strength of the linear relationship
The square of the correlation (r2) is the variation in the values of y that is explained by x.
___%(r2) of the variation of ______ (y) is explained by _____ (x).
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AP Statistics, Section 3.3, Part 1 10
r2 “coefficient of explanation”
In the regression of ERA vs. WINS, we find a r2 value of .4512
We say “45% of the variation in WINS can be explained by ERA”
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AP Statistics, Section 3.3, Part 1 11
Outliers vs. Influential Data
An outlier is an observation outside the overall pattern
If an observation is influential it has a large effect on the regression line. Removing the observation markedly changes the calculation.
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AP Statistics, Section 3.3, Part 1 12
Outliers vs. Influential Data
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AP Statistics, Section 3.3, Part 1 13
Residuals
It is important to note that the observed value almost never match the predicted values exactly
The difference between the observed value and predicted has a special name: residual
Observed Value (y): 5.3 ERA, 43 Wins
Predicted Value ( ): 5.3 ERA 67.03 Wins
y
Residual:
43-67.03=-24.03ˆy y
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AP Statistics, Section 3.3, Part 1 14
Residuals Residuals are
negative when the observed value is below the predicted value
Residuals are positive when the observed value is higher than the predicted value
Observed Value (y): 5.3 ERA, 43 Wins
Predicted Value ( ): 5.3 ERA 67.03 Wins
y
Residual:
43-67.03=-24.03ˆy y
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AP Statistics, Section 3.3, Part 1 15
Residual Plots
You can plot the residuals to see if the there is any trends with the quality of the predictive model
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AP Statistics, Section 3.3, Part 1 16
Residual Plots
This residual shows no tendencies. It is equally bad throughout.
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AP Statistics, Section 3.3, Part 1 17
“Under predicts on the ends”
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AP Statistics, Section 3.3, Part 1 18
“Predictive accuracy decreases”
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AP Statistics, Section 3.3, Part 1 19
“Well Distributed”
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AP Statistics, Section 3.3, Part 1 20
Assignment
Exercises: 3.38, 3.40 for Tuesday Exercises: 3.42, 3.43, 3.46, 3.47, 3.49,
3.53, 3.55, 3.57, 3.61 for Thursday Chapter Review for Monday: 3.63, 3.67,
3.71, 3.73, 3.75, 3.77 Sample Test due Monday Chapter 3 Test take home due on Monday