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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 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data supports 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
Based on the calculation, for scenario 'a' we reject the null hypothesis indicating mean pen lifetime is less than 10 hours. However, for scenarios 'b' and 'c', we fail to reject the null hypothesis due to insufficient evidence that the mean pen lifetime is under 10 hours.

Step by step solution

01

Understanding the Hypothesis Test

The objective of this test is to check the validity of two opposite hypotheses. Here \(H_{0}: \mu=10\) represents null hypothesis suggesting that the mean pen lifetime is equal to 10 hours. The alternative hypothesis \(H_{a}: \mu<10\) suggests that the mean pen lifetime is less than 10 hours. The t-values provided are the test statistics.
02

Finding Critical Value for a

The critical value for the one-tailed t test can be found in the t-table at (n - 1) degree of freedom. In this case, degree of freedom is (18 - 1) = 17. If not found directly in the table, we should take the closest lower degree of freedom. For the alpha level \(\alpha=0.05\), the critical value is approximately -1.740 and for \(\alpha=0.01\) the critical value is approximately -2.567.
03

Comparison and Conclusion for a

Compare the given t-value with the calculated critical value. Here, the critical value is -1.740, and the t-value is -2.3. Since -2.3 is to the left of -1.740 on a normal curve, the result lies into the critical region. So, we conclude by rejecting the null hypothesis \(H_{0}\). It appears that the mean pen lifetime is less than 10 hours.
04

Comparison and Conclusion for b

Here, the critical value for \(\alpha=0.01\) is -2.567, and the given t-value is -1.83. Since -1.83 is to the right of -2.567 on a normal curve, the observed value falls in the acceptance region. Therefore, we fail to reject the null hypothesis \(H_{0}\) for this case. It cannot be stated that the mean pen lifetime is less than 10 hours.
05

Comparison and Conclusion for c

The t-value given here is 0.47 which lies in the acceptance region for any reasonable value of \(\alpha\). Therefore, we again fail to reject the null hypothesis \(H_{0}\). There is insufficient evidence to conclude that the mean pen lifetime is less than 10 hours.

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Key Concepts

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

Hypothesis Testing
To understand the essence of a one-sample t-test, one must first grasp the concept of hypothesis testing. Hypothesis testing is a method used in statistics to decide between two competing hypotheses regarding a population parameter, based on sampled data.

In the context provided, the objective was to determine whether the average writing lifetime of a pen is at least 10 hours. To accomplish this, the exercise takes us through a process where we have a null hypothesis (\(H_0\)) that asserts the pen’s lifetime is 10 hours, and an alternative hypothesis (\(H_a\)) that it is less than 10 hours. The verification of these claims is done by using the test statistic, in this case, a t-value, and comparing it to a critical value that corresponds with a pre-chosen level of significance, denoted by \(\alpha\).
  • If the test statistic falls into the critical region (usually a portion of the t-distribution tails), the null hypothesis is rejected.
  • If the test statistic falls outside the critical region, the null hypothesis is not rejected.
Statistical Significance
Statistical significance is a term that helps us determine if the result of a hypothesis test is not due to random chance. In the presented exercise, this is analyzed by setting a significance level noted as \(\alpha\).

The significance level, often chosen as 0.05 or 0.01, represents a threshold beyond which we may conclude the observed data would be very unlikely if the null hypothesis were true.
  • For \(\alpha = 0.05\), we are saying there is a 5% risk we are rejecting the null hypothesis when it is actually true (Type I error).
  • For a more stringent \(\alpha = 0.01\), the risk is reduced to 1%.
Choosing a suitable \(\alpha\) level is critical as it affects whether we should reject or not reject the null hypothesis, as seen in the various scenarios of the exercise.
Null Hypothesis
The null hypothesis (\(H_0\)) plays a pivotal role in hypothesis testing. It is a statement about the population parameter that indicates no effect, no difference, or a state of skepticism. Essentially, it's the default assumption to test against.

In the exercise we've observed, the null hypothesis was that the mean writing lifetime of the pens is exactly 10 hours (\(H_0: \mu = 10\)). Unless the sample data provides sufficient evidence to the contrary, we do not discard this assumption. The null hypothesis thus provides a benchmark for comparing the evidence provided by the sample data. If the evidence is strong enough to prove that the parameter is different from the value stated in \(H_0\), only then we consider rejecting \(H_0\).
Alternative Hypothesis
Complementing the null hypothesis is the alternative hypothesis (\(H_a\)), which represents what we aim to support by the test. It's a statement that contradicts the null hypothesis, and in this test scenario, it posits that the average writing lifetime of the pens is less than 10 hours (\(H_a: \mu < 10\)).

The alternative hypothesis becomes the focus if we find enough evidence to reject the null hypothesis. In the given exercise, when the test statistic was -2.3 with \(\alpha = 0.05\), we had sufficient evidence to support \(H_a\) and concluded that the writing lifetime is indeed less than 10 hours. Alternatively, for other given \(t\)-values with different \(\alpha\) levels, we did not find enough evidence to reject \(H_0\) and hence, could not substantiate \(H_a\).

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Most popular questions from this chapter

In a survey conducted by CareerBuilder.com, employers were asked if they had ever sent an employee home because they were dressed inappropriately (June 17 . 2008 , www.careerbuilder.com). A total of 2765 employers responded to the survey, with 968 saying that they had sent an employee home for inappropriate attire. In a press release, CareerBuilder makes the claim that more than one- third of employers have sent an employee home to change clothes. Do the sample data provide convincing evidence in support of this claim? Test the relevant hypotheses using \(\alpha=.05 .\) For purposes of this exercise, assume that it is reasonable to regard the sample as representative of employers in the United States.

The city council in a large city has become concerned about the trend toward exclusion of renters with children in apartments within the city. The housing coordinator has decided to select a random sample of 125 apartments and determine for each whether children are permitted. Let \(p\) be the proportion of all apartments that prohibit children. If the city council is convinced that \(p\) is greater than 0.75 , it will consider appropriate legislation. a. If 102 of the 125 sampled apartments exclude renters with children, would a level .05 test lead you to the conclusion that more than \(75 \%\) of all apartments exclude children? b. What is the power of the test when \(p=.8\) and \(\alpha=.05 ?\)

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.

Ann Landers, in her advice column of October 24,1994 (San Luis Obispo Telegram-Tribune), described the reliability of DNA paternity testing as follows: "To get a completely accurate result, you would have to be tested, and so would (the man) and your mother. The test is \(100 \%\) accurate if the man is not the father and \(99.9 \%\) accurate if he is." a. Consider using the results of DNA paternity testing to decide between the following two hypotheses: \(H_{0}:\) a particular man is the father \(H_{a}:\) a particular man is not the father In the context of this problem, describe Type I and Type II errors. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) b. Based on the information given, what are the values of \(\alpha,\) the probability of a Type I error, and \(\beta,\) the probability of a Type II error? c. Ann Landers also stated, "If the mother is not tested, there is a \(0.8 \%\) chance of a false positive." For the hypotheses given in Part (a), what is the value of \(\beta\) if the decision is based on DNA testing in which the mother is not tested?

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.

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