chapter correlation and regression 1 of 84 9 © 2012 pearson education, inc. all rights reserved

ChapterCorrelation and Regression

1 of 84

9

© 2012 Pearson Education, Inc.All rights reserved.

Chapter Outline

• 9.1 Correlation

• 9.2 Linear Regression

• 9.3 Measures of Regression and Prediction Intervals

• 9.4 Multiple Regression

© 2012 Pearson Education, Inc. All rights reserved. 2 of 84

Section 9.1

Correlation


Section 9.1 Objectives

• Introduce linear correlation, independent and dependent variables, and the types of correlation

• Find a correlation coefficient

• Test a population correlation coefficient ρ using a table

• Perform a hypothesis test for a population correlation coefficient ρ

• Distinguish between correlation and causation


Correlation

Correlation

• A relationship between two variables.

• The data can be represented by ordered pairs (x, y) x is the independent (or explanatory) variable y is the dependent (or response) variable


Correlation

x 1 2 3 4 5

y – 4 – 2 – 1 0 2

A scatter plot can be used to determine whether a linear (straight line) correlation exists between two variables.

x

2 4

–2

– 4

y

2

6

Example:


Types of Correlation

x

y

Negative Linear Correlation

x

y

No Correlation

x

y

Positive Linear Correlation

x

y

Nonlinear Correlation

As x increases, y tends to decrease.

As x increases, y tends to increase.


Example: Constructing a Scatter PlotAn economist wants to determine whether there is a linear relationship between a country’s gross domestic product (GDP)

and carbon dioxide (CO2) emissions. The data are shown in the table. Display the data in a scatter plot and determine whether there appears to be a positive or negative linear correlation or no linear correlation. (Source: World Bank and U.S. Energy Information Administration)

GDP(trillions of $),

x

CO2 emission (millions of metric tons),

y1.6 428.23.6 828.84.9 1214.21.1 444.60.9 264.02.9 415.32.7 571.82.3 454.91.6 358.71.5 573.5


Solution: Constructing a Scatter Plot

Appears to be a positive linear correlation. As the gross domestic products increase, the carbon dioxide emissions tend to increase.


Example: Constructing a Scatter Plot Using Technology

Old Faithful, located in Yellowstone National Park, is the world’s most famous geyser. The duration (in minutes) of several of Old Faithful’s eruptions and the times (in minutes) until the next eruption are shown in the table. Using a TI-83/84, display the data in a scatter plot. Determine the type of correlation.

Durationx

Time,y

Durationx

Time,y

1.80 56 3.78 79

1.82 58 3.83 85

1.90 62 3.88 80

1.93 56 4.10 89

1.98 57 4.27 90

2.05 57 4.30 89

2.13 60 4.43 89

2.30 57 4.47 86

2.37 61 4.53 89

2.82 73 4.55 86

3.13 76 4.60 92

3.27 77 4.63 91

3.65 77


Solution: Constructing a Scatter Plot Using Technology

• Enter the x-values into list L1 and the y-values into list L2.

• Use Stat Plot to construct the scatter plot.STAT > Edit… STATPLOT

From the scatter plot, it appears that the variables have a positive linear correlation.

1 550

100


Correlation Coefficient

Correlation coefficient

• A measure of the strength and the direction of a linear relationship between two variables.

• The symbol r represents the sample correlation coefficient.

• A formula for r is

• The population correlation coefficient is represented by ρ (rho). 2 22 2

n xy x yr

n x x n y y

n is the number of data pairs


Correlation Coefficient

• The range of the correlation coefficient is –1 to 1.

-1 0 1

If r = –1 there is a perfect negative

correlation

If r = 1 there is a perfect positive

correlation

If r is close to 0 there is no linear

correlation


Linear Correlation

Strong negative correlation

Weak positive correlation

Strong positive correlation

Nonlinear Correlation

x

y

x

y

x

y

x

y

r = –0.91 r = 0.88

r = 0.42 r = 0.07


Calculating a Correlation Coefficient

1. Find the sum of the x-values.

2. Find the sum of the y-values.

3. Multiply each x-value by its corresponding y-value and find the sum.

x

y

xy

In Words In Symbols


Calculating a Correlation Coefficient

4. Square each x-value and find the sum.

5. Square each y-value and find the sum.

6. Use these five sums to calculate the correlation coefficient.

2x

2y

2 22 2

n xy x yr

n x x n y y

In Words In Symbols


Example: Finding the Correlation Coefficient

Calculate the correlation coefficient for the gross domestic products and carbon dioxide emissions data. What can you conclude?

