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Chapter 15: Problem 9

A manufacturer of a gas additive claims that it improves gas mileage. A random sample of 30 drivers tests this claim by determining their gas mileage for a full tank of gas that contains the additive \(\left(X_{1}\right)\) and for a full tank of gas that does not contain the additive \(\left(X_{2}\right)\). The sample mean difference, \(\bar{D}\), equals 2.12 miles (in favor of the additive), and the estimated standard error equals 1.50 miles. (a) Using \(t\), test the null hypothesis at the .05 level of significance. (b) Specify the \(p\) -value for this result. (c) Are there any special precautions that should be taken with the present experimental design?

Short Answer

Expert verified
Here are the solutions to the problems: (a) The t-statistic is calculated to be 1.413. (b) The p-value is approximately 0.17. (c) There are different precautions that should be taken into account during the experiment such as ensuring random assignment of tests and controlling for external variables like individual driving habits and the model of the car used.

Step by step solution

01

State the Hypotheses

The null hypothesis \(H_{0}\) is that the mean difference between the gas mileage for a full tank of gas with the additive and without the additive is zero, i.e., \(H_{0} : D = 0\). The alternative hypothesis \(H_{1}\) or \(H_{a}\) is that the mean difference between the gas mileage of the two scenarios is not equal to zero, or \(H_{1} : D \neq 0\).
02

Calculate the T-Statistic

Use the formula for the t-statistic which is \(\(t\) = \frac{\(\bar{D}\) - D_{0}}{SE}\), where \(\bar{D}\) = 2.12 and SE = 1.50 are given, and \(D_{0}\) is the population mean difference. Since in the null hypothesis, it's assumed that the mean difference is zero, hence \(D_{0} = 0\). So, \(\(t\) = \frac{2.12 - 0}{1.50} = 1.413\).
03

Determining the P-Value

Using the t-distribution table or calculator, the p-value can be identified. The degree of freedom for this test will be \(n - 1 = 30 - 1 = 29\), where n is the sample size. So, p-value is approximately 0.17 which is more than the level of significance, 0.05.
04

Statistically Testing the Hypothesis

Since the calculated p-value is more than the significance level, we do not reject the null hypothesis \(H_{0}\). This means that the evidence available from the sample is not strong enough to reject the null hypothesis that the gas additive does not improve gas mileage.
05

Precautions for Experimental Design

One should ensure that the tests are randomly assigned, to avoid bias. Also, individuals driving habits and the model of the car used can affect the gas mileage so researchers should also control for these factors during the experiment.

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