Speed vs Gas Mileage An economist wanted to analyze the relationship between the

Question

Speed vs Gas Mileage
An economist wanted to analyze the relationship between the speed of a car (x) and its gas mileage (y). As
an experiment a car is operated at several different speeds and for each speed the gas mileage is
measured. These data are shown below.
Speed 25 35 45 50 60 65 70
Gas Mileage 40 39 37 33 30 27 25
1. {Car Speed and Gas Mileage Narrative} Determine the least squares regression line.
2. {Car Speed and Gas Mileage Narrative} Estimate the gas mileage of a car traveling 70 mph.
3. The following 10 observations of variables x and y were collected.
x 1 2 3 4 5 6 7 8 9 10
y 25 22 21 19 14 15 12 10 6 2
Find the least squares regression line, and the estimated value of y when x = 3.
4. A scatter diagram includes the following data points:
x 3 2 5 4 5
y 8 6 12 10 14
Two regression models are proposed: (1) , and (2) . Using the least squares method,
which of these regression models provides the better fit to the data? Why?
5. Consider the following data values of variables x and y.
x 2 4 6 8 10 13
y 7 11 17 21 27 36
a. Determine the least squares regression line.
b. Find the predicted value of y for x = 9.
c. What does the value of the slope of the regression line tell you?
Sunshine and Melanoma
A medical researcher wanted to examine the relationship between the amount.

Explanation / Answer

Sol:

For

Speed vs Gas Mileage

Regression in R :

Enter the below code in R

Speed <- c(25,35,45,50,60,65,70)
Gas_Mileage <- c(40, 39, 37, 33, 30, 27, 25)
regmod1 <- lm(Gas_Mileage~Speed)
summary(regmod1)

Output:

summary(regmod1)

Call:

lm(formula = Gas_Mileage ~ Speed)

Residuals:

1 2 3 4 5

-1.828e+00 7.031e-01 2.234e+00 5.218e-15 5.312e-01

6 7

-7.031e-01 -9.375e-01

Coefficients:

Estimate Std. Error t value Pr(>|t|)   

(Intercept) 50.6562 1.8910 26.788 1.36e-06 ***

Speed -0.3531 0.0362 -9.754 0.000193 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.448 on 5 degrees of freedom

Multiple R-squared: 0.9501, Adjusted R-squared: 0.9401

F-statistic: 95.15 on 1 and 5 DF, p-value: 0.0001926

Regression eq is

gasmileage=50.6562-0.3531(speed)

slope=-0.3531

y intercept=50.6562

For speed=70

reg eq is

gasmileage=50.6562-0.3531(speed)

gasmileage=50.6562-0.3531(70)

=25.9392

=26(rounding to nearest integer)

gasmileage=26

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