1. Section 12.2 Linear Regression
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HAWKES LEARNING SYSTEMS
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Section 12.2
Linear Regression

2. HAWKES LEARNING SYSTEMS
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Regression, Inference, and Model Building
12.2 Linear Regression
Definitions:
If r (correlation coefficient) is statistically significant, then a regression line can be used to make predictions regarding the data.
Regression Line – the line from which the average variation from the data is the smallest. Also known as the line of best fit. There is only one regression line for each data set.
b0 = y-intercept
b1 = slope ^
Σ(Y – Y)2 = 24

3. HAWKES LEARNING SYSTEMS
math courseware specialists
Regression, Inference, and Model Building
12.2 Linear Regression
Slope:
When calculating the slope, round your answers to three decimal places.
y-Intercept:
When calculating the y-intercept, round your answers to three decimal places.

4. We can use a calculator or the equations to find the regression line.
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12.2 Linear Regression
Determine the regression line:
The local school board wants to evaluate the relationship between class size and performance on the state achievement test. They decide to collect data from various schools in the county. A portion of the data collected is listed below. Each pair represents the class size and corresponding average score on the achievement test for 8 schools.
Class Size 15 17 18 20 21 24 26 29
Average Score
85.3
86.2
85.0
82.7
81.9
78.8
75.3
72.1
Solution:
First decide which variable should be the x variable and which variable should be y.
Class size is the independent variable, x, and average score is the dependent variable, y.
We can use a calculator or the equations to find the regression line.

5. Copy Data from Hawkes to Excel and Paste in cell A1

6. Using the equations for slope and y-intercept:
HAWKES LEARNING SYSTEMS
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Regression, Inference, and Model Building
12.2 Linear Regression
Solution (continued):
Using the equations for slope and y-intercept:
n = 8, ∑x = 170, ∑y = 647.3, ∑xy = , ∑x 2 = 3772
Now place the slope and y-intercept into the regression line equation:
–1.043

7. A prediction should not be made with a regression model if:
HAWKES LEARNING SYSTEMS
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Regression, Inference, and Model Building
12.2 Linear Regression
Predictions:
A prediction should not be made with a regression model if:
The correlation is not statistically significant.
You are using a value outside of the range of the sample data.
The population is different than that of the sample data.

8. Use the equation of the regression line,
HAWKES LEARNING SYSTEMS
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Regression, Inference, and Model Building
12.2 Linear Regression
Calculate the prediction:
Use the equation of the regression line,
to predict what the average achievement test score will be for the following class size: 16 19 25 45
Since 45 is outside the range of the original data of 15 to 29, we cannot predict the test score.
86.390
83.260
77.000

9. RUN REGRESSION ANALYSIS
Copy Data from Hawkes to Excel and Paste in cell E1.
Select Data E1:M2, Copy and Paste Special, (Select option Transpose), in cell A1.
RUN REGRESSION ANALYSIS

11. HAWKES LEARNING SYSTEMS math courseware specialists
Regression, Inference, and Model Building
12.2 Linear Regression
Putting it all together:
The table below gives the average monthly temperature and corresponding monthly precipitation totals for one year in Key West, Florida.
Temperatures (in °F) and Precipitation (in inches) in Key West, FL
Average Temperature 75 76 79 82 85 88 89 90 81 77
Inches in Precipitation
2.22
1.51
1.86
2.06
3.48
4.57
3.27
5.40
5.45
4.34
2.64
2.14
Create a scatter plot.
Calculate the correlation coefficient, r.
Verify that the correlation coefficient is statistically significant at the 0.05 level of significance.
Calculate the equation of the line of best fit.
Calculate and interpret the coefficient of determination, r 2.
If appropriate, make a prediction for the monthly precipitation for a month in which the average temperature is 80 degrees.
If appropriate, make a prediction for the monthly precipitation in Destin, FL for a month in which the average temperature is 83 degrees.

12. The scatter plot shows a positive trend in data.
HAWKES LEARNING SYSTEMS
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Regression, Inference, and Model Building
12.2 Linear Regression
Solution (continued):
Create a scatter plot.
The scatter plot shows a positive trend in data.

13. RUN REGRESSION ANALYSIS
Copy Data from Hawkes to Excel and Paste in cell E1.
Select Data E1:Q2, Copy and Paste Special, (Select option Transpose), in cell A1.
RUN REGRESSION ANALYSIS

15. Calculate the correlation coefficient, r.
HAWKES LEARNING SYSTEMS
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Regression, Inference, and Model Building
12.2 Linear Regression
Solution (continued):
Calculate the correlation coefficient, r.
n = 12, ∑x = 995, ∑y = 38.94, ∑xy = , ∑x2 = 82815, ∑y2 =
 0.859

17. Statistically significant if |r | > r .
HAWKES LEARNING SYSTEMS
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Regression, Inference, and Model Building
12.2 Linear Regression
Solution (continued):
Verify that the correlation coefficient is statistically significant at the 0.05 level of significance.
n = 12, r  0.859, a = 0.05
r =
Statistically significant if |r | > r .
Therefore, r is statistically significant at the 0.05 level.
0.576

20. Calculate the equation of the line of best fit.
HAWKES LEARNING SYSTEMS
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Regression, Inference, and Model Building
12.2 Linear Regression
Solution (continued):
Calculate the equation of the line of best fit.
n = 12, ∑x = 995, ∑y = 38.94, ∑xy = , ∑x2 = 82815
Now place the slope and y-intercept into the regression line equation:
0.225
–15.424

22. Calculate and interpret the coefficient of determination, r 2.
HAWKES LEARNING SYSTEMS
math courseware specialists
Regression, Inference, and Model Building
12.2 Linear Regression
Solution (continued):
Calculate and interpret the coefficient of determination, r 2.
r  0.859
r 2 
So about 73.8% of the variation in precipitation can be attributed to the linear relationship between temperature and precipitation. The remaining 26.2% of the variation is from unknown sources.
0.738

24. HAWKES LEARNING SYSTEMS
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Regression, Inference, and Model Building
12.2 Linear Regression
Solution (continued):
If appropriate, make a prediction for the monthly precipitation for a month in which the average temperature is 80 degrees.
2.588
Thus a reasonable estimate for the precipitation for a month in which the average temperature is 80 degrees is about 2.59 inches.
If appropriate, make a prediction for the monthly precipitation in Destin, FL for a month in which the average temperature is 83 degrees.
The data was collected in Key West, not Destin, FL. Therefore, it is not appropriate to use the linear regression line to make predictions regarding the precipitation in Destin.

27. Copy Data from Hawkes to Excel and Paste in cell E1.
Select Data E1:J2, Copy and Paste Special, (Select option Transpose), in cell A1.

30. -0.806

31. Ho: Correlation of the population is 0 (zero)
HA: Correlation of the population is Not 0 (zero)
If Reject => All points predicated by the Model fall on the same line

37. Copy Data from Hawkes to Excel and Paste in cell E1.
Select Data E1:L2, Copy and Paste Special, (Select option Transpose), in cell A1.

38.
y= x
-0.711

39. ---------------- -------------- y= -0.711 + 0.13027x

40.
y= x