Math Problem Statement

An owner of a large car lot believes that fuel prices are going to rise significantly and wonders how this rise might affect demand for the high-performance vehicles. Specifically, the owner is investigating a link between how fast a car can go from
0 to
60 miles per hour (measured in seconds) and the car's economy as measured in miles traveled per gallon used (mpg). If fast cars, which are normally high in demand, are associated with higher mpg then there will be much less demand if gas prices rise as predicted. The owner gathers data on
20 vehicles. The data is provided below. Use Rguroo to calculate the correlation coefficient
r between the two variables. Round your answer to two decimal places.

"mpg" 0 to 60 time (seconds) 28 7.7 25 8.2 25 8.6 22 7.4 22 8 21 6.9 21 7.5 21 7.4 21 7.8 21 8.8 20 6.1 20 6.9 20 7.2 20 7.5 20 7.5 20 7.5 20 7.7 19 6.7 19 7.9 19 8.5

Provide your answer below:

Solution

To calculate the correlation coefficient rr between the two variables "mpg" and "0 to 60 time (seconds)", we can use the following formula for Pearson's correlation coefficient:

r=n∑(xiyi)−∑xi∑yi(n∑xi2−(∑xi)2)(n∑yi2−(∑yi)2)r = \frac{n \sum (x_i y_i) - \sum x_i \sum y_i}{\sqrt{\left(n \sum x_i^2 - (\sum x_i)^2\right)\left(n \sum y_i^2 - (\sum y_i)^2\right)}}

Where:

  • xix_i is the value of the mpg for each vehicle,
  • yiy_i is the value of the 0 to 60 time (in seconds) for each vehicle,
  • nn is the number of vehicles (in this case, n=20n = 20).

Let me calculate this for you using the provided data.The correlation coefficient rr between "mpg"

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation
Linear Relationships

Formulas

Pearson's correlation coefficient formula: r = [n∑(x_iy_i) - (∑x_i)(∑y_i)] / sqrt{[n∑x_i^2 - (∑x_i)^2][n∑y_i^2 - (∑y_i)^2]}

Theorems

Pearson's correlation theorem

Suitable Grade Level

Grades 11-12, Undergraduate

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