Chapter 10

Lecture 2

Section: 10.3

We analyzed paired data with the goal of determining whether thereis a linear correlation between two variables. The main objective of

this section is to describe the relationship between two variables by finding the graph and equation of the straight line that represents the relationship. This straight line is called the regression line, and its equation is called the regression equation.

The regression equation expresses a relationship between the variable x and the variable y. Just as y = mx+b.

• x is called the independent variable, predictor variable, or explanatory variable.

• y is the dependent variable, or response variable.

0 1 0 1

The typical equation of a straight line is expressed in the

ˆform of , where is the y-intercept and is the slope.

y mx b

y b b x b b

Assumptions:1. We are investigating only linear relationships.2. For each x-value, y is a random variable having a normal distribution.

All of these y distributions have the same variance. Also, for a given value of x, the distribution of y-values has a mean that lies on the regression line. Results are not seriously affected if departures from normal distributions and equal variances are not too extreme.

0 1

2

0 122

1 22

ˆFor , y b b x

y x x xyb y b x

n x x

n xy x yb

n x x

Using the Regression Equation for Predictions:Regression equations can be helpful when used for predicting the value of one variable, given some particular value of the other variable. If the regression line fits the data quite well, then it makes sense to use its equation for predictions, provided that we don’t go beyond the scope of the available values. However, we should use the equation of the regression line only if r indicates that there is a linearcorrelation. In the absence of a linear correlation, we should not use the regression equation for projecting or predicting; instead, our best estimate of the second variable is simply its sample mean. Which will be y.

Thus our guide lines in predicting a value of y based on some given value of x are:1. If there is not a linear correlation, the best predicted y-value is y .2. If there is a linear correlation, the best predicted y-value is foundby substituting the x-value into the regression equation.

1. The accompanying table lists monthly income and their food expenditures for the month of December.

Income: $5,500 $8,300 $3,800 $6,100 $3,300 $4,900 $6,700 Food Expenditure: $1,400 $2,400 $1,300 $1,600 $900 $1,500

$1,700

MINITAB OutputRegression Analysis: FoodEx versus income The regression equation isFoodEx = 151 + 0.252 income

Predictor Coef SE Coef T PConstant 150.7 217.4 0.69 0.519income 0.25246 0.03788 6.66 0.001

S = 159.508 R-Sq = 89.9% R-Sq(adj) = 87.9%

income

FoodEx

9000800070006000500040003000

2400

2200

2000

1800

1600

1400

1200

1000

1700

1500

900

1600

1300

2400

1400

Scatterplot of FoodEx vs income

2.NECK(X) 15.8 18.0 16.3 17.5 16.6 17.2 16.5

Arm Length(Y)33.536.835.235.034.735.734.8

Compute the regression equation and find the arm length if the neck size is 16.0.

2. Compute the regression equation and find the GPA of a student if their SAT score is 1000.

SAT GPA

1591 4.33

1530 3.96

1322 3.74

1169 3.12

979 2.80

825 2.70

791 2.54

766 2.35

743 2.32

633 2.07

4. The accompanying table lists weights in pounds of paper discarded by a sample of households, along with the size of the household.

Paper: 2.41 7.57 9.55 8.82 8.72 6.96 6.83 11.42HSize : 2 3 3 6 4 2 1 5What is the best predicted size of a household that discards 10lbs. of paper.

Paper

HS

ize

111098765432

6

5

4

3

2

1

S 1.40073R-Sq 39.6%R-Sq(adj) 29.6%

Fitted Line PlotHSize = 0.152 + 0.3979 Paper

Paper

HSiz

e

111098765432

6

5

4

3

2

1

Scatterplot of HSize vs Paper

Estimate the blood pressure of some one who is 40 years of age.

Age 38 41 42 45 50 52 55 60 62 65

BloodPressure

120 115 130 120 132 135 140 145 140 149

5. The following is the age and the corresponding blood pressure of 10 subjects randomly selected subjects from a large city

Age 17.2 43.5 30.7 53.1 37.2 21.0 27.6 46.3

BAC 0.19 0.20 0.26 0.16 0.24 0.20 0.18 0.23

6. A study was conducted to investigate the relationship between age (in years) and BAC (blood alcohol concentration) measured when convicted DWI (driving while intoxicated) jail inmates were first arrested. What is the best predicted BAC of a person who is 22 years of age who has been convicted and jailed for a DWI?