Linear Regression

Essentials: Regression(Predictions based upon the known.)

Understand what the regression process does - prediction.

Be able to state the steps we use leading up to the decision to conduct regression.

Be able to calculate the slope of a line and the y-intercept.

Be able to calculate a regression equation and apply it to the prediction of other values. Know that these are estimates, not necessarily the actual values that might occur.

Know what the Least Squares Property and Line of Best Fit. Residual – what’s that?

1 2 3 4

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-4 -3 -2 -1 x

y

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-1

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-5

-4 -3 -2 -1 x

y

..

(-2, 2)

(-1, 4)

-x

-y

A Linear Equation in One Independent Variable

y = mx + b

y is the dependentvariable (also called the response

variable). Its value depends on thevalue of x.

m is the slope of the line.The slope indicates how much the y-value

increases (or decreases if the slope is negative)when the x-value increases by 1 unit. When m is

positive, the line will have an upward slope. Whenm is negative, the line will have a downward slope.

b is they-intercept (the point at which theline intersects the y-axis). It is the

value of y when x = 0.

x is theindependent variable

(also known as the predictorvariable.)

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y=mx+by=2x+1

RegressionDefinitions

Regression EquationGiven a collection of paired data, the regression equation

Regression Line (line of best fit or least-squares line)

The regression line is the graph of the regression equation

y = b0 + b1x^

algebraically describes the relationship between the two variables

Always Look at a Scatterplot First

You should be able to “see” a straight line being passed through the data points.

Regression Line Plotted on Scatterplot

The Regression Line is calculated to minimize the distance of the line from the observed values.

The Regression Equation

x is the independent variable (predictor variable)

y is the dependent variable (response variable)^

y = b0 +b1x ^

(recall, y = mx +b )

Where: b0 = y intercept

b1 = slope

Notation for Regression Equation

y-intercept of regression equation 0 b0

Slope of regression equation 1 b1

Equation of the regression line y = 0 + 1x y = b0 + b1x1

PopulationParameter

SampleStatistic

^

Formulas for b0 and b1

Slope:

y-intercept: xbyb 10

221 )()(

))((

xxn

yxxynb

22

2

0 )()(

))(())((

xxn

xyxxyb

NOTE: If you do not find b1 first, then b0 may be determined by:

The Regression Line

y = b0 +b1x

Fits the sample points best. Distances between this line and the sample

points are at a minimum.

^

When is it reasonable to do Regression

Start by asking the following: Does it make sense to look at the relationship between these two

variables? Does a scatter plot present a relationship (either positive or negative)? If yes to both, calculate r (the correlation).

Is the correlation statistically significant? Yes - go on to regression No – best estimate becomes the mean of the y variable

Conduct regression analysis (if yes above) Use the regression equation to calculate (estimate) a y-value given a

specific x-value.

In predicting a value of y based on some given value of x ...

1. If there is not a significant linear correlation, the best predicted y-value is y.

2. If there is a significant linear correlation, the best predicted y-value is found by substituting the

x-value into the regression equation.

Predictions

Predicting the Value of a Variable

Use the regressionequation to makepredictions. Substitutethe given value in theregression equation.

Calculate the value of rand test the hypothesis

that = 0

Isthere a

significant linearcorrelation

?

Given any value of onevariable, the best predictedvalue of the other variableis its sample mean.

Yes

No

Start

If there is no significant linear correlation, don’t use the regression equation to make predictions.

When using the regression equation for predictions, stay within the scope of the available sample data.

A regression equation based on old data is not necessarily valid now.

Don’t make predictions about a population that is different from the population from which the sample data was drawn.

Guidelines for Using The Regression Equation

Definitions

Marginal Change the amount a variable changes when the other

variable changes by exactly one unit Outlier a point lying far away from the other data

points Influential Points points which strongly affect the graph of the

regression line

Definitions Residual

For a sample of paired (x,y) data, the difference (y - y) between an observed sample y-value and the value of y-hat, which is the value of y that is predicted by using the regression equation.

Least-Squares PropertyA straight line satisfies this property if the sum of the squares of the residuals is the smallest sum possible.

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Residuals and the Least-Squares Property

Residuals and the Least-Squares Property

x 1 2 4 5y 4 24 8 32

y = 5 + 4x

02468

101214161820222426283032

1 2 3 4 5

•

•

•

x

yResidual = 7

Residual = -13Residual = -5

Residual = 11

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•

Example : Orion Cars

Orion Cars: The age and price for a sample of 11 Orions are noted below. Calculate a correlation coefficient and , if appropriate, a regression equation for the relationship. Determine the value of cars that are 4.5 years and 10 years old.

Car Age (yrs.) Price ($100’s) 1 5 85 2 4 103 3 6 70 4 5 82 5 5 89 6 5 98 7 6 66 8 6 95 9 2 169 10 7 70 11 7 48

Example : Orion Cars

Example : Orion Cars

Example : Orion Cars

Statistics

11 11

12 12

5.27 88.64

5.00 85.00

1.421 31.159

2 48

7 169

Valid

Missing

N

Mean

Median

Std. Deviation

Minimum

Maximum

Orion Car AgeOrion Car

Price

Model Summary

.924a .853 .837 12.577

Model

1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), Orion Car Agea.

Coefficientsa

195.468 15.240 12.826 .000

-20.261 2.800 -.924 -7.237 .000

(Constant)

Orion Car Age

Model

1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: Orion Car Pricea.

(Price in thousands)

Example : Orion Cars

(Price in thousands)

Example : Orion Cars

(Price in thousands)

(Price in thousands)

With influential point Without influential point