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Chapter 19: Problem 6

Which of the following statements are true? If false, explain briefly. a. It is better to use an alpha level of 0.05 than an alpha level of 0.01 . b. If we use an alpha level of 0.01 , then a P-value of 0.001 is statistically significant. c. If we use an alpha level of \(0.01,\) then we reject the null hypothesis if the \(\mathrm{P}\) -value is 0.001 d. If the P-value is 0.01 , we reject the null hypothesis for any alpha level greater than 0.01 .

Short Answer

Expert verified
Statement a is False because choosing an alpha level depends on the context and cannot universally be determined as 'better'. Statements b, c, and d are True as they align with the correct interpretation of P-values and alpha levels in hypothesis testing

Step by step solution

01

Analysis of Statement a

The statement implies that an alpha level of 0.05 is superior to 0.01. This is subjective and depends on the context or the nature of the study. Having a higher alpha level (0.05) increases the chance of rejecting the null hypothesis (i.e., higher chance of making a Type I Error), which could be risky in certain contexts. Conversely, a lower alpha (0.01) reduces the chance of making a Type I Error but increases the chance of a Type II Error (failing to reject a false null hypothesis). Therefore, one cannot decisively say 'it's better' without specific context.
02

Analysis of Statement b

This statement is True. If the alpha level is set at 0.01, then a P-value of 0.001 (which is less than the alpha) would be considered statistically significant, leading to the rejection of the null hypothesis.
03

Analysis of Statement c

This statement is True. If the alpha level is 0.01, a P-value of 0.001, being less than the alpha level, will lead to the rejection of the null hypothesis. Hence, we reject the null hypothesis if the P-value is 0.001.
04

Analysis of Statement d

This statement is True. If the P-value is 0.01, then any alpha level greater than this P-value (i.e. 0.02, 0.05, 0.1, etc.) will lead to the rejection of the null hypothesis, as the condition for rejecting the null hypothesis is when the P-value is less than or equal to the alpha level. Hence, if the alpha level is higher than the P-value, the null hypothesis would be rejected.

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Key Concepts

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

Alpha Level
In hypothesis testing, the alpha level, often denoted as \( \alpha \), is the threshold for statistical significance. It signifies the probability of making a Type I Error. This is when you reject a true null hypothesis. For example, an alpha level of 0.05 means you are willing to risk a 5% chance of wrongly rejecting the null hypothesis.

Different alpha levels come with different risks:
  • A higher alpha level (like 0.10) increases the chance of making a Type I Error, meaning you might conclude a finding is significant when it isn鈥檛.
  • A lower alpha level (like 0.01) reduces this chance, but it might increase the likelihood of a Type II Error, missing a significant finding.
Choosing the right alpha level depends on the context of your study and how you balance the risks of different errors.
P-value
The P-value is a crucial part of hypothesis testing. It tells us the probability of observing our data, or something more extreme, under the assumption that the null hypothesis is true.

Here's how you can interpret the P-value in the context of your alpha level:
  • If the P-value is less than or equal to the alpha level, you reject the null hypothesis because the observed data is unlikely under the null hypothesis.
  • If the P-value is greater than the alpha level, you fail to reject the null hypothesis, meaning the data is not surprising enough to support a conclusion of statistical significance.
A smaller P-value indicates stronger evidence against the null hypothesis. If you have an alpha level of 0.01, a P-value of 0.001 leads to rejection of the null hypothesis because it is sufficiently small.
Type I Error
A Type I Error happens when you wrongly reject a true null hypothesis. The alpha level directly relates to the probability of committing a Type I Error.

To think about it simply:
  • An alpha level of 0.05 means there is a 5% chance you will incorrectly reject the null hypothesis.
  • An alpha level of 0.01 decreases this risk to 1%.
Balancing the risk of a Type I Error is crucial in planning your study and setting your alpha level. High-stakes research often opts for a smaller alpha level to avoid false positives, which could lead to invalid conclusions or costly mistakes.
Type II Error
A Type II Error occurs when you fail to reject a false null hypothesis. This means that you miss a truly significant effect. While the alpha level dictates the chance of a Type I Error, it indirectly affects the rate of Type II Errors.

Consider these points:
  • A stricter alpha level (e.g., 0.01) makes it harder to reject the null hypothesis, which can increase the likelihood of a Type II Error.
  • A higher alpha level (e.g., 0.10) might reduce Type II Errors but increase Type I Errors.
Researchers need to evaluate the trade-offs between these errors critically. An adequate sample size can also reduce Type II Errors, giving enough statistical power to detect true effects.

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Most popular questions from this chapter

A clean air standard requires that vehicle exhaust emissions not exceed specified limits for various pollutants. Many states require that cars be tested annually to be sure they meet these standards. Suppose state regulators double-check a random sample of cars that a suspect repair shop has certified as okay. They will revoke the shop's license if they find significant evidence that the shop is certifying vehicles that do not meet standards. a. In this context, what is a Type I error? b. In this context, what is a Type II error? c. Which type of error would the shop's owner consider more serious? d. Which type of error might environmentalists consider more serious?

Highway safety engineers test new road signs, hoping that increased reflectivity will make them more visible to drivers. Volunteers drive through a test course with several of the new- and old-style signs and rate which kind shows up the best. a. Is this a one-tailed or a two-tailed test? Why? b. In this context, what would a Type I error be? c. In this context, what would a Type II error be? d. In this context, what is meant by the power of the test? e. If the hypothesis is tested at the \(1 \%\) level of significance instead of \(5 \%\), how will this affect the power of the test? f. The engineers hoped to base their decision on the reactions of 50 drivers, but time and budget constraints may force them to cut back to 20 . How would this affect the power of the test? Explain.

In January \(2016,\) at the end of his time in office, President Obama's approval rating stood at \(57 \%\) in Gallup's daily tracking poll of 1500 randomly surveyed U.S. adults. (www.gallup.com/poll/113980/gallup-daily-obama- jobapproval.aspx) a. Make a \(95 \%\) confidence interval for his approval rating by all U.S. adults. b. Based on the confidence interval, test the null hypothesis that Obama's approval rating was essentially the same as his approval rating of \(52 \%\) when he was elected to his second term.

You are in charge of shipping computers to customers. You learn that a faulty chip was put into some of the machines. There's a simple test you can perform, but it's not perfect. All but \(4 \%\) of the time, a good chip passes the test, but unfortunately, \(35 \%\) of the bad chips pass the test, too. You have to decide on the basis of one test whether the chip is good or bad. Make this a hypothesis test. a. What are the null and alternative hypotheses? b. Given that a computer fails the test, what would you decide? What if it passes the test? c. How large is \(\alpha\) for this test? d. What is the power of this test?

Testing for Alzheimer's disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. A group of researchers (Solomon et al., 1998 ) devised a 7-minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false-positive rate of \(4 \%\) and a false-negative rate of \(8 \%\). a. Put this in the context of a hypothesis test. What are the null and alternative hypotheses? b. What would a Type I error mean? c. What would a Type II error mean? d. Which is worse here, a Type I or Type II error? Explain. e. What is the power of this test?
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