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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
All textbooksLarson 8th EditionCh. 9 - Correlation and RegressionProblem 9.T.7a
Chapter 9, Problem 9.T.7a

"The U.S. Food and Drug Administration (FDA) requires nutrition labeling for most foods. Un FDA regulations, manufacturers are required to list the amounts of certain nutrients in their foods, such as calories, sugar, fat, and carbohydrates. This nutritional information is displayed in the ""Nutrition Facts"" panel on the food's package.
The table shows the nutritional content below for one cup of each of 21 different breakfast
cereals.
C = calories
S = sugar in grams
F = fat in grams
R = carbohydrates in grams

7. Use the equations from Exercise 6 to predict the calories in 1 cup of cereal that has 7 grams of sugar, 0.5 gram of fat, and 31 grams of carbohydrates."

Verified step by step guidance
1
Identify the regression equation from Exercise 6, which predicts calories (C) based on sugar (S), fat (F), and carbohydrates (R). The general form of the multiple linear regression equation is: C = b_0 + b_1S + b_2F + b_3R, where b_0 is the intercept and b_1, b_2, b_3 are the coefficients for sugar, fat, and carbohydrates respectively.
Substitute the given values for sugar, fat, and carbohydrates into the regression equation: S = 7, F = 0.5, R = 31.
Calculate the predicted calories by multiplying each coefficient by its corresponding variable and adding the intercept: C = b_0 + b_1 imes 7 + b_2 imes 0.5 + b_3 imes 31.
Perform the arithmetic operations step-by-step: multiply each coefficient by the given nutrient amount, then sum all these products with the intercept.
The final sum from step 4 will give the predicted number of calories in 1 cup of cereal with 7 grams of sugar, 0.5 gram of fat, and 31 grams of carbohydrates.

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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 is a statistical method used to model the relationship between one dependent variable and two or more independent variables. It estimates coefficients that quantify the effect of each predictor on the response, allowing prediction of the dependent variable based on given values of the predictors.
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Intro to Least Squares Regression

Interpretation of Regression Coefficients

Regression coefficients represent the expected change in the dependent variable for a one-unit increase in an independent variable, holding other variables constant. Understanding these coefficients is essential for making predictions and interpreting the influence of each nutrient (sugar, fat, carbohydrates) on calories.
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Coefficient of Determination

Prediction Using Regression Equations

Once the regression equation is established, it can be used to predict the dependent variable's value for new observations by substituting the given independent variable values. This process is crucial for estimating calories in cereals with specified amounts of sugar, fat, and carbohydrates.
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Related Practice
Textbook Question

"In Exercises 13-16, 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?

13. r =- 0.450"

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Textbook Question

"In Exercises 13-16, 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?

15. r = 0.642"

140
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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."

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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)

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

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Textbook Question

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

21. Construct a 95% prediction interval for the number of hours of sleep for an adult in Exercise 11 who is 45 years old."

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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"

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