GDP(trillions of

$), x

CO2 emission

(millions of metric tons), y

1.6 428.23.6 828.84.9 1214.21.1 444.60.9 264.02.9 415.32.7 571.82.3 454.91.6 358.71.5 573.5


Solution: Finding the Correlation Coefficient

x y xy x2 y2

1.6 428.2 685.12 2.56 183,355.243.6 828.8 2983.68 12.96 686,909.444.9 1214.2 5949.58 24.01 1,474,281.641.1 444.6 489.06 1.21 197,669.160.9 264.0 237.6 0.81 69,6962.9 415.3 1204.37 8.41 172,474.092.7 571.8 1543.86 7.29 326,955.242.3 454.9 1046.27 5.29 206,934.011.6 358.7 573.92 2.56 128,665.691.5 573.5 860.25 2.25 328,902.25

Σx = 23.1 Σy = 5554 Σxy = 15,573.71 Σx2 = 67.35 Σy2 = 3,775,842.76© 2012 Pearson Education, Inc. All rights reserved. 18 of 84

Solution: Finding the Correlation Coefficient

2 22 2

n xy x yr

n x x n y y

2 2

10(15,573.71) 23.1 5554

10(67.35) 23.1 10(3,775,842.76) 5554

27,439.7 0.882139.89 6,911,511.6

Σx = 23.1 Σy = 5554 Σxy = 15,573.71 Σx2 = 32.44

Σy2 = 3,775,842.76

r ≈ 0.882 suggests a strong positive linear correlation. As the gross domestic product increases, the carbon dioxide emissions also increase.


Example: Using Technology to Find a Correlation Coefficient

Use a technology tool to calculate the correlation coefficient for the Old Faithful data. What can you conclude?

Durationx

Time,y

Durationx

Time,y

1.8 56 3.78 79

1.82 58 3.83 85

1.9 62 3.88 80

1.93 56 4.1 89

1.98 57 4.27 90

2.05 57 4.3 89

2.13 60 4.43 89

2.3 57 4.47 86

2.37 61 4.53 89

2.82 73 4.55 86

3.13 76 4.6 92

3.27 77 4.63 91

3.65 77


Solution: Using Technology to Find a Correlation Coefficient

STAT > CalcTo calculate r, you must first enter the DiagnosticOn command found in the Catalog menu

r ≈ 0.979 suggests a strong positive correlation.


Using a Table to Test a Population Correlation Coefficient ρ

• Once the sample correlation coefficient r has been calculated, we need to determine whether there is enough evidence to decide that the population correlation coefficient ρ is significant at a specified level of significance.

• Use Table 11 in Appendix B.

• If |r| is greater than the critical value, there is enough evidence to decide that the correlation coefficient ρ is significant.



• Determine whether ρ is significant for five pairs of data (n = 5) at a level of significance of α = 0.01.

• If |r| > 0.959, the correlation is significant. Otherwise, there is not enough evidence to conclude that the correlation is significant.

Number of pairs of data in sample

level of significance



1. Determine the number of pairs of data in the sample.

2. Specify the level of significance.

3. Find the critical value.

Determine n.

Identify α.

Use Table 11 in Appendix B.

In Words In Symbols



In Words In Symbols

4. Decide if the correlation is significant.

5. Interpret the decision in the context of the original claim.

If |r| > critical value, the correlation is significant. Otherwise, there is not enough evidence to support that the correlation is significant.


Example: Using a Table to Test a Population Correlation Coefficient ρ

Using the Old Faithful data, you used 25 pairs of data to find r ≈ 0.979. Is the correlation coefficient significant? Use α = 0.05.

Durationx

Time,y

Durationx

Time,y

1.8 56 3.78 79

1.82 58 3.83 85

1.9 62 3.88 80

1.93 56 4.1 89

1.98 57 4.27 90

2.05 57 4.3 89

2.13 60 4.43 89

2.3 57 4.47 86

2.37 61 4.53 89

2.82 73 4.55 86

3.13 76 4.6 92

3.27 77 4.63 91

3.65 77


Solution: Using a Table to Test a Population Correlation Coefficient ρ

• n = 25, α = 0.05

• |r| ≈ 0.979 > 0.396

• There is enough evidence at the 5% level of significance to conclude that there is a significant linear correlation between the duration of Old Faithful’s eruptions and the time between eruptions.


