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Chapter 9: Problem 50

The manager of a service station claims that the mean amount spent on gas by its customers is \(\$ 15.90\) per visit. You want to test if the mean amount spent on gas at this station is different from \(\$ 15.90\) per visit. Briefly explain how you would conduct this test when \(\sigma\) is not known.

Short Answer

Expert verified
The Student's t-test will be conducted to test if there is a significant difference between the mean amount spent on gas from the claimed \$15.90. Reject or accept the null hypothesis based on where the calculated t-statistic falls in relation with the t-critical value.

Step by step solution

01

State the Hypotheses

The first step in any hypothesis testing is to state the null and alternative hypothesis. Here the null hypothesis \(H_0\) is that the mean amount spent on gas is exactly \$15.90. The alternative hypothesis \(H_a\) is that the mean amount spent on gas is different from \$15.90. In mathematical terms - \(H_0: \mu = \$15.90\) and \(H_a: \mu \neq \$15.90\).
02

Choose the Significance Level

The significance level of the test must be decided. This is usually at 0.05 or 5%, but depends on the circumstances. This level (denoted as \(\alpha\)) is the probability of rejecting the null hypothesis when it is true. The lower the significance level, the stronger the evidence required to reject the null hypothesis.
03

Calculate the Test Statistic

Since the standard deviation \(\sigma\) is not given, a sample will be taken, and the sample standard deviation \(s\) and the sample mean \(\bar{x}\) will be calculated. These calculations will be used to compute the t statistic with the following formula: \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\), where \(\mu_0\) is the claimed population mean (\$15.90) and \(n\) is the sample size.
04

Determine the critical value and the rejection regions

Based on the significance level and degrees of freedom (which is \(n - 1\) here), find the critical value from the t-table. As this is a two-tailed test (we are interested in a difference, whether more or less), there will be two rejection regions, below \(-t_{critical}\) and above \(+t_{critical}\).
05

Make a Decision

If the calculated t value falls in the rejection region, reject the null hypothesis favoring the alternative one which indicates that the mean amount spent on gas is not \$15.90. If it falls within the non-rejection region, do not reject the null hypothesis, which indicates that the difference is not statistically significant and hence we consider the mean to be \$15.90.

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Most popular questions from this chapter

Consider the null hypothesis \(H_{0}=p=.25 .\) Suppose a random sample of 400 observations is taken to perform this test about the population proportion. Using \(\alpha=.01\), show the rejection and nonrejection regions and find the critical value(s) of \(z\) for a a. left-tailed test b. two-tailed test c. right-tailed test

A random sample of 8 observations taken from a population that is normally distributed produced a sample mean of \(44.98\) and a standard deviation of \(6.77\). Find the critical and observed values of \(t\) and the range for the \(p\) -value for each of the following tests of hypotheses, using \(\alpha=.05\). a. \(H_{0}: \mu=50\) versus \(H_{1}: \mu \neq 50\) b. \(H_{0}: \mu=50\) versus \(H_{1}: \mu<50\)

According to the U.S. Census Bureau, in \(2014,62 \%\) of Americans age 18 and older were married. A recent sample of 2000 Americans age 18 and older showed that \(58 \%\) of them are married. Can you reject the null hypothesis at a \(1 \%\) significance level in favor of the alternative that the percentage of current population of Americans age 18 and older who are married is lower than \(62 \%\) ? Use both the \(p\) -value and the critical-value approaches.

Consider the null hypothesis \(H_{0}: \mu=625 .\) Suppose that a random sample of 29 observations is taken from a normally distributed population with \(\sigma=32 .\) Using a significance level of \(.01\), show the rejection and nonrejection regions on the sampling distribution curve of the sample mean and find the critical value(s) of \(z\) when the alternative hypothesis is as follows. a. \(H_{1}: \mu \neq 625\) b. \(H_{1}: \mu>625\) c. \(H_{1}: \mu<625\)

Two years ago, \(75 \%\) of the customers of a bank said that they were satisfied with the services provided by the bank. The manager of the bank wants to know if this percentage of satisfied customers has changed since then. She assigns this responsibility to you. Briefly explain how you would conduct such a test.
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