Ch. 9 - Correlation and Regression

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 9 - Correlation and Regression

Problem 9.Q.9

Chapter 9, Problem 9.Q.9

"9. Stock Price The equation used to predict the stock price (in dollars) at the end of the year for a restaurant chain is y=- 86+7.46x_1 - 1.61x_2
where x_1 is the total revenue (in billions of dollars) and x_2 is the shareholders' equity (in
billions of dollars). Use the multiple regression equation to predict the y-values for the
values of the independent variables.
a. x_1 = 27.6, x_2 = 15.3
b. x_1 = 24.1, x_2 = 14.6
c. x_1 = 23.5, x_2 = 13.4
d. x_1 = 22.8, x_2 =15.3"

Verified step by step guidance

Identify the multiple regression equation given:

y = - 86 + 7.46 - 1.61

, where

x_1

is total revenue and

x_2

is shareholders' equity.

For each set of values of

x_1

and

x_2

, substitute the numbers into the regression equation. For example, for part (a), substitute

x_1 = 27.6

and

x_2 = 15.3

Calculate the product of the coefficient and the corresponding independent variable for each term: multiply 7.46 by

x_1

and multiply -1.61 by

x_2

Add the constant term (-86) to the sum of the two products calculated in the previous step to find the predicted stock price

y

Repeat steps 2 to 4 for each set of values (b, c, and d) to find the predicted stock prices for all given independent variable values.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Multiple Linear Regression

Multiple linear regression models the relationship between one dependent variable and two or more independent variables by fitting a linear equation. It estimates how changes in each independent variable affect the dependent variable, holding others constant. This method helps predict outcomes based on several predictors simultaneously.

Interpreting Regression Coefficients

Regression coefficients represent the expected change in the dependent variable for a one-unit increase in an independent variable, assuming other variables remain constant. For example, a coefficient of 7.46 for x₁ means the stock price increases by $7.46 billion for each additional billion in revenue, all else equal.

Using the Regression Equation for Prediction

To predict the dependent variable, substitute given values of independent variables into the regression equation and calculate the result. This process provides estimated outcomes, such as predicted stock prices, based on specific input values for revenue and shareholders' equity.

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Related Practice

Textbook Question

In Exercise 26, add data for an international soccer player who can perform the half squat with a maximum of 210 kilograms and can sprint 10 meters in 2.00 seconds. Describe how this affects the correlation coefficient r.

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Textbook Question

"In Exercises 7-10, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?

10. r =0.881"

117

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Textbook Question

"[APPLET] For Exercises 1–8, use the data in the table, which shows the average annual salaries (both in thousands of dollars) for secondary and elementary school teachers, excluding special and vocational education teachers, in the United States for 11 years. (Source: U.S. Bureau of Labor Statistics)

8. Construct a 95% prediction interval for the average annual salary of elementary school teachers when the average annual salary of secondary school teachers is \$63,500. Interpret the results."

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Textbook Question

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.

24. Construct a 99% prediction interval for the price of a gas grill in Exercise 18 with a usable cooking area of 900 square inches."

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Textbook Question

"In Exercises 17 and 18, use the data to (a) find the coefficient of determination r^2 and interpret

the result, and (b) find the standard error of estimate s_e and interpret the result.

18. [APPLET] The table shows the cooking areas (in square inches) of 18 gas grills and their prices (in dollars). The regression equation is y = 1.501x - 341.501. (Source: Lowe's)

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Textbook Question

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.

23. Construct a 99% prediction interval for the top speed of an electric car in Exercise 17 that takes 5.9 seconds to accelerate from 0 to 60 miles per hour."

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