Question

Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the miles per gallon were recorded each time the gas tank was filled, and the computer was then reset. In addition to the computer's calculations of miles per gallon, the driver also recorded the miles per gallon by dividing the miles driven by the number of gallons at each fill-up. The following data are the differences between the computer's and the driver's calculations for that random sample of 20 records. The driver wants to determine if these calculations are different. Assume that the standard deviation of a difference is

σ = 3.0.

5.0	6.5	−0.6	1.8	3.7	4.5	8.0	2.2	4.9	3.0
4.4	0.4	3.0	1.4	1.4	6.0	2.1	3.3	−0.6	−4.2

(a) State the appropriate

H₀

and

H_a

to test this suspicion.

H₀: μ = 3 mpg;    H_a: μ ≠ 3 mpg

H₀: μ > 0 mpg;    H_a: μ < 0 mpg

H₀: μ > 3 mpg;    H_a: μ < 3 mpg

H₀: μ = 0 mpg;    H_a: μ ≠ 0 mpg

H₀: μ < 0 mpg;    H_a: μ > 0 mpg

(b) Carry out the test. Give the P-value. (Round your answer to four decimal places.)

Interpret the result in plain language.

We conclude that μ = 3 mpg; that is, we have strong evidence that the computer's reported fuel efficiency does not differ from the driver's computed values.

We conclude that μ ≠ 0 mpg; that is, we have strong evidence that the computer's reported fuel efficiency differs from the driver's computed values.

We conclude that μ ≠ 3 mpg; that is, we have strong evidence that the computer's reported fuel efficiency differs from the driver's computed values.

We conclude that μ ≠ 0 mpg; that is, we have strong evidence that the computer's reported fuel efficiency does not differ from the driver's computed values.

We conclude that μ = 0 mpg; that is, we have strong evidence that the computer's reported fuel efficiency differs from the driver's computed values.

Answer 1

Solution:

a) The appropriate null and alternative hypotheses are as follows:

H0: μ = 0 mpg; Ha: μ ≠ 0 mpg

The test statistic to test the hypothesis is given as follows:

Where, is sample mean, μ is population mean specified under H_{​​​​​​0} and σ is population standard deviation and n is sample size.

We have, n = 20

We have, μ = 0 mpg,  σ = 3 mpg

The value of the test statistic is 4.1889.

Our test is two-tailed test, therefore we shall calculate two-tailed p-value for the test statistic. The two-tailed p-value is given as follows:

p-value = 2P(Z < value of the test statistic)

p-value = 2P(Z < 4.1889)

p-value = 0.0000

We make decision rule as follows:

If p-value is less than the significance level then we reject the H_{​​​​​​0}.

If p-value is greater than the significance level of then we fail to reject H_{​​​​​​0}.

In our question significance level is not given. Generally significance level of 0.05 or 0.01 is used. We shall use the significance level of 0.01.

p-value = 0.0000 and significance level = 0.01

(0.0000 < 0.05)

Since, p-value is less than the significance level of 0.01, therefore we shall reject H_{​​​​​​0}.

Conclusion:

We conclude that μ ≠ 0 mpg; that is, we have strong evidence that the computer's reported fuel efficiency differs from the driver's computed values.