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Chapter 10: Problem 1

Explain why the statement \(\bar{x}=50\) is not a legitimate hypothesis.

Short Answer

Expert verified
\(\bar{x}=50\) is not a legitimate hypothesis because hypotheses in statistics must make claims about population parameters, not sample statistics. \(\bar{x}=50\) is an observation from a sample, not a claim about a population.

Step by step solution

01

Understanding Hypotheses

In statistical terms, a hypothesis is an assertion or claim about a population parameter, such as the population mean or population proportion. This is typically denoted as \(H_0\) (null hypothesis) or \(H_1\) or \(H_a\) (alternative hypothesis). The goal of hypothesis testing is to provide evidence to either support or contradict the null hypothesis.
02

Understanding the Statement \(\bar{x}=50\)

The statement \(\bar{x}=50\) is referring to a sample mean. The symbol \(\bar{x}\) denotes a sample mean, which is the average of a set of data points, while '50' is simply a particular numerical value of that sample mean. This is not a claim about a population parameter, but rather an observation from a data sample.
03

Conclude why \(\bar{x}=50\) Is Not a Legitimate Hypothesis

Because \(\bar{x}=50\) is an observation of a sample and not an assertion about a population parameter, it is not a valid hypothesis. Hypotheses must make claims about population parameters, not sample statistics. Therefore, the statement \(\bar{x}=50\) is not a legitimate hypothesis.

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Most popular questions from this chapter

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most \(150^{\circ} \mathrm{F}\), there will be no negative effects on the river's ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above \(150^{\circ} \mathrm{F}\), researchers will take 50 water samples at randomly selected times and record the temperature of each sample. The resulting data will be used to test the hypotheses \(H_{0}: \mu=150^{\circ} \mathrm{F}\) versus \(H_{a}: \mu>150^{\circ} \mathrm{F}\). In the context of this example, describe Type I and Type II errors. Which type of error would you consider more serious? Explain.

Researchers have postulated that because of differences in diet, Japanese children have a lower mean blood cholesterol level than U.S. children do. Suppose that the mean level for U.S. children is known to be 170 . Let \(\mu\) represent the true mean blood cholesterol level for Japanese children. What hypotheses should the researchers test?

Teenagers (age 15 to 20 ) make up \(7 \%\) of the driving population. The article "More States Demand Teens Pass Rigorous Driving Tests" (San Luis Obispo Tribune, January 27,2000 ) described a study of auto accidents conducted by the Insurance Institute for Highway Safety. The Institute found that \(14 \%\) of the accidents studied involved teenage drivers. Suppose that this percentage was based on examining records from 500 randomly selected accidents. Does the study provide convincing evidence that the proportion of accidents involving teenage drivers differs from \(.07\), the proportion of teens in the driving population?

The article "Poll Finds Most Oppose Return to Draft, Wouldn't Encourage Children to Enlist" (Associated Press, December 18,2005 ) reports that in a random sample of 1000 American adults, 700 indicated that they oppose the reinstatement of a military draft. Is there convincing evidence that the proportion of American adults who oppose reinstatement of the draft is greater than twothirds? Use a significance level of \(.05\).

A credit bureau analysis of undergraduate students credit records found that the average number of credit cards in an undergraduate's wallet was \(4.09\) ("Undergraduate Students and Credit Cards in 2004," Nellie Mae, May 2005). It was also reported that in a random sample of 132 undergraduates, the sample mean number of credit cards carried was 2.6. The sample standard deviation was not reported, but for purposes of this exercise, suppose that it was \(1.2\). Is there convincing evidence that the mean number of credit cards that undergraduates report carrying is less than the credit bureau's figure of \(4.09 ?\)
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