Hypothesis Testing for a Population Correlation Coefficient ρ

• A hypothesis test can also be used to determine whether the sample correlation coefficient r provides enough evidence to conclude that the population correlation coefficient ρ is significant at a specified level of significance.

• A hypothesis test can be one-tailed or two-tailed.


Hypothesis Testing for a Population Correlation Coefficient ρ

• Left-tailed test

• Right-tailed test

• Two-tailed test

H0: ρ ≥ 0 (no significant negative correlation)

Ha: ρ < 0 (significant negative correlation)

H0: ρ ≤ 0 (no significant positive correlation)

Ha: ρ > 0 (significant positive correlation)

H0: ρ = 0 (no significant correlation)

Ha: ρ ≠ 0 (significant correlation)


The t-Test for the Correlation Coefficient

• Can be used to test whether the correlation between two variables is significant.

• The test statistic is r.

• The standardized test statistic

follows a t-distribution with d.f. = n – 2.

• In this text, only two-tailed hypothesis tests for ρ are considered.

212

r

r rtr

n


Using the t-Test for ρ

1. State the null and alternative hypothesis.

2. Specify the level of significance.

3. Identify the degrees of freedom.

4. Determine the critical value(s) and rejection region(s).

State H0 and Ha.

Identify α.

d.f. = n – 2


In Words In Symbols


Using the t-Test for ρ

5. Find the standardized test statistic.

6. Make a decision to reject or fail to reject the null hypothesis.

7. Interpret the decision in the context of the original claim.

In Words In Symbols

If t is in the rejection region, reject H0. Otherwise fail to reject H0.

212

rtr

n


Example: t-Test for a Correlation Coefficient

Previously you calculated r ≈ 0.882. Test the significance of this correlation coefficient. Use α = 0.05.

GDP(trillions of

$), x

CO2 emission

(millions of metric tons), y

1.6 428.23.6 828.84.9 1214.21.1 444.60.9 264.02.9 415.32.7 571.82.3 454.91.6 358.71.5 573.5


Solution: t-Test for a Correlation Coefficient

• H0:

• Ha:

• α

• d.f. =

• Rejection Region:

• Test Statistic:

0.0510 – 2 = 8

2

0.8825.294

1 (0.882)10 2

t

ρ = 0ρ ≠ 0

• Decision:At the 5% level of significance, there is enough evidence to conclude that there is a significant linear correlation between gross domestic products and carbon dioxide emissions.

Reject H0


Correlation and Causation

• The fact that two variables are strongly correlated does not in itself imply a cause-and-effect relationship between the variables.

• If there is a significant correlation between two variables, you should consider the following possibilities.1. Is there a direct cause-and-effect relationship

between the variables?• Does x cause y?


Correlation and Causation

2. Is there a reverse cause-and-effect relationship between the variables?• Does y cause x?

3. Is it possible that the relationship between the variables can be caused by a third variable or by a combination of several other variables?

4. Is it possible that the relationship between two variables may be a coincidence?


Section 9.1 Summary

• Introduced linear correlation, independent and dependent variables and the types of correlation

• Found a correlation coefficient

• Tested a population correlation coefficient ρ using a table

• Performed a hypothesis test for a population correlation coefficient ρ

• Distinguished between correlation and causation


Section 9.2

Linear Regression



• Find the equation of a regression line

• Predict y-values using a regression equation


Regression lines

• After verifying that the linear correlation between two variables is significant, next we determine the equation of the line that best models the data (regression line).

• Can be used to predict the value of y for a given value of x.

x

y


Residuals

Residual

• The difference between the observed y-value and the predicted y-value for a given x-value on the line.

For a given x-value,

di = (observed y-value) – (predicted y-value)

x

y

}d1

}d2

d3{d4{ }d5

d6{

Predicted y-value

Observed y-value


Regression line (line of best fit)

• The line for which the sum of the squares of the residuals is a minimum.

• The equation of a regression line for an independent variable x and a dependent variable y is

ŷ = mx + b

Regression Line

Predicted y-value for a given x-value

Slopey-intercept


The Equation of a Regression Line

• ŷ = mx + b where

• is the mean of the y-values in the data

• is the mean of the x-values in the data

• The regression line always passes through the point

22

n xy x ym

n x x

y xb y mx mn n

y

x

,x y


Example: Finding the Equation of a Regression Line

Find the equation of the regression line for the gross domestic products and carbon dioxide emissions data.

