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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 researcher developing scanners to search for hidden weapons at airports has concluded that a new device is significantly better than the current scanner. He made this decision based on a test using \(\alpha=0.05 .\) Would he have made the same decision at \(\alpha=0.10 ?\) How about \(\alpha=0.01 ?\) Explain.

For each of the following situations, state whether a Type I, a Type II, or neither error has been made. a. A test of \(\mathrm{H}_{0}: \mu=25\) vs. \(\mathrm{H}_{\mathrm{A}}: \mu>25\) rejects the null hypothesis. Later it is discovered that \(\mu=24.9\). b. A test of \(\mathrm{H}_{0}: p=0.8\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.8\) fails to reject the null hypothesis. Later it is discovered that \(p=0.9\). c. A test of \(\mathrm{H}_{0}: p=0.5\) vs. \(\mathrm{H}_{\mathrm{A}}: p \neq 0.5\) rejects the null hypothesis. Later it is discovered that \(p=0.65\). d. A test of \(\mathrm{H}_{0}: p=0.7\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.7\) fails to reject the null hypothesis. Later it is discovered that \(p=0.6\).

Yahoo surveyed 2400 U.S. men. 1224 of the men identified themselves as the primary grocery shopper in their household. a. Estimate the percentage of all American males who identify themselves as the primary grocery shopper. Use a \(98 \%\) confidence interval. Check the conditions first. b. A grocery store owner believed that only \(45 \%\) of men are the primary grocery shopper for their family, and targets his advertising accordingly. He wishes to conduct a hypothesis test to see if the fraction is in fact higher than \(45 \% .\) What does your confidence interval indicate? c. What is the level of significance of this test? Explain.

Environmentalists concerned about the impact of high-frequency radio transmissions on birds found that there was no evidence of a higher mortality rate among hatchlings in nests near cell towers. They based this conclusion on a test using \(\alpha=0.05\). Would they have made the same decision at \(\alpha=0.10 ?\) How about \(\alpha=0.01 ?\) Explain.

Have harsher penalties and ad campaigns increased seat-belt use among drivers and passengers? Observations of commuter traffic failed to find evidence of a significant change compared with three years ago. Explain what the study's P-value of 0.17 means in this context.
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