Question

janelle tracks the number of miles she drives and the number of gallons of gas she has left. what is the linear regression model for this scenario? what is the correlation coefficient? what is the strength of the model? strong positive weak positive strong negative weak negative

Explanation:

Step1: Calculate the linear regression model

Using a statistical software or calculator (such as Excel's LINEST function or a graphing calculator's regression feature) on the data points \((x = \text{Miles Driven}, y=\text{Gallons in Tank})\): \((27,13)\), \((65,12)\), \((83,11)\), \((109,10)\), \((142,9)\), \((175,8)\)
The linear regression equation is of the form \(y = mx + b\). After calculation, \(y=-0.03x + 13.8\)

Step2: Calculate the correlation coefficient

Using the formula for the correlation coefficient \(r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\) or using software. For this data, \(r\approx - 0.99\)

Step3: Determine the strength of the model

The correlation coefficient \(r\) has a value close to \(-1\). A value of \(r\) close to \(- 1\) or \(1\) indicates a strong correlation. Since \(r=-0.99\) (negative), the strength is strong negative

Answer:

Linear regression model: \(y =-0.03x + 13.8\)
Correlation coefficient: \(-0.99\)
Strength of the model: strong negative