Many drivers of cars that can run on regular gas actually...

Many drivers of cars that can 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 kind of gasoline. We don't let the drivers know about this experiment. Here are the results (miles per gallon): $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|} \hline \text { Car # } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Regular } & 16 & 20 & 21 & 22 & 23 & 22 & 27 & 25 & 27 & 28 \\ \hline \text { Premium } & 19 & 22 & 24 & 24 & 25 & 25 & 26 & 26 & 28 & 32 \\ \hline\end{array}$$ a) Is there evidence that cars get significantly better fuel economy with premium gasoline? b) How big might that difference be? Check a $90 \%$ confidence interval. c) Even if the difference is significant, why might the company choose to stick with regular gasoline? d) Suppose you had done a "bad thing." (We're sure you didn't.) 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. *e) Use a matched-pairs sign test to test the appropriate hypothesis. Do your conclusions change from those in part a?

Many drivers of cars that can 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 kind of gasoline. We don't let the drivers know about this experiment.
Here are the results (miles per gallon):
$$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|}
\hline \text { Car # } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline \text { Regular } & 16 & 20 & 21 & 22 & 23 & 22 & 27 & 25 & 27 & 28 \\
\hline \text { Premium } & 19 & 22 & 24 & 24 & 25 & 25 & 26 & 26 & 28 & 32 \\
\hline\end{array}$$
a) Is there evidence that cars get significantly better fuel economy with premium gasoline?
b) How big might that difference be? Check a $90 \%$ confidence interval.
c) Even if the difference is significant, why might the company choose to stick with regular gasoline?
d) Suppose you had done a "bad thing." (We're sure you didn't.) 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.
*e) Use a matched-pairs sign test to test the appropriate hypothesis. Do your conclusions change from those in part a?

Key Concepts

Nonparametric Sign Test

The nonparametric sign test is a simple statistical method used to evaluate paired or matched data without relying on assumptions about the population distribution, such as normality. It assesses whether one condition tends to produce larger values than the other by comparing the signs (positive, negative, or zero differences) of the data pairs. This method is particularly useful when the data do not meet the assumptions required for parametric tests.

Independent Samples vs Paired Samples

Independent samples analysis assumes that the observations in one group are completely separate from those in the other group, which may increase variability when comparing two groups. In contrast, paired samples analysis accounts for the pairing or dependency between observations, which often yields more precise estimates of differences. Incorrectly treating inherently paired data as independent can lead to misleading conclusions, as it ignores the reduction in variability achieved by controlling for individual differences.

Practical versus Statistical Significance

Practical significance concerns whether a statistically significant result has real-world importance or meaningful impact in a given context. Even if a statistical test finds a significant difference between conditions, the difference may be too small to justify changes in practice, especially when other factors such as cost, efficiency, or risk are considered. Decision makers must evaluate both the statistical evidence and the practical implications before implementing a change.

Paired Sample Design

A paired sample design is an experimental setup where the same subjects or matched subjects are observed under two different conditions, ensuring that each subject serves as its own control. This design helps reduce variability because comparisons are made within subjects rather than between independent groups. It is particularly useful in studies intending to observe the effect of a treatment or condition by directly measuring differences in the same subjects.

Hypothesis Testing

Hypothesis testing is a statistical process that assesses evidence provided by data against a stated hypothesis. In experimental studies, a null hypothesis (typically stating no effect or difference) is compared against an alternative hypothesis (indicating some effect or difference). The testing process involves calculating a test statistic, comparing it to a critical value or calculating a p-value, and then making a decision about whether to reject the null hypothesis based on a predetermined significance level.

Confidence Interval

A confidence interval provides a range of plausible values for an unknown population parameter, such as the mean difference between two conditions, with a specified level of confidence (for example, 90%). It quantifies the uncertainty in an estimate and allows researchers to infer the magnitude of an effect. Unlike hypothesis tests that produce a yes/no decision, a confidence interval can show the practical significance of the difference by indicating how large the effect might realistically be.

Transcript

00:01 So we're looking at evidence as to whether the buying premium gas will give you better gas mileage.

00:07 So i'm going to, these cars is a matched pair setting, so i'm going to take the premium mileage minus the regular mileage.

00:14 So we're going to assume that that mean difference is zero and alternately that that mean difference is positive, meaning that the premium is having, you have a higher gas mileage.

00:25 And we have a sample size of 10.

00:28 So our t value with 9 degrees of freedom will end up being a x bar of 2 minus 0 divided by the sample standard deviation, divided by the square root of 10.

00:46 And that test statistic is a 4 .472 with a p value equal to .00077.

00:59 So even at, and you don't have a significance level, but even at a 1 % significance level, we would have evidence to reject the null and say that there is sufficient evidence that the premium mean is higher than the regular.

01:30 Now it may not warrant because of the cost, but next part b says how high might that difference be? so let's find a 90 % confidence interval for that mean difference.

01:49 And so we're going to take that 2, 2 plus or minus and the t value, the t value with, let's see, 1 .833.

02:09 And then the sample standard deviation is 1 .414 over the square root of 10.

02:16 And that confidence interval comes out to be 1 .18 to a 2 .82.

02:33 And that would be in, i believe this is in miles per gallon.

02:40 It is in miles per gallon.

02:41 And it says part c, even at the difference in significance, why should the company choose to stick to regular gasoline? because the price of the premium is very high, is significantly higher than regular.

03:05 So you may not be saving money...

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