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Chapter 10. Lecture 2 Section: 10.3. We analyzed paired data with the goal of determining whether there

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chapter 10

Chapter 10

Lecture 2

Section: 10.3

slide2

We analyzed paired data with the goal of determining whether there

  • is 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.
slide3

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.

slide4

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 linear

correlation.

In the absence of a linear correlation, we shouldnot usethe 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 found

by substituting the x-value into the regression equation.

slide5

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 Output

Regression Analysis:

FoodEx versus income

The regression equation is

FoodEx = 151 + 0.252 income

Predictor Coef SE Coef T P

Constant 150.7 217.4 0.69 0.519

income 0.25246 0.03788 6.66 0.001

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

slide6

2.

NECK(X)

15.8

18.0

16.3

17.5

16.6

17.2

16.5

Arm Length(Y)

33.5

36.8

35.2

35.0

34.7

35.7

34.8

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

slide8

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 11.42

HSize : 2 3 3 6 4 2 1 5

What is the best predicted size of a household that discards 10lbs. of paper.

slide9

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

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

slide10

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?