stats chapter 5 - least squares regression definition of a regression line: a regression line is a...
TRANSCRIPT
![Page 1: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/1.jpg)
Stats Chapter 5 - Least Squares Regression
Definition of a regression line:A regression line is a straight line that describes how a
response variable (y) changes as an explanatory variable (x) changes…
• Used to predict a y value given an x value.
• Requires an explanatory and a response variable.
• Given as an equation of a line in slope intercept form:
y = a + bx
Read as: “y-hat” a = y-intercept b = slope
![Page 2: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/2.jpg)
How It Works:
^y = a + bx
x
y
Using the regression line to predict a y-value
![Page 3: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/3.jpg)
Vertical Distance
Observed y
Predicted yVertical Distance = Observed - Predicted
Close-Up: We are trying to find a line that minimizesthe squares of the vertical distances…
y = negative
y = positive
![Page 4: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/4.jpg)
Least-Squares Regression Line:
• The least-squares regression line of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible.
• The slope is the amount of change in y when x increases by one unit.
^
• The intercept of the line is the predicted value of y when x = 0.
^
![Page 5: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/5.jpg)
Calculator Procedure
1) Enter Data into lists…
List 1 List 2 List 1 List 2
![Page 6: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/6.jpg)
Run Stat > Calc > 8Run Stat > Calc > 8
y = 1.089 + .189x
Write regression line from
Calculated a and b values
Write regression line from
Calculated a and b values
y = a + bx
y-int = gas used when degree
days = 0
slope = increase in gas used when
degree days increase by one
![Page 7: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/7.jpg)
Correlation vs. Regression
• The square of the correlation (r2) is the fraction of the variation in the values of y that is explained by the least squares regression line of y on x.
• 0 < r2 < 1
• When reporting a regression, give r2 as a measure of how successful the regression was in explaining the response.
• ex: 5.4 pg 134.
![Page 8: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/8.jpg)
Residuals
• Residual = The difference between observed & predicted y-values.
• Residual = y - y
• Residual Plot - plots the residual values on the y-axis vs. the explanatory variable on the x-axis.
• Makes patterns easier to see.
-1
1
0 55• Used to assess the fit of a regression line.
![Page 9: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/9.jpg)
Calculator Procedure
1) Run regression
2) Go into Stat Plot 1
3) Set Y-List to ‘RESID’
4) Set window values to match data range
5) Graph
2nd > STAT > 72nd > STAT > 7
![Page 10: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/10.jpg)
Residual Patterns
Ideal:
Curved:
Spread:
![Page 11: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/11.jpg)
Outliers:
Vertical Horizontal
![Page 12: Stats Chapter 5 - Least Squares Regression Definition of a regression line: A regression line is a straight line that describes how a response variable](https://reader036.vdocument.in/reader036/viewer/2022062409/5697bff11a28abf838cbb6d1/html5/thumbnails/12.jpg)
Cautions About Correlation and Regression
• Both describe LINEAR relationships
• Both are affected by outliers
• Always plot your data before interpreting
• Beware of EXTRAPOLATION
• Beware of LUKRING VARIABLES
CORRELATION DOES NOTIMPLY CAUSATION!!!