Questions: The following table shows the weights, in hundreds of pounds, for six selected cars. Also shown is the corresponding fuel efficiency, in miles per gallon (mpg), for the car in city driving. Use the data provided in the table to complete parts a) through d). Weight (hundreds of pounds) 29 31 36 35 29 29 Fuel efficiency (mpg) 22 22 20 22 24 21 a) Determine the correlation coefficient between the weight of a car and the fuel efficiency. The correlation coefficient between the weight of a car and the fuel efficiency is r=-0.539 b) Determine whether a correlation exists at α=0.01. Choose the correct answer below. A. A linear correlation exists. B. A linear correlation does not exist.

Solution Steps

The covariance between the weight of a car \( X \) and the fuel efficiency \( Y \) is calculated as follows:

\[ \text{Cov}(X,Y) = -2.3 \]

The standard deviation of the weights \( X \) is given by:

\[ \sigma_X = 3.209 \]

The standard deviation of the fuel efficiency \( Y \) is given by:

\[ \sigma_Y = 1.329 \]

The correlation coefficient \( r \) is calculated using the formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

Substituting the values:

\[ r = \frac{-2.3}{3.209 \times 1.329} \approx -0.539 \]

Thus, the correlation coefficient rounded to three decimal places is:

\[ \text{Correlation coefficient (rounded to 3 decimal places)}: -0.539 \]

To determine whether a linear correlation exists at \( \alpha = 0.01 \), we compare the absolute value of the correlation coefficient with the critical value for \( n = 6 \). The critical value is approximately \( 0.834 \).

Since:

\[ |r| = 0.539 < 0.834 \]

we conclude that a linear correlation does not exist.

Final Answer