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Chapter 11

- Introduction to Linear Regression and Correlation

Analysis

Chapter 11 - Chapter Outcomes

- After studying the material in this chapter, you

should be able to - Calculate and interpret the simple correlation

between two variables. - Determine whether the correlation is significant.
- Calculate and interpret the simple linear

regression coefficients for a set of data. - Understand the basic assumptions behind

regression analysis. - Determine whether a regression model is

significant.

Chapter 11 - Chapter Outcomes(continued)

- After studying the material in this chapter, you

should be able to - Calculate and interpret confidence intervals for

the regression coefficients. - Recognize regression analysis applications for

purposes of prediction and description. - Recognize some potential problems if regression

analysis is used incorrectly. - Recognize several nonlinear relationships between

two variables.

Scatter Diagrams

- A scatter plot is a graph that may be used to

represent the relationship between two variables.

Also referred to as a scatter diagram.

Dependent and Independent Variables

- A dependent variable is the variable to be

predicted or explained in a regression model.

This variable is assumed to be functionally

related to the independent variable.

Dependent and Independent Variables

- An independent variable is the variable related

to the dependent variable in a regression

equation. The independent variable is used in a

regression model to estimate the value of the

dependent variable.

Two Variable Relationships(Figure 11-1)

Y

X

(a) Linear

Two Variable Relationships(Figure 11-1)

Y

X

(b) Linear

Two Variable Relationships(Figure 11-1)

Y

X

(c) Curvilinear

Two Variable Relationships(Figure 11-1)

Y

X

(d) Curvilinear

Two Variable Relationships(Figure 11-1)

Y

X

(e) No Relationship

Correlation

- The correlation coefficient is a quantitative

measure of the strength of the linear

relationship between two variables. The

correlation ranges from 1.0 to - 1.0. A

correlation of ? 1.0 indicates a perfect linear

relationship, whereas a correlation of 0

indicates no linear relationship.

Correlation

- SAMPLE CORRELATION COEFFICIENT
- where
- r Sample correlation coefficient
- n Sample size
- x Value of the independent variable
- y Value of the dependent variable

Correlation

- SAMPLE CORRELATION COEFFICIENT
- or the algebraic equivalent

Correlation(Example 11-1)

(Table 11-1)

Correlation(Example 11-1)

Correlation(Example 11-1)

Correlation between Years and Sales

Excel Correlation Output (Figure 11-5)

Correlation

- TEST STATISTIC FOR CORRELATION
- where
- t Number of standard deviations r is from 0
- r Simple correlation coefficient
- n Sample size

Correlation Significance Test(Example 11-1)

Rejection Region ? /2 0.025

Rejection Region ? /2 0.025

Since t4.752 gt 2.048, reject H0, there is a

significant linear relationship

Correlation

- Spurious correlation occurs when there is a

correlation between two otherwise unrelated

variables.

Simple Linear Regression Analysis

- Simple linear regression analysis analyzes the

linear relationship that exists between a

dependent variable and a single independent

variable.

Simple Linear Regression Analysis

- SIMPLE LINEAR REGRESSION MODEL (POPULATION MODEL)
- where
- y Value of the dependent variable
- x Value of the independent variable
- Populations y-intercept
- Slope of the population regression line
- Error term, or residual

Simple Linear Regression Analysis

- The simple linear regression model has four

assumptions - Individual values if the error terms, ?i, are

statistically independent of one another. - The distribution of all possible values of ? is

normal. - The distributions of possible ?i values have

equal variances for all value of x. - The means of the dependent variable, for all

specified values of the independent variable, y,

can be connected by a straight line called the

population regression model.

Simple Linear Regression Analysis

- REGRESSION COEFFICIENTS
- In the simple regression model, there are two

coefficients the intercept and the slope.

Simple Linear Regression Analysis

- The interpretation of the regression slope

coefficient is that is gives the average change

in the dependent variable for a unit increase in

the independent variable. The slope coefficient

may be positive or negative, depending on the

relationship between the two variables.

Simple Linear Regression Analysis

- The least squares criterion is used for

determining a regression line that minimizes the

sum of squared residuals.

Simple Linear Regression Analysis

- A residual is the difference between the actual

value of the dependent variable and the value

predicted by the regression model.

Simple Linear Regression Analysis

Y

390

400

Sales in Thousands

300

312

200

Residual 312 - 390 -78

100

X

4

Years with Company

Simple Linear Regression Analysis

- ESTIMATED REGRESSION MODEL
- (SAMPLE MODEL)
- where
- Estimated, or predicted, y value
- b0 Unbiased estimate of the regression

intercept - b1 Unbiased estimate of the regression slope
- x Value of the independent variable

Simple Linear Regression Analysis

- LEAST SQUARES EQUATIONS
- algebraic equivalent
- and

Simple Linear Regression Analysis

- SUM OF SQUARED ERRORS

Simple Linear Regression Analysis (Midwest

Example)

(Table 11-3)

Simple Linear Regression Analysis (Table 11-3)

The least squares regression line is

Simple Linear Regression Analysis(Figure 11-11)

Excel Midwest Distribution Results

Least Squares Regression Properties

- The sum of the residuals from the least squares

regression line is 0. - The sum of the squared residuals is a minimum.
- The simple regression line always passes through

the mean of the y variable and the mean of the x

variable. - The least squares coefficients are unbiased

estimates of ?0 and ?1.

