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Chapter 10: Problem 91

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 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

01

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., \( \mu < 30 \).
02

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.
03

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.
04

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.
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null and Alternative Hypotheses
In hypothesis testing, we begin by setting up two hypotheses: the null hypothesis and the alternative hypothesis. These act as the foundation for statistical testing.
The **Null Hypothesis** (\( H_0 \)) is a statement that there is no effect or no difference, and it proposes that any difference observed is due to sampling or random chance. In our example, the null hypothesis is: "The car's true average fuel efficiency is at least 30 mpg," or mathematically, \( \mu \geq 30 \).
On the other hand, the **Alternative Hypothesis** (\( H_1 \)) is what you might want to prove. It indicates that the effect or difference you鈥檙e testing is real. For our automobile test, the alternative hypothesis is: "The car's true average fuel efficiency is less than 30 mpg," or \( \mu < 30 \).
Setting these hypotheses clearly helps establish a binary structure where you either reject the null hypothesis in favor of the alternative or you fail to reject the null hypothesis.
t-test
The t-test is a statistical test used to evaluate the difference between the sample mean and the population mean. In our scenario, we use it to determine whether the car's reported fuel efficiency of 30 mpg stands true with the sample data.

To perform a t-test:
  • Calculate the **sample mean** (\( \bar{x} \)) and **sample standard deviation** (\( s \)). These statistics provide an overview of the data's central tendency and dispersion.
  • Determine the **t statistic** using the formula: \( t = (\bar{x} - \mu) / (s / \sqrt{n}) \). Here, \( \bar{x} \) is the sample mean, \( \mu \) is the claimed population mean (30 mpg), \( s \) is the sample standard deviation, and \( n \) is the sample size.
  • This t statistic helps to compare the sample and population mean and determine if the observed differences are statistically significant.
The t-test assumes that the sample is drawn from a normally distributed population, which fits the conditions here.
p-value
The p-value is a probability that measures evidence against the null hypothesis. It tells us the likelihood of observing our sample data, assuming the null hypothesis is true.

To interpret the p-value:
  • After calculating the t statistic, refer to a **t-distribution table** to find the p-value.
  • Compare this p-value with a pre-defined significance level, typically 0.05.
  • If the **p-value** is less than the significance level, it indicates strong evidence against the null hypothesis, leading us to reject it.
  • In contrast, a **p-value** greater than the significance level suggests insufficient evidence to reject the null hypothesis.
For our test, a low p-value would suggest rejecting the claim that the car averages at least 30 mpg, while a higher p-value would mean that our sample data does not contradict this claim.

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Most popular questions from this chapter

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Suppose that you are an inspector for the Fish and Game Department and that you are given the task of determining whether to prohibit fishing along part of the Oregon coast. You will close an area to fishing if it is determined that fish in that region have an unacceptably high mercury content. a. Assuming that a mercury concentration of \(5 \mathrm{ppm}\) is considered the maximum safe concentration, which of the following pairs of hypotheses would you test: $$ H_{0}: \mu=5 \text { versus } H_{a}: \mu>5 $$ or $$ H_{0}: \mu=5 \text { versus } H_{a}: \mu<5 $$ Give the reasons for your choice. b. Would you prefer a significance level of \(.1\) or \(.01\) for your test? Explain.

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