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Chapter 10: Problem 80

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

Expert verified
The short answer would depend on the calculated values. If the computed t-value is less than the critical value, then yes, the data contradict the claim that the true average fuel efficiency is at least 30 mpg. If the computed t-value is greater than the critical value, then no, the data do not provide enough evidence to contradict the claim.

Step by step solution

01

Calculate Sample Mean and Standard Deviation

First, compute the sample mean (average) and the sample standard deviation of the six fuel efficiencies:\[\overline{x} = \frac{27.2 + 29.3 + 31.2 + 28.4 + 30.3 + 29.6}{6}\]and \[s = \sqrt{\frac{(27.2-\overline{x})^2 + (29.3-\overline{x})^2 + (31.2-\overline{x})^2 + (28.4-\overline{x})^2 + (30.3-\overline{x})^2 + (29.6-\overline{x})^2}{6 - 1}}\]Here, \(\overline{x}\) stands for the sample mean and \(s\) is the sample standard deviation.
02

Compute t-value

Next, calculate the test statistic (t-value) using the formula: \[t = \frac{\overline{x} - \mu_0}{s/ \sqrt{n}}\]where \(\mu_0=30\) is the claimed value of the population mean, \(\overline{x}\) is the computed sample mean, \(s\) is the calculated sample standard deviation and \(n=6\) is the sample size.
03

Determine Critical Value and Make Decision

Determine the critical value from the t-distribution table for \(df = n - 1 = 5\) and a significance level of \(\alpha = 0.05\). This is a one-tailed test because the question is whether the fuel efficiency is less than 30 mpg, not simply different from 30 mpg. If the computed t-value is less than the critical value, reject the null hypothesis, meaning the data contradict the claim that the true average fuel efficiency is at least 30 mpg. If the computed t-value is greater than the critical value, do not reject the null hypothesis, meaning the data do not provide enough evidence to contradict the claim.

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