Question

Interpreting the Coefficient of Determination. In Exercises 5–8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Pizza and Subways r = 0.992 (x = cost of a slice of pizza, y = subway fare in New York City

Short answer

The coefficient of determination is 0.984.

The percentage of variation that can be explained by the linear association between the cost of a slice of pizza and the subway fare in New York City is 98.4%.

1.6% of the variation in the response variable (subway fare in New York city) is explained by other factors and random variation.

Step-by-step solution

Given information

The linear correlation coefficient between the cost of a slice of pizza and the subway fare in New York City is 0.992.

Coefficient of determination

The coefficient of determination is the square of the linear correlation coefficient between the two variables.

Here, the linear correlation coefficient (r) between the cost of a slice of pizza and the subway fare in New York City is 0.992.

Thus,

\(\begin{array}{c}{\rm{Coefficient}}\;{\rm{of}}\;{\rm{determination}} = {r^2}\\ = {0.992^2}\\ = 0.984\end{array}\)

Therefore, the value of the coefficient of determination is 0.984.

Percentage of variation

Here,

\(\begin{array}{c}{r^2} = 0.984\\ = \frac{{98.4}}{{100}} \times 100\% \\ = 98.4\% \end{array}\)

Therefore, the percentage of the variation explained by the linear association between the cost of a slice of pizza and the subway fare in New York City is 98.4%.

The remaining \(100\% - 98.4\% = 1.6\% \) variation is explained by other factors and random variation.