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Chapter 10: Problem 28

Give an example of a situation where you would want to select a small significance level.

Short Answer

Expert verified
A clinical trial conducted to test the safety and effectiveness of a new drug is an example of a situation requiring a small significance level, such as 0.01 or 0.001. In this scenario, the null hypothesis (H0) is that the drug has no significant effect, and the alternative hypothesis (H1) is that the drug has a significant effect on patients' conditions. A small significance level reduces the likelihood of making a type I error (falsely approving an ineffective or harmful drug), which can have severe consequences for patients.

Step by step solution

01

Identify the situation requiring a small significance level

A clinical trial conducted to test the safety and effectiveness of a new drug is a situation that would require a small significance level, as the consequences of making a type I error (falsely approving an ineffective or harmful drug) can have severe consequences for patients receiving the drug.
02

Determine the null and alternative hypotheses

In this scenario, the null hypothesis (H0) would be that the drug has no significant effect or improvement on the patients' condition. The alternative hypothesis (H1) would be that the drug has a significant effect on the patients' condition and is effective for treating the targeted ailment.
03

Explain why a small significance level is necessary

A small significance level is necessary in this situation because we want to minimize the probability of making a type I error. The smaller the significance level, the more stringent the criteria for rejecting the null hypothesis. Since approving an ineffective or potentially harmful drug can lead to severe consequences, it is crucial to minimize this type of error. Using a small significance level, such as 0.01 or 0.001, increases the certainty of our decision and reduces the likelihood of making a type I error. In conclusion, a clinical trial to test the safety and effectiveness of a new drug is an example of a situation where a small significance level is preferred to minimize the risk of false approval of an ineffective or harmful drug, which can have severe consequences for patients. The smaller the significance level, the more stringent the criteria for rejecting the null hypothesis, thus reducing the likelihood of making a type I error.

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Key Concepts

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

Clinical Trials
Clinical trials are a fundamental part of medical research, playing a crucial role in developing new treatments and ensuring patient safety. In these trials, researchers test new drugs or treatments to determine if they are safe and effective for human use. Clinical trials typically follow a structured process, consisting of several phases that aim to assess various aspects of the treatment, such as its safety, efficacy, and optimal dosage.
  • Phase 1: Tests the drug on a small group of people to evaluate its safety and determine a safe dosage range.
  • Phase 2: Expands the study to a larger group to assess the drug's effectiveness and continue safety evaluation.
  • Phase 3: Involves even larger groups to confirm the drug's effectiveness, monitor side effects, and compare it to commonly used treatments.
  • Phase 4: Takes place after the drug is approved. It gathers additional information on the drug's risks, benefits, and optimal use in various populations.
These trials must have strictly defined protocols to ensure reliable results. One of the key aspects is setting a significance level, which influences the threshold for message results as statistically significant. This is crucial to avoid errors that could lead to ineffective or harmful drugs being approved.
Type I Error
A Type I error, also known as a false positive, occurs when a study incorrectly rejects a true null hypothesis. In simpler terms, it means that the study suggests a treatment effect or difference exists, even though it does not. The probability of making a Type I error is denoted by the significance level (伪). For instance, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is none.
  • This type of error is particularly critical in clinical trials, as a Type I error implies that an ineffective or harmful treatment is considered beneficial.
  • Thus, a low significance level is often chosen to minimize this risk, ensuring that only truly effective treatments are approved.
Minimizing Type I errors is crucial to avoid the harmful consequences of approving unsafe drugs. Researchers must balance this with the risk of a Type II error, which is the opposite鈥攆ailing to identify a real effect.
Null Hypothesis
The null hypothesis (H0) is a fundamental concept in statistical hypothesis testing. It represents a default position that there is no effect or no difference. When conducting a clinical trial, the null hypothesis typically states that the new treatment has no effect on the patients compared to the standard treatment.
  • The aim of a clinical trial is to challenge this hypothesis by collecting evidence and statistically analyzing the data.
  • If the data provide enough evidence, the null hypothesis is rejected, suggesting that there might be a real effect of the treatment.
The null hypothesis is crucial in the structured approach of hypothesis testing as it helps focus the research on seeking proof of actual effects. By setting a small significance level, researchers ensure that the evidence required to reject the null hypothesis is substantial, thereby safeguarding against Type I errors.
Alternative Hypothesis
The alternative hypothesis (H1) is the counter-statement to the null hypothesis. In clinical trials, the alternative hypothesis suggests that the treatment under study does have a significant effect on the patients. This hypothesis provides the potential for discovery in the research, offering a contrast to the null hypothesis.
  • Researchers aim to gather sufficient data to support the alternative hypothesis, indicating that the treatment could be more effective than the status quo.
  • In hypothesis testing, rejecting the null hypothesis typically suggests acceptance of the alternative hypothesis, supporting an effect that warrants further consideration or validation.
In practice, by carefully setting the significance level, researchers determine how strongly the data must contradict the null hypothesis before embracing the alternative. This ensures a robust standard for claiming new findings, essential to scientific credibility and ethics in clinical trials.

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Most popular questions from this chapter

Refer to the instructions given prior to Exercise \(10.57 .\) The paper referenced in the previous exercise also reported that when each of the 1178 students who participated in the study was asked if he or she played video games at least once a day, 271 responded "yes." The researchers were interested in using this information to decide if there is convincing evidence that more than \(20 \%\) of students age 8 to 18 play video games at least once a day.

Which of the following specify legitimate pairs of null and alternative hypotheses? a. \(H_{0}: p=0.25 \quad H_{i}: p>0.25\) b. \(H_{0}: p<0.40 \quad H_{i}: p>0.40\) c. \(H_{0}: p=0.40 \quad H: p<0.65\) d. \(H_{0}: p \neq 0.50 \quad H_{i}: p=0.50\) e. \(H_{\mathrm{n}}: p=0.50 \quad H_{i}: p>0.50\) f. \(H_{0}: \hat{p}=0.25 \quad H_{e^{\prime}} \hat{p}>0.25\)

How accurate are DNA paternity tests? By comparing the DNA of the baby and the DNA of a man that is being tested, one maker of DNA paternity tests claims that their test is \(100 \%\) accurate if the man is not the father and \(99.99 \%\) accurate if the man is the father (IDENTIGENE, www.dnatesting .com/paternity-test-questions/paternity-test-accuracy/, retrieved November 16,2016 ). a. Consider using the result of this DNA paternity test to decide between the following two hypotheses: \(H_{0}:\) a particular man is not the father \(H:\) a particular man is the father In the context of this problem, describe Type I and Type II errors. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) b. Based on the information given, what are the values of \(\alpha\), the probability of a Type I error, and \(\beta\), the probability of a Type II error?

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample \(z\) test appropriate? a. \(H_{0}: p=0.2, n=25\) b. \(H_{0}: p=0.6, n=200\) c. \(H_{0}: p=0.9, n=100\) d. \(H_{0}: p=0.05, n=75\)

In a survey of 1005 adult Americans, \(46 \%\) indicated that they were somewhat interested or very interested in having web access in their cars (USA TODAY, May 1,2009 ). Suppose that the marketing manager of a car manufacturer claims that the \(46 \%\) is based only on a sample and that \(46 \%\) is close to half, so there is no reason to believe that the proportion of all adult Americans who want car web access is less than \(0.50 .\) Is the marketing manager correct in his claim? Provide statistical evidence to support your answer. For purposes of this exercise, assume that the sample can be considered representative of adult Americans.
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