Question

An automobile manufacturer who wishes to advertise that one of its models achieves \(30 \mathrm{mpg}\) (miles per gallon) decides to carry out a fuel efficiency test. Six nonprofessional drivers are selected, and each one drives a car from Phoenix to Los Angeles. The resulting fuel efficiencies (in miles per gallon) are: \(\begin{array}{llllll}27.2 & 29.3 & 31.2 & 28.4 & 30.3 & 29.6\end{array}\) Assuming that fuel efficiency is normally distributed under these circumstances, do the data contradict the claim that true average fuel efficiency is (at least) \(30 \mathrm{mpg}\) ?

Short answer

The exact answer would depend on the calculated t statistic, and hence the p-value. If the p-value is less than the significance level (0.05), then we would reject the null hypothesis and conclude that the true average fuel efficiency is less than 30 mpg. However, if the p-value is higher, we cannot reject the null hypothesis and conclude that the data does not contradict the claim of at least 30 mpg.

Step-by-step solution

State the null and alternative hypotheses

Null hypothesis (\( H_0 \)): The car's true average fuel efficiency is at least 30 mpg, i.e., \( \mu \geq 30 \). \n Alternative hypothesis (\( H_1 \)): The car's true average fuel efficiency is less than 30 mpg, i.e., \( \mu < 30 \).

Calculate the sample mean and standard deviation

Using the provided data set, calculate the sample mean (\( \bar{x} \)) and the sample standard deviation (\( s \)). The sample mean will give us an idea about where the data is centered and the standard deviation will provide a measure of the spread of the data.

Conduct the t-test

We will proceed by creating a t statistic, which is defined as \( t = (\bar{x} - \mu) / (s / \sqrt{n}) \), where \(\bar{x}\) is the sample mean, \( \mu \) is the population mean, \( s \) is the sample standard deviation and \( n \) is the sample size.

Find the p-value

Refer to a t-table to find the p-value corresponding to your obtained t statistic. If the p-value is lower than your chosen significance level, in this case, a popular choice is 0.05, then this would mean we reject our null hypothesis in favor of our alternative one.

Interpret the results

If the p-value is less than the significance level, that means we have enough evidence to reject the null hypothesis and accept the alternative hypothesis, which means that the true average fuel efficiency is less than 30 mpg. However, if the p-value is greater than the significance level, we cannot reject the null hypothesis and we conclude that the data does not contradict the claim that the true average fuel efficiency is at least 30 mpg.