Question

data were recorded for a cars fuel efficiency, in miles per gallon (mpg), and corresponding speed, in miles per hour (mph). given the least - squares regression line, \\( \ln ( \text { fuel efficiency } ) = 1.437 + 0.541 \ln ( \text { speed } ) \\), what is the predicted fuel efficiency for a speed of 30 mph?
17.67 mpg
26.50 mpg
30.00 mpg
37.74 mpg

Explanation:

Step1: Substitute speed value into the equation

Given the equation \( \ln(\text{Fuel Efficiency})=1.437 + 0.541\ln(\text{Speed})\), and speed \( = 30\) mph. First, find \( \ln(30)\).
Using a calculator, \( \ln(30)\approx3.4012\).

Step2: Calculate the value of \( \ln(\text{Fuel Efficiency})\)

Substitute \( \ln(30)\) into the regression line equation:
\( \ln(\text{Fuel Efficiency})=1.437+0.541\times3.4012\)
\( =1.437 + 1.8400\)
\( =3.277\)

Step3: Find the fuel efficiency

Since \( y = \ln(x)\) implies \( x = e^{y}\), then Fuel Efficiency \(=e^{3.277}\)
Using a calculator, \( e^{3.277}\approx26.50\)

Answer:

26.50 mpg