Question

Determining the Strength of a Regression Model
A 2-column table with 6 rows. The first column is labeled miles driven with entries 27, 65, 83, 109, 142, 175. The second column is labeled gallons in tank with entries 13, 12, 11, 10, 9, 8.

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?

Answer 1

Compute the least-squares line for points (x = miles, y = gallons):

Summary statistics
- x̄ = 601/6 = 100.1667
- ȳ = 63/6 = 10.5
- SS_xx = 14312.8333, SS_xy = −498.5, SS_yy = 17.5

Linear regression model
- slope b = SS_xy / SS_xx ≈ −0.03483
- intercept a = ȳ − b x̄ ≈ 13.99

So the model is
gallons ≈ 13.99 − 0.03483·(miles driven)
(rounded: gallons ≈ 14.0 − 0.0348·miles)

Correlation coefficient
r = SS_xy / sqrt(SS_xx·SS_yy) ≈ −0.996 (r^2 ≈ 0.992)

Strength of the model
There is a very strong negative linear relationship (about 99.2% of the variation in gallons is explained by miles driven).