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Chapter 10: Q2BSC (page 468)

What is the difference between the regression equation\(\hat y = {b_0} + {b_1}x\)and the regression equation\(y = {\beta _0} + {\beta _1}x\)?

Short Answer

Expert verified

The equation \(\hat y = {b_0} + {b_1}x\) represents the regression equation for sample statistics where the predicted measure of y variable is obtained while the equation \(y = {\beta _0} + {\beta _1}x\) is the regression equation is based on the population parameter.

Step by step solution

01

Given information

The two regression equations are provided as,

\(\hat y = {b_0} + {b_1}x\)

\(y = {\beta _0} + {\beta _1}x\)

02

State the interpretation of symbols

The regression equation is given as,

\(\hat y = {b_0} + {b_1}x\)

Where,

\(\hat y\)is the predicted value of the dependent variable,\({b_0}\)is the y-intercept for the regression equation,\({b_1}\)is the slope of the regression equation and x is the independent variable.

For, the regression equation,

\(y = {\beta _0} + {\beta _1}x\)

Where,

y is the dependent variable,\({\beta _0}\)is thepopulation parameter for the y-intercept for the regression equation,\({\beta _1}\)is thepopulation slope parameter for the independent variable x.

03

State the difference between the two regression equations

From the provided regression equations, it can be observed that \(\hat y = {b_0} + {b_1}x\) is the regression equation based on sample statistics while \(y = {\beta _0} + {\beta _1}x\) is the regression equation for the population based on parameters.

Also, the resultant values provide thepredicted value of y \(\left( {\hat y} \right)\)and the observed value of y (y).

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Most popular questions from this chapter

Critical Thinking: Is the pain medicine Duragesic effective in reducing pain? Listed below are measures of pain intensity before and after using the drug Duragesic (fentanyl) (based on data from Janssen Pharmaceutical Products, L.P.). The data are listed in order by row, and corresponding measures are from the same subject before and after treatment. For example, the first subject had a measure of 1.2 before treatment and a measure of 0.4 after treatment. Each pair of measurements is from one subject, and the intensity of pain was measured using the standard visual analog score. A higher score corresponds to higher pain intensity.

Pain Intensity Before Duragesic Treatment

1.2

1.3

1.5

1.6

8

3.4

3.5

2.8

2.6

2.2

3

7.1

2.3

2.1

3.4

6.4

5

4.2

2.8

3.9

5.2

6.9

6.9

5

5.5

6

5.5

8.6

9.4

10

7.6










Pain Intensity After Duragesic Treatment

0.4

1.4

1.8

2.9

6

1.4

0.7

3.9

0.9

1.8

0.9

9.3

8

6.8

2.3

0.4

0.7

1.2

4.5

2

1.6

2

2

6.8

6.6

4.1

4.6

2.9

5.4

4.8

4.1










Correlation Use the given data to construct a scatterplot, then use the methods of Section 10-1 to test for a linear correlation between the pain intensity before and after treatment. If there does appear to be a linear correlation, can we conclude that the drug treatment is effective?

Testing for a Linear Correlation. In Exercises 13鈥�28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of A = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

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The display is based on Data Set 5 鈥淔amily Heights鈥� in Appendix B.

Identify the multiple regression equation that expresses the height of a son in terms of the height of his father and mother.

Interpreting the Coefficient of Determination. In Exercises 5鈥�8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Pizza and Subways r = 0.992 (x = cost of a slice of pizza, y = subway fare in New York City

let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 10-5 on page 493. Use a 0.05 significance level.

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