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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 9.3.32
Chapter 9, Problem 9.3.32

"Old Vehicles In Exercises 31–34, use the figure shown at the left.
Table showing the average age of vehicles on U.S. roads from 2014 to 2021, with years and corresponding ages in years.
Regression Line Find and draw the regression line."

Verified step by step guidance
1
Step 1: Organize the data into two variables: the independent variable (year, x) and the dependent variable (average age of vehicles, y). The data points are: (2014, 11.4), (2015, 11.5), (2016, 11.6), (2017, 11.7), (2018, 11.7), (2019, 11.8), (2020, 11.9), (2021, 12.1).
Step 2: Convert the years into a simpler numerical format for regression calculations. For example, let x = 0 represent 2014, x = 1 represent 2015, and so on up to x = 7 for 2021. This simplifies the data points to: (0, 11.4), (1, 11.5), (2, 11.6), (3, 11.7), (4, 11.7), (5, 11.8), (6, 11.9), (7, 12.1).
Step 3: Use the formula for the slope of the regression line, m = (Σ(x*y) - n*mean(x)*mean(y)) / (Σ(x^2) - n*mean(x)^2), where n is the number of data points, Σ represents summation, and mean(x) and mean(y) are the averages of x and y respectively.
Step 4: Calculate the y-intercept of the regression line using the formula b = mean(y) - m*mean(x), where m is the slope calculated in Step 3.
Step 5: Write the equation of the regression line in the form y = mx + b, where m is the slope and b is the y-intercept. Plot the regression line on a graph with the given data points to visualize the trend.

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Key Concepts

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

Regression Line

A regression line is a statistical tool used to model the relationship between two variables by fitting a linear equation to observed data. In this context, it helps to predict the average age of vehicles based on the year. The line is determined by minimizing the distance between the data points and the line itself, providing insights into trends over time.
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Dependent and Independent Variables

In regression analysis, the dependent variable is the outcome we are trying to predict or explain, while the independent variable is the predictor or input. Here, the average age of vehicles (y) is the dependent variable, and the year (x) is the independent variable. Understanding this distinction is crucial for interpreting the regression results correctly.
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Correlation

Correlation measures the strength and direction of a linear relationship between two variables. A positive correlation indicates that as one variable increases, the other also tends to increase. In this scenario, analyzing the correlation between the year and the average age of vehicles can help determine if older vehicles are becoming more common over time.
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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?

16. r = 0.795"

Textbook Question

4. Give examples of two variables that have perfect positive linear correlation and two variables that have perfect negative linear correlation.

Textbook Question

3. Explain how to predict y-values using the equation of a regression line.

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.

19. Value of home and life span"

Textbook Question

"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.

4. Sorghum Yield The equation used to predict the annual sorghum yield (in bushels per

acre) is y = 80.1-20.2x_1 +21.2x_2

where x_1 is the number of acres planted (in millions) and x_2 is the number of acres harvested (in millions). (Adapted from United States Department of Agriculture)

a. x_1 = 5.5, x_2 = 3.9

b. x_1 = 8.3, x_2 = 7.3

c. x_1 = 6.5, x_2 = 5.7

d. x_1 = 9.4, x_2= 7.8"