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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

An experiment was planned to compare the mean time (in days) to recover from a common cold for people given a daily dose of 4 milligrams (mg) of vitamin C versus those who were not. Suppose that 35 adults were randomly selected for each treatment category and that the mean recovery times and standard deviations for the two groups were as follows: $$ \begin{array}{lcc} \hline & \begin{array}{l} \text { No Vitamin } \\ \text { Supplement } \end{array} & \begin{array}{c} 4 \mathrm{mg} \\ \text { Vitamin C } \end{array} \\ \hline \text { Sample Size } & 35 & 35 \\ \text { Sample Mean } & 6.9 & 5.8 \\ \text { Sample Standard Deviation } & 2.9 & 1.2 \end{array} $$ a. If you want to show that the use of vitamin \(\mathrm{C}\) reduces the mean time to recover from a common cold, give the null and alternative hypotheses for the test. Is this a one- or a two-tailed test? b. Conduct the statistical test of the null hypothesis in part a and state your conclusion. Test using \(\alpha=.05\)

What is normal, when it comes to people's body temperatures? A random sample of 130 human body temperatures, provided by Allen Shoemaker in the Journal of Statistical Education, had a mean of \(98.25^{\circ} \mathrm{F}\) and a standard deviation of \(0.73^{\circ} \mathrm{F}\). Does the data indicate that the average body temperature for healthy humans is different from \(98.6^{\circ} \mathrm{F}\), the usual average temperature cited by physicians and others? a. Test using the \(p\) -value approach with \(\alpha=.05\). b. Test using the critical value approach with \(\alpha=.05\). c. Compare the conclusions from parts a and b. Are they the same? d. The 98.6 standard was derived by a German doctor in 1868 , who claimed to have recorded 1 million temperatures in the course of his research. \({ }^{5}\) What conclusions can you draw about his research in light of your conclusions in parts a and b?

An experimenter has prepared a drug-dose level that he claims will induce sleep for at least \(80 \%\) of people suffering from insomnia. After examining the dosage we feel that his claims regarding the effectiveness of his dosage are too high. In an attempt to disprove his claim, we administer his prescribed dosage to 50 insomniacs and observe that 37 of them have had sleep induced by the drug dose. Is there enough evidence to refute his claim at the \(5 \%\) level of significance?

A drug manufacturer claimed that the mean potency of one of its antibiotics was \(80 \%\). A random sample of \(n=100\) capsules was tested and produced a sample mean of \(\bar{x}=79.7 \%\) with a standard deviation of \(s=.8 \%\). Do the data present sufficient evidence to refute the manufacturer's claim? Let \(\alpha=.05\) a. State the null hypothesis to be tested. b. State the alternative hypothesis. c. Conduct a statistical test of the null hypothesis and state your conclusion.

What is the power of a test and how is it related to \(\beta ?\)
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