Questions: Which of the following statements is an example of a negative correlation between two variables?
A As a runner's speed increases, the time to reach the finish line decreases.
B As a company's advertising budget decreases, the number of customers decreases.
C The price of an album does not change based on the number of tracks on that album.
D As the outdoor temperature increases, there is an increase in ice cream sales.
Transcript text: Which of the following statements is an example of a negative correlation between two variables?
A As a runner's speed increases, the time to reach the finish line decreases.
B As a company's advertising budget decreases, the number of customers decreases.
C The price of an album does not change based on the number of tracks on that album.
D As the outdoor temperature increases, there is an increase in ice cream sales.
Solution
Solution Steps
Step 1: Understanding the Correlation Coefficient
The correlation coefficient \( r \) quantifies the degree of linear relationship between two variables \( X \) and \( Y \). It is calculated using the formula:
\[
r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}
\]
where:
\( \text{Cov}(X,Y) \) is the covariance between \( X \) and \( Y \),
\( \sigma_X \) is the standard deviation of \( X \),
\( \sigma_Y \) is the standard deviation of \( Y \).
Step 2: Calculating Covariance and Standard Deviations
From the calculations, we have:
\( \text{Cov}(X,Y) = -27.75 \)
\( \sigma_X = 7.91 \)
\( \sigma_Y = 3.89 \)
Step 3: Computing the Correlation Coefficient
Substituting the values into the correlation formula:
\[
r = \frac{-27.75}{7.91 \times 3.89}
\]
Calculating the denominator:
\[
7.91 \times 3.89 \approx 30.7499
\]
Thus, the correlation coefficient becomes:
\[
r = \frac{-27.75}{30.7499} \approx -0.9
\]
Step 4: Interpreting the Result
The correlation coefficient \( r = -0.9 \) indicates a strong negative correlation between the two variables. This means that as one variable (runner's speed) increases, the other variable (time to reach the finish line) decreases.
Final Answer
The statement that exemplifies a negative correlation is: