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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 9.1.19
Chapter 9, Problem 9.1.19

"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"

Verified step by step guidance
1
Understand the concept of correlation: Correlation measures the strength and direction of a linear relationship between two variables. It does not imply causation, meaning one variable does not necessarily cause the other to change.
Identify the two variables in the problem: The variables are 'value of home' and 'life span.' These are correlated, but there is no direct cause-and-effect relationship between them.
Consider external factors or confounding variables: A possible reason for the correlation could be socioeconomic status. Individuals with higher socioeconomic status may afford homes with higher values and also have access to better healthcare, nutrition, and living conditions, which can contribute to a longer life span.
Explore shared environmental influences: Another reason for the correlation might be the geographic location. Areas with higher property values may also have better infrastructure, cleaner environments, and access to resources that promote healthier lifestyles, indirectly affecting life span.
Summarize the reasoning: The correlation between 'value of home' and 'life span' is likely due to shared external factors such as socioeconomic status or environmental conditions, rather than a direct cause-and-effect relationship between the two variables.

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Key Concepts

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

Correlation vs. Causation

Correlation refers to a statistical relationship between two variables, indicating that they tend to move together in some way. However, this does not imply that one variable causes the other to change. Understanding this distinction is crucial, as many correlations can arise from confounding factors or coincidental relationships rather than direct causation.
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Confounding Variables

Confounding variables are external factors that may influence both variables being studied, leading to a spurious correlation. For example, in the case of home value and life span, factors such as socioeconomic status, access to healthcare, and neighborhood safety could affect both, creating a misleading association between the two without a direct causal link.
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Spurious Correlation

A spurious correlation occurs when two variables appear to be related but are actually influenced by a third variable or are coincidentally correlated. This concept highlights the importance of careful analysis in statistics, as it emphasizes that observed relationships may not reflect true connections, necessitating further investigation to uncover underlying factors.
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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

"Old Vehicles In Exercises 31–34, use the figure shown at the left.

Scatter Plot Construct a scatter plot of the data. Show y and x on the graph."

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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

"Old Vehicles In Exercises 31–34, use the figure shown at the left.

Regression Line Find and draw the regression line."

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"