GDP(trillions of $),

x

CO2 emission (millions of

metric tons), y1.6 428.23.6 828.84.9 1214.21.1 444.60.9 264.02.9 415.32.7 571.82.3 454.91.6 358.71.5 573.5


Solution: Finding the Equation of a Regression Line

x y xy x2 y2

1.6 428.2 685.12 2.56 183,355.243.6 828.8 2983.68 12.96 686,909.444.9 1214.2 5949.58 24.01 1,474,281.641.1 444.6 489.06 1.21 197,669.160.9 264.0 237.6 0.81 69,6962.9 415.3 1204.37 8.41 172,474.092.7 571.8 1543.86 7.29 326,955.242.3 454.9 1046.27 5.29 206,934.011.6 358.7 573.92 2.56 128,665.691.5 573.5 860.25 2.25 328,902.25

Σx = 23.1 Σy = 5554 Σxy = 15,573.71 Σx2 = 67.35Σy2 =

3,775,842.76

Recall from section 9.1:



Σx = 23.1 Σy = 5554 Σxy = 15,573.71 Σx2 = 67.35 Σy2 = 3,775,842.76

22

n xy x ym

n x x

b y mx

210(15,573.71) (23.1)(5554)

10(67.35) 23.1

27,439.7 196.151977139.89

5554 23.1(196.151977)10 10

555.4 (196.151977)(2.31) 102.2889

ˆ 196.152 102.289y x Equation of the regression line



• To sketch the regression line, use any two x-values within the range of the data and calculate the corresponding y-values from the regression line.


Example: Using Technology to Find a Regression Equation

Use a technology tool to find the equation of the regression line for the Old Faithful data.

Durationx

Time,y

Durationx

Time,y

1.8 56 3.78 79

1.82 58 3.83 85

1.9 62 3.88 80

1.93 56 4.1 89

1.98 57 4.27 90

2.05 57 4.3 89

2.13 60 4.43 89

2.3 57 4.47 86

2.37 61 4.53 89

2.82 73 4.55 86

3.13 76 4.6 92

3.27 77 4.63 91

3.65 77


Solution: Using Technology to Find a Regression Equation

550

100

1© 2012 Pearson Education, Inc. All rights reserved. 49 of 84

ˆ 12.481 33.683y x

Example: Predicting y-Values Using Regression Equations

The regression equation for the gross domestic products (in trillions of dollars) and carbon dioxide emissions (in millions of metric tons) data is ŷ = 196.152x + 102.289. Use this equation to predict the expected carbon dioxide emissions for the following gross domestic products. (Recall from section 9.1 that x and y have a significant linear correlation.)

1. 1.2 trillion dollars




Solution: Predicting y-Values Using Regression Equations

ŷ = 196.152x + 102.289


When the gross domestic product is $1.2 trillion, the CO2 emissions are about 337.671 million metric tons.

ŷ =196.152(1.2) + 102.289 ≈ 337.671


When the gross domestic product is $2.0 trillion, the CO2 emissions are 494.595 million metric tons.

ŷ =196.152(2.0) + 102.289 = 494.593


Solution: Predicting y-Values Using Regression Equations


When the gross domestic product is $2.5 trillion, the CO2 emissions are 592.669 million metric tons.

ŷ =196.152(2.5) + 102.289 = 592.669

Prediction values are meaningful only for x-values in (or close to) the range of the data. The x-values in the original data set range from 0.9 to 4.9. So, it wouldnot be appropriate to use the regression line to predictcarbon dioxide emissions for gross domestic products such as $0.2 or $14.5 trillion dollars.


Section 9.2 Summary

• Found the equation of a regression line

• Predicted y-values using a regression equation


Section 9.3

Measures of Regression and Prediction Intervals



• Interpret the three types of variation about a regression line

• Find and interpret the coefficient of determination

• Find and interpret the standard error of the estimate for a regression line

• Construct and interpret a prediction interval for y


Variation About a Regression Line

• Three types of variation about a regression line Total variation Explained variation Unexplained variation

• To find the total variation, you must first calculate The total deviation The explained deviation The unexplained deviation



iy yîy y

î iy y

(xi, ŷi)

x

y (xi, yi)

(xi, yi)

Unexplained deviation

î iy yTotal

deviationiy y Explained

deviationîy y

y

x

Total Deviation =

Explained Deviation =

Unexplained Deviation =


Total variation

• The sum of the squares of the differences between the y-value of each ordered pair and the mean of y.

