Ch. 9 - Correlation and Regression

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 9 - Correlation and Regression

Problem 9.T.6

Chapter 9, Problem 9.T.6

"[APPLET] For Exercises 2–9, use the data in the table, which shows the average annual salaries (both in thousands of dollars) for librarians and postsecondary library science teachers in the United States for 12 years. (Source: U.S. Bureau of Labor Statistics)

6. Use the regression equation that you found in Exercise 5 to predict the average annual salary of postsecondary library science teachers when the average annual salary of librarians is \$61,000."

Verified step by step guidance

Recall the regression equation from Exercise 5, which should be in the form

y = b + m x

, where

y

is the predicted salary of library science teachers,

x

is the salary of librarians,

b

is the y-intercept, and

m

is the slope of the regression line.

Identify the values of the slope

m

and intercept

b

from the regression equation you found in Exercise 5. These values represent the relationship between the salaries of librarians and library science teachers.

Substitute the given librarian salary value,

x = 61

(since the salary is \$61,000, and the data is in thousands), into the regression equation in place of

x

Calculate the predicted salary of postsecondary library science teachers by performing the arithmetic operations in the regression equation after substitution, but do not finalize the numeric result here as per instructions.

Interpret the predicted value as the estimated average annual salary (in thousands of dollars) for postsecondary library science teachers when librarians earn \$61,000 on average.

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Key Concepts

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

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one independent variable by fitting a linear equation to observed data. In this context, it helps predict the salary of library science teachers based on librarians' salaries. The regression equation typically has the form y = mx + b, where m is the slope and b is the intercept.

Prediction Using Regression Equation

Once the regression equation is established, it can be used to predict the value of the dependent variable for any given independent variable. Here, by substituting the librarian's salary ($61,000) into the regression equation, we can estimate the corresponding salary of postsecondary library science teachers. This process assumes the linear relationship holds within the data range.

Interpretation of Data and Variables

Understanding the variables and their units is crucial: 'x' represents the average annual salary of librarians (in thousands of dollars), and 'y' represents the average annual salary of library science teachers. Correct interpretation ensures accurate use of the regression model and meaningful predictions, emphasizing the importance of data context in statistical analysis.

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Introduction to Collecting Data

Related Practice

Textbook Question

"Finding the Coefficient of Determination and the Standard Error of Estimate In Exercises 11-20, use the data to (a) find the coefficient of determination r^2 and interpret the result,

12. [APPLET] Median and Mean Hourly Wages The table shows the median and mean hourly wages (in dollars) in 10 states in a recent year. The equation of the regression line is y = 1.208x + 1.495. (Source: U.S. Census Bureau)

Textbook Question

Writing Use an appropriate research source to find a real-life data set with the indicated cause-and-effect relationship. Write a paragraph describing each variable and explain why you think the variables have the indicated cause-and-effect relationship.

a. Direct Cause-and-Effect: Changes in one variable cause changes in the other variable.

Textbook Question

"[APPLET] For Exercises 2–9, use the data in the table, which shows the average annual salaries (both in thousands of dollars) for librarians and postsecondary library science teachers in the United States for 12 years. (Source: U.S. Bureau of Labor Statistics)

7. Find the coefficient of determination r^2 and interpret the result."

Textbook Question

8. Find the standard error of estimate Se and interpret the result."

Textbook Question

b. Other Factors: The relationship between the variables is caused by a third variable.

Textbook Question

"1. Net Sales The equation used to predict the net sales (in millions of dollars) for a fiscal

year for a clothing retailer is y=23,769 + 9.18x_1 - 8.41x_2

where x_1 is the number of stores open at the end of the fiscal year and x_2 is the average

square footage per store. Use the multiple regression equation to predict the y-values for

the values of the independent variables.

a. x_1 = 1057, x_2 = 3698

b. x_1 = 1012, x_2 = 3659

c. x_1 = 952, x_2 = 3601

d. x_1 = 914, x_2 = 3594"