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Chapter 10: Problem 50

A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least \(10 \mathrm{hr}\). A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data support the use of a one-sample \(t\) test. The relevant hypotheses are \(H_{0}: \mu=10\) versus \(H_{a}: \mu<10\). a. If \(t=-2.3\) and \(\alpha=.05\) is selected, what conclusion is appropriate? b. If \(t=-1.83\) and \(\alpha=.01\) is selected, what conclusion is appropriate? c. If \(t=0.47\), what conclusion is appropriate?

Short Answer

Expert verified
a. The pen has an average writing lifetime of less than 10 hours. b. The pen has an average writing lifetime of 10 hours. c. The pen has an average writing lifetime of 10 hours.

Step by step solution

01

Understand the Hypotheses

In this exercise, the null hypothesis (\(H_{0}\)) states that the pen has an average writing lifetime of 10 hours (\(\mu = 10\)). The alternative hypothesis (\(H_{a}\)) states that the pen has an average writing lifetime of less than 10 hours (\(\mu < 10\)).
02

Determine Critical t Value

For a one-tail t test, we look up the critical t value according to the significance level (\(\alpha\)) in a t-distribution table. For \(\alpha = .05\), and degrees of freedom \(df = n-1 = 18 - 1 = 17\), the critical t value is approximately -1.7396. For \(\alpha= .01\), the critical t value is approximately -2.5669.
03

Compare Test Statistic with Critical t Value (a)

The t statistic calculated from the data is -2.3. We compare this against the critical t value for \(\alpha = .05\) which is -1.7396. Since -2.3 < -1.7396, we reject \(H_{0}\) and conclude that the pen has an average writing lifetime of less than 10 hours.
04

Compare Test Statistic with Critical t Value (b)

The t statistic calculated from the data is -1.83. We compare this against the critical t value for \(\alpha = .01\) which is -2.5669. Since -1.83 > -2.5669, we fail to reject \(H_{0}\) and assume that the pen has an average writing lifetime of 10 hours.
05

Compare Test Statistic with Critical t Value (c)

The t statistic calculated from the data is 0.47. Since this is greater than both of our previous critical t-values, we fail to reject \(H_{0}\) and assume that the pen has an average writing lifetime of 10 hours.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
In statistics, the null hypothesis (\( H_0 \)) is a statement that assumes there is no effect or no difference. Think of it as the position we start from, assuming nothing unusual is happening. In our specific problem, the null hypothesis is that the average writing lifetime of the pens is 10 hours (\( \mu = 10 \)).
This hypothesis serves as a reference point. We compare our sample data against this assumption, aiming to determine if there is strong enough evidence to reject it.
  • The null hypothesis is usually what researchers aim to disprove or find evidence against.
  • It is considered true until statistical evidence suggests otherwise.
The null hypothesis is essential because it provides a baseline for any testing procedure, allowing researchers to determine if their findings have statistical significance.
Alternative Hypothesis
The alternative hypothesis (\( H_a \)) is what researchers want to prove. It suggests there is an effect or difference present, contrary to the null hypothesis. In our case, the alternative hypothesis claims that the average writing lifetime of the pens is less than 10 hours (\( \mu < 10 \)).
This hypothesis opens the door to new findings and discoveries.
When there's enough evidence gathered from the data, the null hypothesis is rejected in favor of the alternative hypothesis.
  • The alternative hypothesis is what you accept when the null hypothesis is rejected.
  • It represents a deviation from the established norm presented in the null hypothesis.
Rejecting the null hypothesis with a high level of confidence allows researchers to adopt the alternative hypothesis with reasonable certainty.
Critical Value
The critical value is a threshold used in hypothesis testing. It helps decide whether to reject or fail to reject the null hypothesis. This value is determined by the significance level and the degrees of freedom in the data.
Think of the critical value as a borderline. If our test statistic falls beyond this boundary, it indicates that our observed data is far enough from the expected data under the null hypothesis.
In this problem, for a one-tailed test:
  • For a significance level (\( \alpha \)) of 0.05, the critical t value is approximately -1.7396.
  • For \( \alpha = 0.01 \), the critical t value is approximately -2.5669.
By comparing our test statistic to these critical values, we determine whether there's enough evidence to support the alternative hypothesis.
Significance Level
The significance level (\( \alpha \)) represents the probability of rejecting the null hypothesis when it is actually true. It's a threshold for determining whether the test statistic is extreme enough to reject the null hypothesis. This level is chosen before conducting the test.
In our scenario:
  • \( \alpha = 0.05 \) implies a 5% risk of concluding that a difference exists when there is none.
  • \( \alpha = 0.01 \) implies a 1% risk of making that error.
Choosing a smaller significance level reduces the chance of a false positive but increases the risk of a false negative.
This balance ensures that any changes are genuinely due to a significant effect rather than by random chance.

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Most popular questions from this chapter

A student organization uses the proceeds from a particular soft-drink dispensing machine to finance its activities. The price per can had been \(\$ 0.75\) for a long time, and the average daily revenue during that period had been \(\$ 75.00\). The price was recently increased to \(\$ 1.00\) per can. A random sample of \(n=20\) days after the price increase yielded a sample average daily revenue and sample standard deviation of \(\$ 70.00\) and \(\$ 4.20\), respectively. Does this information suggest that the true average daily revenue has decreased from its value before the price increase? Test the appropriate hypotheses using \(\alpha=.05\).

When a published article reports the results of many hypothesis tests, the \(P\) -values are not usually given. Instead, the following type of coding scheme is frequently used: \(^{*} p<.05,^{* *} p<.01,^{*} *^{*} p<.001, *\) Which of the symbols would be used to code for each of the following \(P\) -values? a. 037 c. \(.072\) b. \(.0026\) d. \(.0003\)

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.

Students at the Akademia Podlaka conducted an experiment to determine whether the Belgium-minted Euro coin was equally likely to land heads up or tails up. Coins were spun on a smooth surface, and in 250 spins, 140 landed with the heads side up (New Scientist, January 4 , 2002 ). Should the students interpret this result as convincing evidence that the proportion of the time the coin would land heads up is not .5? Test the relevant hypotheses using \(\alpha=.01 .\) Would your conclusion be different if a significance level of \(.05\) had been used? Explain.

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). a. Is this conclusion consistent with testing \(H_{0}:\) concealed weapons laws reduce crime versus \(H_{a}\) : concealed weapons laws do not reduce crime or with testing \(H_{0}\) : concealed weapons laws do not reduce crime versus \(H_{e}:\) concealed weapons laws reduce crime Explain. b. Does the stated conclusion indicate that the null hypothesis was rejected or not rejected? Explain.
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