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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 credible hypothesis since it refers to a sample mean. Hypotheses should be made about population parameters. The correct hypothesis could be \(\mu=50\), where \(\mu\) signifies the population mean.

Step by step solution

01

Understanding hypotheses

A legitimate hypothesis in statistics must be a testable claim about a population parameter, such as the population mean (\(\mu\)), the population proportion (p), the population standard deviation (\(\sigma\)), among others.
02

Wrong reference

\(\bar{x}=50\) is not a legitimate hypothesis as it is a statement about a sample mean, not a population parameter. A sample mean is subject to variability and depends on the specific sample chosen.
03

Right reference

If you want to make a legitimate hypothesis, it should instead refer to a population parameter. For example, a correct hypothesis could be \(\mu=50\), where \(\mu\) represents the population mean.

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Most popular questions from this chapter

In a representative sample of 1000 adult Americans, only 430 could name at least one justice who is currently serving on the U.S. Supreme Court (Ipsos, January 10,2006 ). Using a significance level of \(.01\), carry out ? hypothesis test to determine if there is convincing evidence to support the claim that fewer than half of adult Americans can name at least one justice currently serving on the Supreme Court.

A hot tub manufacturer advertises that with its heating equipment, a temperature of \(100^{\circ} \mathrm{F}\) can be achieved in at most \(15 \mathrm{~min}\). A random sample of 25 tubs is selected, and the time necessary to achieve a \(100^{\circ} \mathrm{F}\) temperature is determined for each tub. The sample average time and sample standard deviation are \(17.5\) min and \(2.2\) min, respectively. Does this information cast doubt on the company's claim? Carry out a test of hypotheses using significance level \(.05 .\)

The state of Georgia's HOPE scholarship program guarantees fully paid tuition to Georgia public universities for Georgia high school seniors who have a B average in academic requirements as long as they maintain a B average in college. Of 137 randomly selected students enrolling in the Ivan Allen College at the Georgia Institute of Technology (social science and humanities majors) in 1996 who had a B average going into college, \(53.2 \%\) had a GPA below \(3.0\) at the end of their first year ("Who Loses HOPE? Attrition from Georgia's College Scholarship Program," Southern Economic Journal [1999]: 379-390). Do these data provide convincing evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship?

The article "Fewer Parolees Land Back Behind Bars" (Associated Press, April 11,2006 ) includes the following statement: "Just over 38 percent of all felons who were released from prison in 2003 landed back behind bars by the end of the following year, the lowest rate since 1979." Explain why it would not be necessary to carry out a hypothesis test to determine if the proportion of felons released in 2003 was less than \(.40\).

An article titled "Teen Boys Forget Whatever It Was" appeared in the Australian newspaper The Mercury (April 21, 1997). It described a study of academic performance and attention span and reported that the mean time to distraction for teenage boys working on an independent task was 4 min. Although the sample size was not given in the article, suppose that this mean was based on a random sample of 50 teenage Australian boys and that the sample standard deviation was \(1.4\) min. Is there convincing evidence that the average attention span for teenage boys is less than 5 min? Test the relevant hypotheses using \(\alpha=.01\).
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