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 hypotheses and the alternative hypothesis
- b) Find the line of best least squares fit.
- c) Test the null hypothesis.
- d) Give a 95% confidence interval for the slope in the regression equation.
- e) Give a 95% confidence interval for the mileage that a car with a 2 liter engine would get.

a) State the null and alternative hypotheses.

H_{o}: Engine size has no effect on gas mileage. r = 0, so
b = 0

H_{a}: 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

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

and the lower confidence limit is

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.

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

and the lower confidence limit is