Questions: Consider the following statement about a correlation. State the correlation clearly (for example, there is a positive correlation between variable A and variable B). Then state whether the correlation is most likely due to coincidence, a common underlying cause, or a direct cause. Explain your answer. When gasoline prices rise, sales of sport utility vehicles decline. There is a correlation between gasoline prices and sport utility vehicle sales. Determine the possible explanation for the correlation. Choose the correct answer below. A. This correlation is due to a possible direct cause because there is no link between gas prices and sales of sport utility vehicles. B. This correlation is due to a common underlying cause because sport utility vehicles require more gasoline than a smaller type of vehicle. C. This correlation is due to coincidence because there is no link between gas prices and sales of sport utility vehicles. D. This correlation is due to a possible direct cause because sport utility vehicles require more gasoline than a smaller type of vehicle. E. This correlation is due to a common underlying cause because the price of gas increases the purchase price of only sports utility vehicles. F. There is no correlation because there is no link between gas prices and sales of sport utility vehicles.

Consider the following statement about a correlation. State the correlation clearly (for example, there is a positive correlation between variable A and variable B). Then state whether the correlation is most likely due to coincidence, a common underlying cause, or a direct cause. Explain your answer. When gasoline prices rise, sales of sport utility vehicles decline. There is a correlation between gasoline prices and sport utility vehicle sales. Determine the possible explanation for the correlation. Choose the correct answer below. A. This correlation is due to a possible direct cause because there is no link between gas prices and sales of sport utility vehicles. B. This correlation is due to a common underlying cause because sport utility vehicles require more gasoline than a smaller type of vehicle. C. This correlation is due to coincidence because there is no link between gas prices and sales of sport utility vehicles. D. This correlation is due to a possible direct cause because sport utility vehicles require more gasoline than a smaller type of vehicle. E. This correlation is due to a common underlying cause because the price of gas increases the purchase price of only sports utility vehicles. F. There is no correlation because there is no link between gas prices and sales of sport utility vehicles.
Transcript text: Consider the following statement about a correlation. State the correlation clearly (for example, there is a positive correlation between variable A and variable B). Then state whether the correlation is most likely due to coincidence, a common underlying cause, or a direct cause. Explain your answer. When gasoline prices rise, sales of sport utility vehicles decline. There is $\square$ correlation between gasoline prices and sport utility vehicle sales. Determine the possible explanation for the correlation. Choose the correct answer below. A. This correlation is due to a possible direct cause because there is no link between gas prices and sales of sport utility vehicles. B. This correlation is due to a common underlying cause because sport utility vehicles require more gasoline than a smaller type of vehicle. C. This correlation is due to coincidence because there is no link between gas prices and sales of sport utility vehicles. D. This correlation is due to a possible direct cause because sport utility vehicles require more gasoline than a smaller type of vehicle. E. This correlation is due to a common underlying cause because the price of gas increases the purchase price of only sports utility vehicles. F. There is no correlation because there is no link between gas prices and sales of sport utility vehicles.
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Solution

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Solution Steps

Step 1: Calculate Covariance and Correlation Coefficient

The covariance between gasoline prices \( X \) and sport utility vehicle sales \( Y \) is calculated as follows:

\[ \text{Cov}(X,Y) = -30.02 \]

The standard deviations of \( X \) and \( Y \) are given by:

\[ \sigma_X = 0.56 \] \[ \sigma_Y = 54.07 \]

Using these values, the correlation coefficient \( r \) is computed using the formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} = \frac{-30.02}{0.56 \times 54.07} = -1.0 \]

Step 2: Interpret the Correlation

The correlation coefficient \( r = -1.0 \) indicates a perfect negative correlation between gasoline prices and sport utility vehicle sales. This means that as gasoline prices increase, the sales of sport utility vehicles decline.

Step 3: Determine the Explanation for the Correlation

Given the negative correlation, we can conclude that this relationship is likely due to a common underlying cause. Specifically, sport utility vehicles typically require more gasoline than smaller vehicles, which makes them less appealing to consumers when gasoline prices rise. Therefore, the increase in gasoline prices leads to a decrease in the sales of sport utility vehicles.

Final Answer

The correlation is negative, and the explanation is that this correlation is due to a common underlying cause because sport utility vehicles require more gasoline than a smaller type of vehicle.

Thus, the answer is:

\(\boxed{B}\)

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