Explained variation

• The sum of the squares of the differences between each predicted y-value and the mean of y.


2iy y Total variation =

Explained variation = 2ˆiy y


Unexplained variation

• The sum of the squares of the differences between the y-value of each ordered pair and each corresponding predicted y-value.


2ˆi iy y Unexplained variation =

The sum of the explained and unexplained variation is equal to the total variation.

Total variation = Explained variation + Unexplained variation


Coefficient of Determination

Coefficient of determination

• The ratio of the explained variation to the total variation.

• Denoted by r2

2 Explained variationTotal variation

r


Example: Coefficient of Determination

22 (0.882)

0.778

r

About 77.8% of the variation in the carbon emissions can be explained by the variation in the gross domestic products. About 22.2% of the variation is unexplained.

The correlation coefficient for the gross domestic products and carbon dioxide emissions data as calculated in Section 9.1 is r ≈ 0.882. Find the coefficient of determination. What does this tell you about the explained variation of the data about the regression line? About the unexplained variation?Solution:


The Standard Error of Estimate

Standard error of estimate

• The standard deviation of the observed yi -values about the predicted ŷ-value for a given xi -value.

• Denoted by se.

• The closer the observed y-values are to the predicted y-values, the smaller the standard error of estimate will be.

2( )ˆ2

i ie

y ys

n

n is the number of ordered pairs in the data set


The Standard Error of Estimate

2

, , , ( ), ˆ ˆ( )ˆ

i i i i i

i i

x y y y yy y

1. Make a table that includes the column headings shown.

2. Use the regression equation to calculate the predicted y-values.

3. Calculate the sum of the squares of the differences between each observed y-value and the corresponding predicted y-value.

4. Find the standard error of estimate.

ˆi iy mx b

2 ( )ˆi iy y

2( )ˆ2

i ie

y ys

n

In Words In Symbols


Example: Standard Error of Estimate

The regression equation for the gross domestic products and carbon dioxide emissions data as calculated in section 9.2 is

ŷ = 196.152x + 102.289

Find the standard error of estimate.

Solution:Use a table to calculate the sum of the squared differences of each observed y-value and the corresponding predicted y-value.


Solution: Standard Error of Estimatex y ŷ i yi – ŷ i (yi – ŷ i)2

1.6 428.2 416.1322 12.0678 145.63179684

3.6 828.8 808.4362 20.3638 414.68435044

4.9 1214.2 1063.4338 150.7662 22,730.44706244

1.1 444.6 318.0562 126.5438 16,013.33331844

0.9 264.0 278.8258 –14.8258 219.80434564

2.9 415.3 671.1298 –255.8298 65,448.88656804

2.7 571.8 631.8994 –60.0994 3611.93788036

2.3 454.9 553.4386 –98.5386 9709.85568996

1.6 358.7 416.1322 –57.4322 3298.45759684

1.5 573.5 396.517 176.983 31,322.982289

Σ = 152,916.020898

unexplained variation© 2012 Pearson Education, Inc. All rights reserved. 65 of 84

Solution: Standard Error of Estimate

• n = 10, Σ(yi – ŷ i)2 = 152,916.020898

2( )ˆ2

i ie

y ys

n

The standard error of estimate of the carbon dioxide emissions for a specific gross domestic product is about 138.255 million metric tons.

152,916.020898 138.25510 2


Prediction Intervals• Two variables have a bivariate normal distribution

if for any fixed value of x, the corresponding values of y are normally distributed and for any fixed values of y, the corresponding x-values are normally distributed.


Prediction Intervals

• A prediction interval can be constructed for the true value of y.

• Given a linear regression equation ŷ = mx + b and x0, a specific value of x, a c-prediction interval for y is

ŷ – E < y < ŷ + E where

• The point estimate is ŷ and the margin of error is E. The probability that the prediction interval contains y is c.

