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 |
7.5 |
−0.6 |
1.6 |
3.7 |
4.5 |
7.0 |
2.2 |
4.8 |
3.0 |

4.4 |
0.2 |
3.0 |
1.4 |
1.4 |
5.0 |
2.1 |
3.5 |
−0.6 |
−4.2 |

(a) State the appropriate

*H*_{0}

and

*H*_{a}

to test this suspicion.

*H*_{0}: *μ* = 3
mpg; *H*_{a}:
*μ* ≠ 3 mpg

*H*_{0}: *μ* = 0
mpg; *H*_{a}:
*μ* ≠ 0 mpg

*H*_{0}: *μ* < 0
mpg; *H*_{a}:
*μ* > 0 mpg

*H*_{0}: *μ* > 3
mpg; *H*_{a}:
*μ* < 3 mpg

*H*_{0}: *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 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.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
does not differ from the driver's computed values.

Answer #1

The statistical software output for this problem is:

**One sample Z hypothesis test:**

μ : Mean of variable

H_{0} : μ = 0

H_{A} : μ ≠ 0

Standard deviation = 3

**Hypothesis test results:**

Variable |
n |
Sample Mean |
Std. Err. |
Z-Stat |
P-value |
---|---|---|---|---|---|

Data | 20 | 2.745 | 0.67082039 | 4.0920044 | <0.0001 |

Hence,

a) H0: μ = 0 mpg; Ha: μ ≠ 0 mpg

**Option B** is correct.

b) P - value = 0.0000

c) 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.

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