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Chapter 9: Problem 2

For a fixed sample size \(n\), what is the effect on \(\beta\) when \(\alpha\) is decreased?

Short Answer

Expert verified
Answer: When the sample size is fixed, decreasing the Type I error rate (伪) will result in an increase in the Type II error rate (尾), and a decrease in the power (1 - 尾) of the hypothesis test. This is because a more stringent test with a lower 伪 requires stronger evidence to reject the null hypothesis, increasing the chance of failing to reject a false null hypothesis (Type II error) and decreasing the power of the test.

Step by step solution

01

Define Type I error rate (伪) and Power (1 - 尾)

The Type I error rate (伪) is the probability of rejecting the null hypothesis when it is true. It represents the likelihood of making a false positive error in hypothesis testing. The power of a test (1 - 尾) is the probability of rejecting the null hypothesis when it is false. Specifically, power is the probability of correctly identifying a true effect and avoiding a Type II error (which occurs when we fail to reject a false null hypothesis).
02

Describe the relationship between 伪 and 尾

There is a trade-off between the Type I error rate (伪) and the power (1 - 尾) of a hypothesis test. In general, a decrease in the Type I error rate (伪) will lead to an increase in the Type II error rate (尾) and, consequently, a decrease in the power (1 - 尾) of the test for a fixed sample size.
03

Describe the effect of decreasing 伪 on 尾 when the sample size is fixed

When the sample size (n) is fixed, decreasing the Type I error rate (伪) will result in a corresponding increase in the Type II error rate (尾). As a result, the power (1 - 尾) of the test will decrease when 伪 is decreased for a fixed sample size. This occurs because by decreasing 伪, we are making the test more stringent, requiring stronger evidence to reject the null hypothesis. This increases the chance of failing to reject a false null hypothesis (Type II error), and thus, decreases the power of the test. In conclusion, when the sample size (n) is fixed, decreasing the Type I error rate (伪) will lead to an increase in the Type II error rate (尾) and a decrease in the power (1 - 尾) of the hypothesis test.

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Key Concepts

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

Type I Error Rate (伪)
Understanding the Type I error rate, denoted by the symbol \(\alpha\), is crucial in hypothesis testing. It represents the probability of incorrectly rejecting a true null hypothesis, thus committing a false positive. Imagine a new drug is being tested for efficacy, and the test concludes it works when, in fact, it does not; this would be a Type I error. The lower the value of \(\alpha\), the less likely you are to make this error; however, this conservatism comes at a cost: it can lead to a higher chance of committing a Type II error, especially if the sample size remains unchanged.
Power of a Test (1 - 尾)
The power of a test, expressed as \(1 - \beta\), is the probability of correctly rejecting a false null hypothesis. When a test has high power, it is more likely to detect an actual effect when there is one. Think of it as the sensitivity of the test. If the drug mentioned earlier truly works, a powerful test would correctly identify its effectiveness. The relationship between power and Type I error rate (\alpha) is inversely proportional in a fixed-sample-size scenario; decreasing \(\alpha\) often results in lower power, meaning a reduced likelihood of catching the mistake of failing to reject a false null hypothesis, which leads to a Type II error.
Sample Size (n)
Sample size () is a fundamental aspect of study design. It's the number of observations or data points used in a statistical test. The size of the sample has direct implications for the validity of results. Larger samples tend to provide more reliable estimates and more statistical power. However, in hypothesis testing, if the sample size is fixed, changes to error rates can affect the results. Balancing sample size with desired error rates and power is a key part of research methodology, making it essential to carefully choose a sample size that aligns with the study objectives and resource constraints.
Null Hypothesis
The null hypothesis, typically denoted as \(H_0\), is a statement of no effect or no difference that serves as a starting presumption in hypothesis testing. It's the skeptic's stance, assuming that any observed effects are due to chance rather than a real effect. In the context of the drug efficacy example, the null hypothesis would be that the drug has no effect. The goal of the test is to determine whether the data provide enough evidence to reject this null hypothesis in favor of the alternative hypothesis, which would claim that the drug does have an effect. The interplay between Type I and Type II errors revolves around whether to accept or reject this null hypothesis.
Statistical Significance
Statistical significance is a determination about the null hypothesis based on a pre-defined significance level (\alpha). It indicates whether the observed effect is likely to be due to an actual difference rather than random chance. When a result is statistically significant, it means that the data are unlikely to have occurred under the null hypothesis, and thus, the null hypothesis can be rejected. However, statistical significance does not guarantee that an effect is practically significant or that it carries real-world importance. It's simply a measure of confidence in the result, based on the chosen threshold for \(\alpha\).

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Most popular questions from this chapter

In Exercise \(13,\) we tried to prove that the national brand tasted better than the store brand. \({ }^{16}\) Perhaps, however, the store brand has a better taste than the national brand! If this is true, then the store brand should be judged as better more than \(50 \%\) of the time. a. State the null and alternative hypotheses to be tested. Is this a one- or a two-tailed test? b. Suppose that, of the 35 food categories used for the taste test, the store brand was found to be better than the national brand in six categories. Use this information to test the hypothesis in part a. Use \(\alpha=.01\). c. In the other 21 food comparisons in this experiment, the tasters could find no difference in taste between the store and national brands. What practical conclusions can you draw from this fact and from the two hypothesis tests in Exercises \(13(\mathrm{~b})\) and \(14(\mathrm{~b}) ?\)

Calculate the p-value for the hypothesis tests given. $$ n=1000 \text { and } x=279 \text { . You wish to show that } p<.3 . $$

A peony plant with red petals was crossed with another plant having streaky petals. A geneticist states that \(75 \%\) of the offspring resulting from this cross will have red flowers. To test this claim. 100 seeds from this cross were collected and germinated, and 58 plants had red petals. a. What hypothesis should you use to test the geneticist's claim? b. Calculate the test statistic and its \(p\) -value. Use the \(p\) -value to evaluate the statistical significance of the results at the \(1 \%\) level.

A random sample of \(n=35\) observations from a quantitative population produced a mean \(\bar{x}=2.4\) and a standard deviation of \(s=.29 .\) Your research objective is to show that the population mean \(\mu\) exceeds 2.3. Use this information to answer the questions. Locate the rejection region for the test using a \(5 \%\) significance level.

In a head-to-head taste test of storebrand foods versus national brands, Consumer Reports found that it was hard to find a taste difference in the two. \(^{16}\) If the national brand is indeed better than the store brand, it should be judged as better more than \(50 \%\) of the time. a. State the null and alternative hypothesis to be tested. Is this a one- or a two-tailed test? b. Suppose that, of the 35 food categories used for the taste test, the national brand was found to be better than the store brand in eight categories. Use this information to test the hypothesis in part a with \(\alpha=.01 .\) What practical conclusions can you draw from the results?
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