202 2

( )11( )c e

n x xE t s

n n x x


Constructing a Prediction Interval for y for a Specific Value of x

1. Identify the number of ordered pairs in the data set n and the degrees of freedom.

2. Use the regression equation and the given x-value to find the point estimate ŷ.

3. Find the critical value tc that corresponds to the given level of confidence c.

ˆi iy mx b


In Words In Symbols

d.f. = n – 2


Constructing a Prediction Interval for y for a Specific Value of x

4. Find the standard error of estimate se.

4. Find the margin of error E.

5. Find the left and right endpoints and form the prediction interval.

In Words In Symbols2( )ˆ

2i i

ey y

sn

202 2

( )11( )c e

n x xE t s

n n x x

Left endpoint: ŷ – E Right endpoint: ŷ + E Interval: ŷ – E < y < ŷ + E


Example: Constructing a Prediction Interval

Construct a 95% prediction interval for the carbon dioxide emission when the gross domestic product is $3.5 trillion. What can you conclude?

Recall, n = 10, ŷ = 196.152x + 102.289, se = 138.255

Solution:Point estimate: ŷ = 196.152(3.5) + 102.289 ≈ 788.821

Critical value:d.f. = n –2 = 10 – 2 = 8 tc = 2.306

223.1, 67.35, 2.31x x x


Solution: Constructing a Prediction Interval

0

2

2

2

2 2

1 10(3.5 2.31)(2.306)(138.255)

(

1 349.42410 10(67.35) (

)11

2 .1

( )

3 )

c en x x

E t sn n x x

Left Endpoint: ŷ – E Right Endpoint: ŷ + E

439.397 < y < 1138.245

788.821 – 349.424= 439.397

788.821 + 349.424= 1138.245

You can be 95% confident that when the gross domestic product is $3.5 trillion, the carbon dioxide emissions will be between 439.397 and 1138.245 million metric tons.


Section 9.3 Summary

• Interpreted the three types of variation about a regression line

• Found and interpreted the coefficient of determination

• Found and interpreted the standard error of the estimate for a regression line

• Constructed and interpreted a prediction interval for y


Section 9.4

Multiple Regression



• Use technology to find a multiple regression equation, the standard error of estimate and the coefficient of determination

• Use a multiple regression equation to predict y-values


Multiple Regression Equation

• In many instances, a better prediction can be found for a dependent (response) variable by using more than one independent (explanatory) variable.

• For example, a more accurate prediction for the carbon dioxide emissions discussed in previous sections might be made by considering the number of cars as well as the gross domestic product.


Multiple Regression Equation

Multiple regression equation • ŷ = b + m1x1 + m2x2 + m3x3 + … + mkxk

• x1, x2, x3,…, xk are independent variables

• b is the y-intercept

• y is the dependent variable

* Because the mathematics associated with this concept is complicated, technology is generally used to calculate the multiple regression equation.


Example: Finding a Multiple Regression Equation

A researcher wants to determine how employee salaries at a certain company are related to the length of employment, previous experience, and education. The researcher selects eight employees from the company and obtains the data shown on the next slide. Use MINITAB to find a multiple regression equation that models the data.


Example: Finding a Multiple Regression Equation

Employee Salary, yEmployment

(yrs), x1

Experience (yrs), x2

Education (yrs), x3

A 57,310 10 2 16B 57,380 5 6 16C 54,135 3 1 12D 56,985 6 5 14E 58,715 8 8 16F 60,620 20 0 12G 59,200 8 4 18H 60,320 14 6 17


Solution: Finding a Multiple Regression Equation

• Enter the y-values in C1 and the x1-, x2-, and x3-values in C2, C3 and C4 respectively.

• Select “Regression > Regression…” from the Stat menu.

• Use the salaries as the response variable and the remaining data as the predictors.


Solution: Finding a Multiple Regression Equation

The regression equation is ŷ = 49,764 + 364x1 + 228x2 + 267x3


Predicting y-Values

• After finding the equation of the multiple regression line, you can use the equation to predict y-values over the range of the data.

• To predict y-values, substitute the given value for each independent variable into the equation, then calculate ŷ.


Example: Predicting y-Values

Use the regression equation ŷ = 49,764 + 364x1 + 228x2 + 267x3

to predict an employee’s salary given 12 years of current employment, 5 years of experience, and 16 years of education.

Solution:ŷ = 49,764 + 364(12) + 228(5) + 267(16) = 59,544

The employee’s predicted salary is $59,544.


Section 9.4 Summary

• Used technology to find a multiple regression equation, the standard error of estimate and the coefficient of determination

• Used a multiple regression equation to predict y-values


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