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Chapter 8: Problem 48

Pairs of \(P\)-values and significance levels, \(\alpha\), are given. For each pair, state whether the observed \(P\)-value would lead to rejection of \(H_{0}\) at the given significance level. a. \(P\)-value \(=.084, \alpha=.05\) b. \(P\)-value \(=.003, \alpha=.001\) c. \(P\)-value \(=.498, \alpha=.05\) d. \(P\)-value \(=.084, \alpha=.10\) e. \(P\)-value \(=.039, \alpha=.01\) f. \(P\)-value \(=.218, \alpha=.10\)

Short Answer

Expert verified
Reject \( H_{0} \) only for pair d; do not reject for pairs a, b, c, e, f.

Step by step solution

01

Understand Hypothesis Testing

In hypothesis testing, a P-value is used to determine the strength of evidence against the null hypothesis, denoted as \( H_{0} \). The significance level, \( \alpha \), is the threshold for this decision-making. To reject \( H_{0} \), the \( P \)-value must be less than \( \alpha \).
02

Analyzing Pair a

For pair (a), the \( P \)-value is 0.084 and \( \alpha \) is 0.05. Since 0.084 is greater than 0.05, we do not reject \( H_{0} \).
03

Analyzing Pair b

For pair (b), the \( P \)-value is 0.003 and \( \alpha \) is 0.001. Since 0.003 is greater than 0.001, we do not reject \( H_{0} \).
04

Analyzing Pair c

For pair (c), the \( P \)-value is 0.498 and \( \alpha \) is 0.05. Since 0.498 is greater than 0.05, we do not reject \( H_{0} \).
05

Analyzing Pair d

For pair (d), the \( P \)-value is 0.084 and \( \alpha \) is 0.10. Since 0.084 is less than 0.10, we reject \( H_{0} \).
06

Analyzing Pair e

For pair (e), the \( P \)-value is 0.039 and \( \alpha \) is 0.01. Since 0.039 is greater than 0.01, we do not reject \( H_{0} \).
07

Analyzing Pair f

For pair (f), the \( P \)-value is 0.218 and \( \alpha \) is 0.10. Since 0.218 is greater than 0.10, we do not reject \( H_{0} \).

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Key Concepts

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

P-value
In hypothesis testing, the P-value is a crucial component that helps us draw conclusions from our data. Simply put, the P-value indicates the probability of observing data at least as extreme as the current sample, assuming the null hypothesis is true. It helps in determining how unexpected the observed data is. A smaller P-value generally indicates stronger evidence against the null hypothesis.

P-values are very useful because:
  • They quantify the evidence against the null hypothesis.
  • Help decide whether to reject or not reject the null hypothesis.
  • Are easy to compare with a predetermined significance level for decision making.
It's essential to remember that the P-value does not measure the size of an effect or the importance of a result, merely the evidence against the null hypothesis. Misinterpretations in this area can lead to incorrect conclusions.
Null Hypothesis
The null hypothesis, often represented as \( H_0 \), is a foundational concept in hypothesis testing. It acts as the starting assumption that there is no effect or no difference in the context of the study. For example, if you were testing a new drug's effect on recovery time, the null hypothesis might state that the drug has no impact on recovery time compared to the current standard.

Understanding the importance of the null hypothesis involves recognizing that:
  • It provides a baseline to test our assumptions against what might be expected purely by chance.
  • It's the hypothesis that we either reject or fail to reject based on the data analysis.
  • Failure to reject \( H_0 \) does not prove its truth, but rather indicates insufficient evidence to support an alternative hypothesis.
Armed with this knowledge, you can use the null hypothesis as a stalwart foundation for scientific reasoning, guiding the decision-making process with clarity and rigor.
Significance Level
The significance level, notated as \( \alpha \), is a user-defined threshold that determines the cutoff for rejecting the null hypothesis. Before testing begins, researchers set the \( \alpha \) level, which represents the probability of rejecting the null hypothesis when it is actually true, commonly known as a Type I error. Typical significance levels are 0.05, 0.01, or 0.10.

Here's why understanding \( \alpha \) is crucial:
  • A smaller \( \alpha \) means stricter criteria for rejecting \( H_0 \), reducing the risk of Type I error but increasing the risk of a Type II error (failing to reject a false \( H_0 \)).
  • It provides a concrete benchmark against which the P-value is compared.
  • Setting the right \( \alpha \) level involves balancing risk of error with the practical consequences of decisions based on the hypothesis test.
Ultimately, the significance level defines how confident you need to be about your results before rejecting the null hypothesis, making it an essential part of any hypothesis testing framework.

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Most popular questions from this chapter

The sample average unrestrained compressive strength for 45 specimens of a particular type of brick was computed to be \(3107 \mathrm{psi}\), and the sample standard deviation was 188 . The distribution of unrestrained compressive strength may be somewhat skewed. Does the data strongly indicate that the true average unrestrained compressive strength is less than the design value of 3200 ? Test using \(\alpha=.001\).

The melting point of each of 16 samples of a certain brand of hydrogenated vegetable oil was determined, resulting in \(\bar{x}=94.32\). Assume that the distribution of the melting point is normal with \(\sigma=1.20\). a. Test \(H_{0}: \mu=95\) versus \(H_{\mathrm{a}}: \mu \neq 95\) using a two- tailed level \(.01\) test. b. If a level \(.01\) test is used, what is \(\beta(94)\), the probability of a type II error when \(\mu=94\) ? c. What value of \(n\) is necessary to ensure that \(\beta(94)=.1\) when \(\alpha=.01\) ?

A regular type of laminate is currently being used by a manufacturer of circuit boards. A special laminate has been developed to reduce warpage. The regular laminate will be used on one sample of specimens and the special laminate on another sample, and the amount of warpage will then be determined for each specimen. The manufacturer will then switch to the special laminate only if it can be demonstrated that the true average amount of warpage for that laminate is less than for the regular laminate. State the relevant hypotheses, and describe the type I and type II errors in the context of this situation.

A plan for an executive travelers' club has been developed by an airline on the premise that \(5 \%\) of its current customers would qualify for membership. A random sample of 500 customers yielded 40 who would qualify. a. Using this data, test at level \(.01\) the null hypothesis that the company's premise is correct against the alternative that it is not correct. b. What is the probability that when the test of part (a) is used, the company's premise will be judged correct when in fact \(10 \%\) of all current customers qualify?

The article "Caffeine Knowledge, Attitudes, and Consumption in Adult Women" (J. of Nutrition Educ., 1992: 179-184) reports the following summary data on daily caffeine consumption for a sample of adult women: \(n=47\), \(\bar{x}=215 \mathrm{mg}, s=235 \mathrm{mg}\), and range \(=5-1176\). a. Does it appear plausible that the population distribution of daily caffeine consumption is normal? Is it necessary to assume a normal population distribution to test hypotheses about the value of the population mean consumption? Explain your reasoning. b. Suppose it had previously been believed that mean consumption was at most \(200 \mathrm{mg}\). Does the given data contradict this prior belief? Test the appropriate hypotheses at significance level . 10 and include a \(P\)-value in your analysis.
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