Simple Linear Regression Analysis

- SUM OF RESIDUALS

SUM OF SQUARED RESIDUALS

Simple Linear Regression Analysis

- TOTAL SUM OF SQUARES
- where
- TSS Total sum of squares
- n Sample size
- y Values of the dependent variable
- Average value of the dependent variable

Simple Linear Regression Analysis

- SUM OF SQUARES ERROR (RESIDUALS)
- where
- SSE Sum of squares error
- n Sample size
- y Values of the dependent variable
- Estimated value for the average of y for

the given x value

Simple Linear Regression Analysis

- SUM OF SQUARES REGRESSION
- where
- SSR Sum of squares regression
- Average value of the dependent variable
- y Values of the dependent variable
- Estimated value for the average of y for

the given x value

Simple Linear Regression Analysis

- SUMS OF SQUARES

Simple Linear Regression Analysis

- The coefficient of determination is the portion

of the total variation in the dependent variable

that is explained by its relationship with the

independent variable. The coefficient of

determination is also called R-squared and is

denoted as R2.

Simple Linear Regression Analysis

- COEFFICIENT OF DETERMINATION (R2)

Simple Linear Regression Analysis(Midwest

Example)

- COEFFICIENT OF DETERMINATION (R2)

69.31 of the variation in the sales data for

this sample can be explained by the linear

relationship between sales and years of

experience.

Simple Linear Regression Analysis

- COEFFICIENT OF DETERMINATION SINGLE INDEPENDENT

VARIABLE CASE - where
- R2 Coefficient of determination
- r Simple correlation coefficient

Simple Linear Regression Analysis

- STANDARD DEVIATION OF THE REGRESSION SLOPE

COEFFICIENT (POPULATION) - where
- Standard deviation of the regression slope

(Called the standard error of the slope) - Population standard error of the estimate

Simple Linear Regression Analysis

- ESTIMATOR FOR THE STANDARD ERROR OF THE ESTIMATE
- where
- SSE Sum of squares error
- n Sample size
- k number of independent variables in the

model

Simple Linear Regression Analysis

- ESTIMATOR FOR THE STANDARD DEVIATION OF THE

REGRESSION SLOPE - where
- Estimate of the standard error of the least

squares slope - Sample standard error of the estimate

Simple Linear Regression Analysis

- TEST STATISTIC FOR TEST OF SIGNIFICANCE OF THE

REGRESSION SLOPE - where
- b1 Sample regression slope coefficient
- ?1 Hypothesized slope
- sb1 Estimator of the standard error of the

slope

Significance Test of Regression Slope(Example

11-5)

Rejection Region ? /2 0.025

Rejection Region ? /2 0.025

Since t4.753 gt 2.048, reject H0 conclude that

the true slope is not zero

Simple Linear Regression Analysis

- MEAN SQUARE REGRESSION
- where
- SSR Sum of squares regression
- k Number of independent variables in the model

Simple Linear Regression Analysis

- MEAN SQUARE ERROR
- where
- SSE Sum of squares error
- n Sample size
- k Number of independent variables in the model

Significance Test(Example 11-6)

Rejection Region ? 0.05

Since F 22.59 gt 4.96, reject H0 conclude that

the regression model explains a significant

amount of the variation in the dependent variable

Simple Regression Steps

- Develop a scatter plot of y and x. You are

looking for a linear relationship between the two

variables. - Calculate the least squares regression line for

the sample data. - Calculate the correlation coefficient and the

simple coefficient of determination, R2. - Conduct one of the significance tests.

Simple Linear Regression Analysis

- CONFIDENCE INTERVAL ESTIMATE FOR THE REGRESSION

SLOPE - or equivalently
- where
- sb1 Standard error of the regression slope

coefficient - s? Standard error of the estimate

Simple Linear Regression Analysis

- CONFIDENCE INTERVAL FOR
- where
- Point estimate of the dependent variable
- t Critical value with n - 2 d.f.
- s? Standard error of the estimate
- n Sample size
- xp Specific value of the independent variable
- Mean of independent variable observations

Simple Linear Regression Analysis

- PREDICTION INTERVAL FOR

Residual Analysis

- Before using a regression model for description

or prediction, you should do a check to see if

the assumptions concerning the normal

distribution and constant variance of the error

terms have been satisfied. One way to do this is

through the use of residual plots.

Key Terms

- Coefficient of Determination
- Correlation Coefficient
- Dependent Variable
- Independent Variable
- Least Squares Criterion
- Regression Coefficients

- Regression Slope Coefficient
- Residual
- Scatter Plot
- Simple Linear Regression Analysis
- Spurious Correlation

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