3. An Environmental Studies class did a survey to determine the relation between engine size and fuel consumption. They obtained the following data for five types of cars.

Car

Size

MPG

A

1.6

38

B

1.8

29

C

2.2

32

D

3.2

24

E

4.7

17

where the engine size is given in liters and the mileage is given in miles per gallon. The question is whether engine size has an effect on gasoline mileage.

a) State the null and alternative hypotheses.

Ho: Engine size has no effect on gas mileage. r = 0, so b = 0

Ha: Engine size has an effect on mileage r is not 0, so b is not zero.

b) Find the line of best least squares fit.

If we do a regression with Data Desk, we get the following result.

So the regression line is

y = 43.6948 - 5.81288x

c) Test the null hypothesis.

The null hypothesis is equivalent to saying that the coefficient of x in the regression equation is 0.

In the Data Desk pirint out, the last row which starts with "Size" has all of the information concerning the coefficient of the x term. The slope is -5.81288, the standard error for the slope is 1.312, and if you divide the standard error into the slope you get the t-score.

If we compare this with the critical valuse of t for 3 degrees of freedom we see that it is between the critical values of t for 1% and 2%, closer to the critical value for 1%. However, since this is a two tailed test, we need to double that to get the p value. Note that the prob, the last number, on the bottom line is about 2.14%.

In this problem it will depend on the level of significance. The null hypothesis would be rejected at a 5% level of significance but it would be accepted at a 2% level or smaller.

While there is a general trend for cars with larger engines to get poorer gas mileage, cars B and C taken by themselves provide a counterexample to this rule. The situation of cars B and C increases the probability that engine size has no effect on mileage to the point where we are getting close to the gray area.

d) Find a 95% confidence interval for the slope.

The critical value of t for a 95% confidence interval for 3 degrees of freedom is 3.182 The confidence interval is given by

The upper confidence limit is then

-5.81288 + 3.182(1.312) = -1.638

and the lower confidence limit is

-5.81288 - 3.182(1.312) = -9.988

So we are at least 95% sure that the slope is not zero. However, the upper end of the confidence interval comes fairly close to 0. o would be in a slightly broader confidence interval.

e) Give a 95% confidence interval for the mileage that a car with a 2 liter engine would get.

To get a 95% confidence interval for the mileage we use the formula

where

s is found on the 4th line of the Data Desk printout.

s = 3.351

The critical value for t is still 3.182. The upper confidence limit is then

32.069 + 3.182(3.351) = 42.73

and the lower confidence limit is

32.069 - 3.182(3.351) = 21.406