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 9.4.5

Chapter 9, Problem 9.4.5

"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.
5. Black Cherry Tree Volume The volume (in cubic feet) of a black cherry tree can be modeled by the equation
y =- 52.2+0.3x_1 +4.5x_2
where x_1 is the tree's height (in feet) and x_2 is the tree's diameter (in inches). (Source: Journal of the Royal Statistical Society)
a. x_1 = 70, x_2 = 8.6
b. x_1 = 65, x_2 = 11.0
c. x_1 = 83, x_2 = 17.6
d. x_1 = 87, x_2 = 19.6"

Verified step by step guidance

Identify the multiple regression equation given:

y = - 52.2 + 0.3 x 1 + 4.5 x 2

, where

x 1

is the tree's height and

x 2

is the tree's diameter.

For each set of values for

x 1

and

x 2

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

x 1 = 70

and

x 2 = 8.6

Calculate the product of each coefficient and its corresponding independent variable: multiply 0.3 by

x 1

and 4.5 by

x 2

Add the results from the multiplications to the intercept (-52.2) to find the predicted volume

y

for each case.

Repeat steps 2 to 4 for parts (b), (c), and (d) using their respective values of

x 1

and

x 2

to find all predicted volumes.

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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 each predictor influences the outcome.

Interpreting Coefficients

Coefficients in a regression equation represent the expected change in the dependent variable for a one-unit increase in the corresponding independent variable, holding others constant. For example, a coefficient of 0.3 for height means each additional foot increases volume by 0.3 cubic feet, assuming diameter stays the same.

Prediction Using Regression

To predict y-values, substitute given values of independent variables into the regression equation and calculate the result. This process estimates the dependent variable based on the model, enabling practical predictions like estimating tree volume from height and diameter measurements.

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Using Regression Lines to Predict Values

Related Practice

Textbook Question

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

26. Voter Turnout Construct a 99% prediction interval for number of ballots cast in Exercise 16 when the voting age population is 210 million."

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Textbook Question

"In Exercises 19-22, two variables are given that have been shown to have correlation but no cause-and-effect relationship. Describe at least one possible reason for the correlation.

20. Alcohol use and tobacco use"

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Textbook Question

"Confidence Intervals for y-Intercept and Slope

You can construct confidence intervals for the y-intercept B and slope M of the regression line y = Mx + B for the population by using the inequalities below.

y-intercept B :

b - E < B < b + E

where

E = t_c s_e \(\sqrt{\frac{1}{n}\) + \(\frac{\overline{x}\)^2}{\(\sum\) x^2 - \(\frac{(\Sigma x)^2}{n}\)}}

slope M :

m - E < M < m + E

where

E = \(\frac{t_c s_e}{\sqrt{\sum x^2 - \frac{(\Sigma x)^2}{n}\)}}

The values of m and b are obtained from the sample data, and the critical value t_c is found using Table 5 in Appendix B with n - 2 degrees of freedom.

In Exercises 37 and 38, construct the indicated confidence intervals for B and M using the gross domestic products and carbon dioxide emissions data found in Example 2.

38. 99% confidence interval"

119

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Textbook Question

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

25. Mean Wage Construct a 99% prediction interval for the mean annual wage in Exercise 15 when the percentage of employment in STEM occupations is 13% in the industry."

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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?

170

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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."

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