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 9.4.3

Chapter 9, Problem 9.4.3

"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.
3. Cauliflower Yield The equation used to predict the annual cauliflower yield (in pounds
per acre) is y=24,791+4.508x_1-4.723x_2
where x_1 is the number of acres planted and x_2 is the number of acres harvested.(Adapted from United States Department of Agriculture)
a. x_1 = 36,500, x_2 = 36,100
b. x_1 = 38,100, x_2 = 37,800
c. x_1 = 39,000, x_2 = 38,800
d. x_1 = 42,200, x_2 = 42,100"

Verified step by step guidance

Identify the multiple regression equation given:

y = 24791 + 4.508x_1 - 4.723x_2

, where

x_1

is the number of acres planted and

x_2

is the number of acres harvested.

For each set of values of

x_1

and

x_2

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

x_1 = 36500

and

x_2 = 36100

Perform the multiplication for each term involving the independent variables: multiply 4.508 by

x_1

and multiply -4.723 by

x_2

Add the constant term 24791 to the results of the multiplications to calculate the predicted value of

y

(the cauliflower yield) for each case.

Repeat steps 2 to 4 for each set of values given in parts (b), (c), and (d) to find the predicted yields for all scenarios.

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Key Concepts

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

Multiple Regression Equation

A multiple regression equation models the relationship between one dependent variable and two or more independent variables. It predicts the dependent variable (y) by combining the independent variables (x₁, x₂, etc.) multiplied by their coefficients, plus a constant term. This allows for understanding how changes in each independent variable affect the outcome.

Interpreting Coefficients in Regression

Each coefficient in a regression equation represents the expected change in the dependent variable for a one-unit increase in the corresponding independent variable, holding other variables constant. Positive coefficients indicate a direct relationship, while negative coefficients indicate an inverse relationship.

Predicting Values Using Regression

To predict y-values, substitute the given values of independent variables into the regression equation and perform the arithmetic operations. This process estimates the dependent variable based on the model, enabling practical forecasting or decision-making.

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Guided course

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Using Regression Lines to Predict Values

Related Practice

Textbook Question

1. What is a residual? Explain when a residual is positive, negative, and zero.

Textbook Question

4. For a set of data and a corresponding regression line, describe all values of x that provide meaningful predictions for y.

Textbook Question

2. Two variables have a positive linear correlation. Is the slope of the regression line for the variables positive or negative?

Textbook Question

1. Two variables have a positive linear correlation. Does the dependent variable increase or decrease as the independent variable increases? What if the variables have a negative linear correlation?

Textbook Question

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.

23. Points Earned Construct a 90% prediction interval for total points earned in Exercise 13 when the number of goals allowed by the team is 140."

Textbook Question

1. Interpret the meaning of the coefficient -8.2 in the multiple regression equation y=112.1+0.43x_1-8.2x_2+29.5x_3.