Gasoline: Many drivers of cars that run on regular gas actually buy premium in the belief that they will get better gas mileage. To test that belief, we use 10 cars from a company fleet in which all the cars run on regular gas. Each car is filled first with either regular or premium gasoline, decided by a coin toss, and the mileage for that tankful is recorded. Then the mileage is recorded again for the same cars for a tankful of the other king of gasoline. We don’t let driers know about this experiment.
Here are the results (miles per gallon)
Car # |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
Regular |
16 |
20 |
21 |
22 |
23 |
22 |
27 |
25 |
27 |
28 |
Premium |
19 |
22 |
24 |
24 |
25 |
25 |
26 |
26 |
28 |
32 |
C. Even if the difference is significant, why might the company choose to stick with regular gasoline?
E) Suppose you had done a “bad thing” Suppose you had mistakenly treated these data as two independent samples instead of matched pairs. What would the significance test have found? Carefully explain why the results are so different.
Is there evidence that cars get significantly better fuel economy with premium gasoline?
The hypothesis being tested is:
H_{0}: µ_{1} - µ_{2} = 0
H_{a}: µ_{1} - µ_{2} < 0
The output is:
23.100 | mean Regular |
25.100 | mean Premium |
-2.000 | mean difference (Regular - Premium) |
1.414 | std. dev. |
0.447 | std. error |
10 | n |
9 | df |
-4.472 | t |
.0008 | p-value (one-tailed, lower) |
-2.820 | confidence interval 90.% lower |
-1.180 | confidence interval 90.% upper |
0.820 | margin of error |
Since the p-value (0.0008) is less than the significance level, we can reject the null hypothesis.
Therefore, we can conclude that cars get significantly better fuel economy with premium gasoline.
How big might that difference be? Check a 90% confidence interval
The 90% confidence interval is between -2.820 and -1.180. Thus, the difference is small.
Even if the difference is significant, why might the company choose to stick with regular gasoline?
The regular gasoline is cheaper than the premium one and this is the first reason to consider.
Suppose you had done a “bad thing” Suppose you had mistakenly treated these data as two independent samples instead of matched pairs. What would the significance test have found? Carefully explain why the results are so different.
Regular | Premium | |
23.10 | 25.10 | mean |
3.73 | 3.45 | std. dev. |
10 | 10 | n |
18 | df | |
-2.000 | difference (Regular - Premium) | |
12.878 | pooled variance | |
3.589 | pooled std. dev. | |
1.605 | standard error of difference | |
0 | hypothesized difference | |
-1.246 | t | |
.1143 | p-value (one-tailed, lower) |
Since the p-value (0.1143) is greater than the significance level, we cannot reject the null hypothesis.
Therefore, we cannot conclude that cars get significantly better fuel economy with premium gasoline.
This happened because we considered that the samples have no relationship and the gasoline is independent of its types.
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