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Chapter 10: Problem 7

Explain why the statement \(\hat{p}>0.50\) is not a legitimate hypothesis.

Short Answer

Expert verified
The statement \(\hat{p}>0.50\) is not a legitimate hypothesis because it's not an exact statement about a population parameter; rather, it's an inequality. Legitimate hypotheses should equate to a specific value, not contain inequalities.

Step by step solution

01

Understanding Hypotheses

In statistical hypothesis testing, a hypothesis is an assumption made about a population parameter. There are two types of statistical hypotheses: Null hypothesis (H0): The null hypothesis asserts that the population parameter equals a particular value. Alternative hypothesis (H1): The alternative hypothesis contradicts the null hypothesis, asserting any other possible conditions other than that which is stated in H0. The null hypothesis and alternative hypothesis must account for all possible outcomes of the population parameter.
02

Identifying the Problem with the Given Statement

Looking at the given statement \(\hat{p}>0.50\), it seems that it's intended to be a hypothesis for a proportion, with the symbol \(\hat{p}\) typically representing a sample proportion. The problem here is the mention of a 'greater than' symbol. Hypotheses should not contain inequalities; they should be clear, concise and exact. They typically equate the population parameter to a specific value. Therefore, the statement \(\hat{p}>0.50\) is not a clear-cut hypothesis.
03

Solution Explanation

Since the hypotheses should not contain inequalities but should equate to a specific value, the statement \(\hat{p}>0.50\) does not constitute a legitimate hypothesis. A correct null hypothesis might be H0: \(\hat{p}=0.50\), and a corresponding alternative could be H1: \(\hat{p}\neq 0.50\). These hypotheses encompass all possible conditions - p is either 0.50, or it's not - thus meeting the criteria for legitimate hypotheses.

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Most popular questions from this chapter

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 1,000 American adults, 700 indicated that they oppose the reinstatement of a military draft. Suppose you want to use this information to decide if there is convincing evidence that the proportion of American adults who oppose reinstatement of the draft is greater than two-thirds. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is \(0.013 .\) What conclusion would you reach if \(\alpha=0.05 ?\)

In a survey of 1,000 women ages 22 to 35 who work full-time, 540 indicated that they would be willing to give up some personal time in order to make more money (USA Today, March 4, 2010). The sample was selected to be representative of women in the targeted age group. a. Do the sample data provide convincing evidence that a majority of women ages 22 to 35 who work fulltime would be willing to give up some personal time for more money? Test the relevant hypotheses using \(\alpha=0.01\) b. Would it be reasonable to generalize the conclusion from Part (a) to all working women? Explain why or why not.

One type of error in a hypothesis test is failing to reject a false null hypothesis. What is the other type of error that might occur when a hypothesis test is carried out?

For which of the following \(P\) -values will the null hypothesis be rejected when performing a test with a significance level of \(0.05 ?\) a. 0.001 d. 0.047 b. 0.021 e. 0.148 c. 0.078

A television station has been providing live coverage of a sensational criminal trial. The station's program director wants to know if more than half of potential viewers prefer a return to regular daytime programming. A survey of randomly selected viewers is conducted. With \(p\) representing the proportion of all viewers who prefer regular daytime programming, what hypotheses should the program director test?
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