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A drug manufacturer claims that fewer than \(10 \%\) of patients who take its new drug for treating Alzheimer's disease will experience nausea. To test this claim, a significance test is carried out of $$ \begin{array}{l} H_{0}: p=0.10 \\ H_{a}: p<0.10 \end{array} $$ You learn that the power of this test at the \(5 \%\) significance level against the alternative \(p=0.08\) is 0.29 . (a) Explain in simple language what "power \(=0.29 "\) means in this setting. (b) You could get higher power against the same alternative with the same \(\alpha\) by changing the number of measurements you make. Should you make more measurements or fewer to increase power? Explain. (c) If you decide to use \(\alpha=0.01\) in place of \(\alpha=0.05\), with no other changes in the test, will the power increase or decrease? Justify your answer. (d) If you shift your interest to the alternative \(p=0.07\) with no other changes, will the power increase or decrease? Justify your answer.

Step by step solution

01

Understanding Power of the Test

**Power = 0.29** means that if the true proportion of patients who experience nausea is actually 0.08, then there is a 29% chance that the test will correctly reject the null hypothesis and conclude that the proportion is indeed less than 0.10.

02

More Measurements to Increase Power

To increase the power of the test, more measurements (or a larger sample size) should be made. Increasing the sample size will enhance the test's ability to detect a true effect (i.e., a proportion genuinely less than 0.10).

03

Effect of Lowering Significance Level

If you change the significance level from \( \alpha = 0.05 \) to \( \alpha = 0.01 \), the power of the test will decrease. This is because a smaller \( \alpha \) means a stricter criterion for rejecting the null hypothesis, which can increase the chance of a Type II error (failing to reject a false null).

04

Impact of Changing the Alternative Hypothesis

If you switch to the alternative hypothesis that \( p = 0.07 \), the power of the test will increase. This is because the true alternative value \( p = 0.07 \) is further from the null value \( p = 0.10 \), making it easier for the test to detect a difference and reject the null hypothesis.

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Key Concepts

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

Power of a Test

The power of a test is a key concept in significance testing. It refers to the probability that the test will correctly reject a false null hypothesis. In other words, it measures the likelihood of detecting a true effect when one exists.
For this exercise, **power = 0.29** indicates that if the true proportion of patients experiencing nausea is actually 0.08, there is a 29% chance that the test will correctly reject the null hypothesis and support the claim that the proportion is indeed less than 0.10.
A higher power means a more sensitive test that is more likely to detect differences from the null hypothesis. Increasing power is crucial because it reduces the chance of a Type II error, where a test fails to detect a real effect.

Sample Size and Power

Sample size plays a critical role in determining the power of a test. The larger the sample size, the greater the power, provided that all other factors remain the same. This is because a larger sample provides more information about the population, improving the test's ability to identify true effects.
In this particular scenario, we can increase the power by collecting more data or increasing the sample size. By doing so, the likelihood of correctly rejecting a false null hypothesis increases, making the hypothesis test more reliable.
Thus, if we are aiming for a more trustworthy conclusion about the drug's side effects, increasing the number of participants in the study would be beneficial.

Significance Level

The significance level, denoted as \( \alpha \), is the threshold used for deciding whether to reject the null hypothesis. It's the probability of making a Type I error, which means rejecting a true null hypothesis.
Choosing a significance level of \( \alpha = 0.05 \) means being willing to accept a 5% risk of incorrect rejection, whereas \( \alpha = 0.01 \) reduces this risk to 1%. However, a lower \( \alpha \) increases the risk of a Type II error, thus decreasing the test's power.
If we switch from \( \alpha = 0.05 \) to \( \alpha = 0.01 \), we would have a stricter criteria for rejecting the null hypothesis. This could lead to more false negatives, causing a decrease in the power of the test and making it less likely that true effects are detected.

Type II Error

A Type II Error occurs when a test fails to reject a false null hypothesis. This error is closely related to the power of a test; higher power means a lower chance of committing a Type II error.
In this context, if the actual proportion of patients that experience nausea is lower than claimed, we want to identify this accurately. If we shift our focus to a more extreme alternative, like \( p = 0.07 \), the test power increases because it becomes easier to distinguish it from the null hypothesis \( p = 0.10 \).
Thus, decreasing the chances of Type II error involves choosing suitable significance levels and maintaining an adequate sample size, which together contribute to a more powerful and effective test.