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Chapter 10: Problem 23

Use the definition of the \(P\) -value to explain the following: a. Why \(H_{0}\) would certainly be rejected if \(P\) -value \(=.0003\) b. Why \(H_{0}\) would definitely not be rejected if \(P\) -value \(=\) \(.350\)

Short Answer

Expert verified
a. \(H_{0}\) is rejected as P-value 0.0003 < \(\alpha\), indicating that the observed data is extremely unlikely given \(H_{0}\).\nb. \(H_{0}\) is not rejected as P-value 0.350 > \(\alpha\), showing that the observed data could be due to chance and is not significantly contradictory to \(H_{0}\).

Step by step solution

01

Understanding P-value

The P-value is a statistical measurement that helps us decide whether the observed data contradict the null hypothesis \(H_{0}\) or not. It's the probability of obtaining the observed data (or something more extreme) assuming the null hypothesis is true.
02

Relating P-value to Hypothesis Rejection

Typically, there's a level of significance, often denoted as \(\alpha\) (e.g., 0.05). If the P-value is less than \(\alpha\), \(H_{0}\) is rejected, indicating that the observed data is unlikely under the null hypothesis. If the P-value is greater than or equal to \(\alpha\), \(H_{0}\) is not rejected, suggesting that the observed data is not contradictory to our null hypothesis.
03

Explanation for Subpoint a

Given the P-value as 0.0003, this value is typically much smaller than most levels of significance like 0.01 or 0.05. Hence, with a smaller P-value, the probability of obtaining such extreme data under the null hypothesis is very low. Therefore, \(H_{0}\) would certainly be rejected.
04

Explanation for Subpoint b

Given the P-value as 0.350, which is much larger than common levels of significance, there isn't strong evidence against \(H_{0}\). The observed data could likely have happened by chance under the null hypothesis, and there is not enough statistical evidence to reject \(H_{0}\).

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Most popular questions from this chapter

Medical personnel are required to report suspected cases of child abuse. Because some diseases have symptoms that mimic those of child abuse, doctors who see a child with these symptoms must decide between two competing hypotheses: \(H_{0}\) " symptoms are due to child abuse \(H_{a}:\) symptoms are due to disease (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) The article "Blurred Line Between Illness, Abuse Creates Problem for Authorities" (Macon Telegraph, February 28,2000 ) included the following quote from a doctor in Atlanta regarding the consequences of making an incorrect decision: "If it's disease, the worst you have is an angry family. If it is abuse, the other kids (in the family) are in deadly danger." a. For the given hypotheses, describe Type I and Type II errors. b. Based on the quote regarding consequences of the two kinds of error, which type of error does the doctor quoted consider more serious? Explain.

Are young women delaying marriage and marrying at a later age? This question was addressed in a report issued by the Census Bureau (Associated Press, June 8 , 1991). The report stated that in 1970 (based on census results) the mean age of brides marrying for the first time was \(20.8\) years. In 1990 (based on a sample, because census results were not yet available), the mean was \(23.9\). Suppose that the 1990 sample mean had been based on a random sample of size 100 and that the sample standard deviation was \(6.4\). Is there sufficient evidence to support the claim that in 1990 women were marrying later in life than in 1970 ? Test the relevant hypotheses using \(\alpha=.01\). (Note: It is probably not reasonable to think that the distribution of age at first marriage is normal in shape.)

Do state laws that allow private citizens to carry concealed weapons result in a reduced crime rate? The author of a study carried out by the Brookings Institution is reported as saying, "The strongest thing I could say is that \(\bar{I}\) don't see any strong evidence that they are reducing crime" (San Luis Obispo Tribune, January 23,2003 ).

The article "Theaters Losing Out to Living Rooms" (San Luis Obispo Tribune, June 17,2005 ) states that movie attendance declined in 2005 . The Associated Press found that 730 of 1000 randomly selected adult Americans preferred to watch movies at home rather than at a movie theater. Is there convincing evidence that the majority of adult Americans prefer to watch movies at home? Test the relevant hypotheses using a \(.05\) significance level.

Let \(\mu\) denote the true average diameter for bearings of a certain type. A test of \(H_{0}: \mu=0.5\) versus \(H_{a}: \mu \neq 0.5\) will be based on a sample of \(n\) bearings. The diameter distribution is believed to be normal. Determine the value of \(\beta\) in each of the following cases: a. \(n=15, \alpha=.05, \sigma=0.02, \mu=0.52\) b. \(n=15, \alpha=.05, \sigma=0.02, \mu=0.48\) c. \(n=15, \alpha=.01, \sigma=0.02, \mu=0.52\) d. \(n=15, \alpha=.05, \sigma=0.02, \mu=0.54\) e. \(n=15, \alpha=.05, \sigma=0.04, \mu=0.54\) f. \(n=20, \alpha=.05, \sigma=0.04, \mu=0.54\) g. Is the way in which \(\beta\) changes as \(n, \alpha, \sigma\), and \(\mu\) vary consistent with your intuition? Explain.
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