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Chapter 8: Problem 4

Consider a test for \(\mu\). If the \(P\) -value is such that you can reject \(H_{0}\) at the \(5 \%\) level of significance, can you always reject \(H_{0}\) at the \(1 \%\) level of significance? Explain.

Short Answer

Expert verified
No, rejecting \(H_0\) at 5% does not guarantee rejection at 1% unless the \(P\)-value is also below 0.01.

Step by step solution

01

Understanding the Hypotheses

Let's begin by defining the problem. We have a null hypothesis, denoted as \(H_0\), and are considering an alternative hypothesis, \(H_a\). The exercise asks us what happens when the \(P\)-value allows us to reject \(H_0\) at the 5% level and whether this is also true at the 1% level.
02

Recall Significance Levels

The significance level is the threshold at which we decide whether to reject the null hypothesis. At 5%, it means there's a 5% risk of rejecting \(H_0\) when it is actually true. Similarly, a 1% level is more stringent, allowing for only a 1% risk.
03

Comparing Significance Levels

If the \(P\)-value is smaller than 0.05, we reject \(H_0\) at the 5% level. However, to reject \(H_0\) at the 1% level, the \(P\)-value must be smaller than 0.01. Therefore, a \(P\)-value of, say, 0.04 leads to rejection at 5% but not at 1%.
04

Conclusion

You cannot always reject \(H_0\) at the 1% level if it can be rejected at the 5% level. It depends on whether the \(P\)-value is also smaller than 0.01. Therefore, rejection at 5% does not guarantee rejection at 1%.

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Key Concepts

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

P-value
When conducting a hypothesis test, the P-value is an essential figure. It tells us the probability of observing a test result as extreme as, or more extreme than, the results seen in the data, under the assumption that the null hypothesis (\(H_0\)) is true. In simpler terms:
  • If the P-value is small, it suggests that the observed data is unlikely under \(H_0\).
  • This provides evidence against the null hypothesis.
Consider the P-value as a measure of how much evidence you have against \(H_0\). The smaller it gets, the more reason you have to reject \(H_0\) in favor of the alternative hypothesis.
Significance Levels
Choosing a significance level is crucial in hypothesis testing. It is a threshold set before conducting the test, which defines how much risk of error you're willing to accept. Common significance levels include 5% (0.05) and 1% (0.01). They mean:
  • 5%: There's a 5% chance of rejecting the null hypothesis when it's actually true, indicating a moderate threshold.
  • 1%: This stricter level reduces that chance to 1%, requiring stronger evidence against \(H_0\).
Lowering the significance level makes it harder to reject \(H_0\), providing more confidence when you do reject it, at the cost of potentially missing a true effect (i.e., making a type II error).
Null Hypothesis
The null hypothesis, denoted \(H_0\), is the starting assumption in hypothesis testing. It usually states that there is no effect or no difference. For example:
  • "The mean of the population is equal to a specified value."
  • "There is no difference between the two groups."
Hypothesis testing starts with the assumption that \(H_0\) is true. You collect data, calculate the P-value, and then decide whether the observed data is improbable enough to reject \(H_0\) based on your chosen significance level.
Alternative Hypothesis
The alternative hypothesis, represented as \(H_a\) or sometimes \(H_1\), is the opposite of the null hypothesis. It suggests that there is an effect or difference, and it's what researchers typically aim to support. For instance:
  • "The mean of the population is not equal to a specified value."
  • "There is a difference between the two groups."
During hypothesis testing, if the evidence strongly contradicts \(H_0\), the \(H_a\) is considered. Proving \(H_a\) effectively means: Disproving or rejecting \(H_0\) based on strong evidence from your data, thereby lending support to the alternative claim.

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Most popular questions from this chapter

What terminology do we use for the probability of rejecting the null hypothesis when it is, in fact, false?

Homser Lake, Oregon, has an Atlantic salmon catch and release program that has been very successful. The average fisherman's catch has been \(\mu=8.8\) Atlantic salmon per day (Source: National Symposium on Catch and Release Fishing, Humboldt State University). Suppose that a new quota system restricting the number of fishermen has been put into effect this season. A random sample of fishermen gave the following catches per day: $$\begin{array}{ccccccc} 12 & 6 & 11 & 12 & 5 & 0 & 2 \\ 7 & 8 & 7 & 6 & 3 & 12 & 12 \end{array}$$ i. Use a calculator with mean and sample standard deviation keys to verify that \(\bar{x}=7.36\) and \(s \approx 4.03.\) ii. Assuming the catch per day has an approximately normal distribution, use a \(5 \%\) level of significance to test the claim that the population average catch per day is now different from \(8.8 .\)

Snow avalanches can be a real problem for travelers in the western United States and Canada. A very common type of avalanche is called the slab avalanche. These have been studied extensively by David McClung, a professor of civil engineering at the University of British Columbia. Slab avalanches studied in Canada have an average thickness of \(\mu=67 \mathrm{cm}\) (Source: Avalanche Handbook by \(\mathrm{D.}\) McClung and \(\mathrm{P.}\) Schaerer). The ski patrol at Vail, Colorado, is studying slab avalanches in its region. A random sample of avalanches in spring gave the following thicknesses (in cm): \(\begin{array}{ccccccc}59 & 51 & 76 & 38 & 65 & 54 & 49 & 62\quad\end{array}\) \(\begin{array}{ccccc}64 & 67 & 63 & 74 & 65 & 79\quad\end{array}\) \(68 \quad 55\) i. Use a calculator with mean and standard deviation keys to verify that \(\bar{x}=61.8\) and \(s \approx 10.6 \mathrm{cm}.\) ii. Assume the slab thickness has an approximately normal distribution. Use a \(1 \%\) level of significance to test the claim that the mean slab thickness in the Vail region is different from that in Canada.

Discuss each of the following topics in class or review the topics on your own. Then write a brief but complete essay in which you answer the following questions. (a) What is a null hypothesis \(H_{0}\) ? (b) What is an alternate hypothesis \(H_{1} ?\) (c) What is a type I error? a type II error? (d) What is the level of significance of a test? What is the probability of a type II error?

When using a Student's \(t\) distribution for a paired differences test with \(n\) data pairs, what value do you use for the degrees of freedom?
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