Question

Two major automobile manufacturers have produced compact cars with engines of the same size. We are interested in determining whether or not there is a significant difference in the mean MPG (miles per gallon) when testing for the fuel efficiency of these two brands of automobiles. A random sample of eight cars from each manufacturer is selected, and eight drivers are selected to drive each automobile for a specified distance. The following data (in miles per gallon) show the results of the test. Assume the population of differences is normally distributed.

Driver Manufacturer A Manufacturer B
1. 32 28
2. 27 22
3. 26 27
4. 26 24
5. 25 24
6. 29 25
7. 31 28
8. 25 27
At a.10, the null hypothesis
a) should not be rejected.
b) should be rejected.
c) should be revised.

Answer 1

Explanation:

Corresponding fuel efficiencies of manufacturer A's car and manufacturer B's car form matched pairs.

The data for the test are the differences between the efficiencies of manufacturer A's car and manufacturer B's car

μd = fuel efficiency of manufacturer A's car minus the​ fuel efficiency of manufacturer B's car.

A B diff

32 28 4

27 22 5

26 27 - 1

26 24 2

25 24 1

29 25 4

31 28 3

25 27 - 2

Sample mean, xd

= (4 + 5 - 1 + 2 + 1 + 4 + 3 - 2)/8 = 2

xd = 2

Standard deviation = √(summation(x - mean)²/n

n = 8

Summation(x - mean)² = (4 - 2)^2 + (5 - 2)^2 + (- 1 - 2)^2 + (2 - 2)^2 + (1 - 2)^2 + (4 - 2)^2 + (3 - 2)^2 + (- 2 - 2)^2 = 44

Standard deviation = √(44/8

sd = 2.35

For the null hypothesis

H0: μd = 0

For the alternative hypothesis

H1: μd ≠ 0

This is a two tailed test and the distribution is a students t. Therefore, degree of freedom, df = n - 1 = 8 - 1 = 7

2) The formula for determining the test statistic is

t = (xd - μd)/(sd/√n)

t = (2 - 0)/(2.35/√8)

t = 2.41

We would determine the probability value by using the t test calculator.

p = 0.047

Since alpha, 0.1 > the p value 0.047, then we would reject the null hypothesis. Therefore, at 1% significance level, we can conclude that there is a significant difference in the mean MPG (miles per gallon) when testing for the fuel efficiency of these two brands of automobiles.