Chapter 7Regression

RegressionA regression line attempts to predict one

variable based on the relationship with another variable (its correlation).

The regression line is placed so that the error (distance from the line to each data point) is minimized.

The placement of the regression line minimizes the total squared predictive error. (That way there are no negative values.)

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PredictionWhat is predictive error?

The amount of error associated with the placement of a best fitting regression line.

The placement of the regression line: Minimizes the total predictive error, and Minimizes the total squared predictive error

3Chapter 7-Regression

Progress Check 7.1

Chapter 7-Regression 4

1 2 3 4

a) Predict the approximate rate of inflation, given an unemployment rate of 5 percent.

b) Predict the approximate rate of inflation, given an unemployment rate of 15 percent.

5 10 15 20

20

15

10

5

Estimating regression and relationshipWitte web demonstration

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Least Squares Regression EquationY = bX + a

Where Y = the predicted valueX = a known valueb = slope of the linea = Y-intercept

Ability to predict an outcome for a variable, given a regression line and a value of a second paired variable.

Chapter 7-Regression 6

Calculating Least Squares RegressionPage 159

1. Calculate SSx, SSy, and r for the data.

2. Substitute numbers into the formula below SSy

b = SSx

3. Find the mean for X and mean for Y4. Solve for a

a = Y-(b)(X)

5. Solve for the predicted valueY’ = (b)(X) + a

Chapter 7-Regression 7

(r)√

Standard error of estimateThis represents a special kind of standard

deviation that reflects the magnitude of predictive error.

It is the difference between known values and predicted values based on the regression equation.

It is how much we over/under estimate a value based on the regression equation, which is related to the strength of the correlation.

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Calculation of standard error of estimate Page 162

Square root of the quantity of sum of squares for Y times one minus r squared divided by n minus 2.

(n-2 because 2 paired variable results in n-2 degrees of freedom)

SSy(1-r2)

Sy|x= n-2

Chapter 7-Regression 9

√

AssumptionsUse of regression equation requires that the

underlying relationship be linear.

Use of the standard error of estimate assumes that except for chance, the dots in the original scatterplot will be dispersed equally about all segments of the regression line. (homoscedasticity)

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Progress Check 7.2 Page 160-1

a) Determine the least squares equation for predicting weekly reading time from educational level.

b) Faith’s education level is 15. What is her predicted reading time?

c) Keegan’s educational level is 11. What is his predicted reading time?

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Educational Level (X) Weekly Reading Time (Y)

X = 13 Y = 8

SSx = 25 SSy = 50

R = .30

Calculate Standard Error of EstimateCalculate the Standard Error of Estimate

using the data in 7.2 on page 160

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Educational Level (X) Weekly Reading Time (Y)

X = 13 Y = 8

SSx = 25 SSy = 50

R = .30

Correlation, Prediction, ErrorX Y X2 Y2 X*Y

5 4

4 6

1 4

2 3

4 5

2 4

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X - mean number of cases of influenza in a month among employeesY - mean number of bacteria x 1 million on a door knob on front door

Regression toward the meanExtreme scores on multiple trials will tend

toward the mean.

The regression fallacy: accepting that regression toward the mean is real, rather than a chance effect.

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Regression toward the meanTversky and KahnemannStudy of Israeli Air Force pilots in 1974Some trainees were praised after good

landings, while others were reprimanded after bad landings.

On their next landings, praised trainees did more poorly and reprimanded trainees did better.

Conclusion:Praise hinders but a reprimand helps

performance!Chapter 7-Regression 15