Question

the table shows a countrys average fuel consumption for \light duty\ vehicles (passenger cars and small trucks) for several years. let x be the number of years since 1985, so that x = 5 stands for 1990 and so forth.
(a) find the linear regression model for these data. what does the slope in the regression model represent?
(b) use the linear regression model to predict the countrys average fuel economy for light duty vehicles in the year 2035.
(a) find the linear regression model for these data. choose the correct answer below.
(use integers or decimals for any numbers in the equation. round to the nearest hundredth as needed)

Explanation:

Step1: Calculate sums

Let \(x_i\) be the number of years since 1985 and \(y_i\) be the fuel - economy values.
We have the data points: \((5,17.3),(10,18.2),(15,18.9),(20,19.6),(25,20.4)\)
\(n = 5\)
\(\sum_{i = 1}^{n}x_i=5 + 10+15+20+25=75\)
\(\sum_{i = 1}^{n}y_i=17.3 + 18.2+18.9+19.6+20.4 = 94.4\)
\(\sum_{i = 1}^{n}x_i^2=5^2+10^2+15^2+20^2+25^2=25 + 100+225+400+625 = 1375\)
\(\sum_{i = 1}^{n}x_iy_i=5\times17.3+10\times18.2+15\times18.9+20\times19.6+25\times20.4=86.5+182+283.5+392+510 = 1454\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression line \(y=mx + b\) is \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\)
\[

$$\begin{align*} m&=\frac{5\times1454 - 75\times94.4}{5\times1375-75^2}\\ &=\frac{7270-7080}{6875 - 5625}\\ &=\frac{190}{1250}\\ &= 0.152 \end{align*}$$

\]

Step3: Calculate intercept \(b\)

The formula for the intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_i-m\sum_{i = 1}^{n}x_i}{n}\)
\[

$$\begin{align*} b&=\frac{94.4-0.152\times75}{5}\\ &=\frac{94.4 - 11.4}{5}\\ &=\frac{83}{5}\\ &=16.6 \end{align*}$$

\]

The linear - regression model is \(y = 0.15x+16.6\) (rounded to the nearest hundredth). The slope represents the average increase in fuel economy (in mpg) per year.

Step4: Predict for 2035

For the year 2035, \(x=2035 - 1985=50\)
Substitute \(x = 50\) into the equation \(y=0.15x + 16.6\)
\(y=0.15\times50+16.6=7.5+16.6 = 24.1\)

Answer:

(a) The linear - regression model is \(y = 0.15x+16.6\). The slope represents the average increase in the country's average fuel economy (in mpg) for light - duty vehicles per year.
(b) The predicted average fuel economy for light - duty vehicles in 2035 is \(24.1\) mpg.