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Chapter 9: Problem 49

Consider the test of \(\mathrm{H}_{0}:\) The new drug is safe against \(\mathrm{H}_{a}:\) the new drug is not safe. a. Explain, in context, the conclusion of the test if \(\mathrm{H}_{0}\) is rejected. b. Describe, in context, a Type I error. c. Explain, in context, the conclusion of the test if you fail to reject \(\mathrm{H}_{0}\) d. Describe, in context, a Type II error.

Short Answer

Expert verified
Rejecting $H_{0}$ means the drug is not safe. A Type I error misidentifies a safe drug as unsafe. Not rejecting $H_{0}$ suggests insufficient evidence of unsafe. A Type II error fails to identify a truly unsafe drug.

Step by step solution

01

Understanding Hypotheses

In hypothesis testing, we have two hypotheses: the null hypothesis ( $ $H _{0}$) and the alternative hypothesis ( $ H _{a}$). Here, $H_{0}$ states that the new drug is safe. The alternative hypothesis, $H_{a}$, claims that the drug is not safe.
02

Conclusion if $H_{0}$ is Rejected

If $H_{0}$ is rejected, it indicates there is sufficient evidence against the claim that the new drug is safe. Thus, the conclusion would be that the drug is not considered safe based on the test results.
03

Definition and Context of Type I Error

A Type I error occurs when the null hypothesis is true, but we incorrectly reject it. In this context, a Type I error means the test concludes that the drug is not safe when, in fact, it is safe.
04

Conclusion if $H_{0}$ is Not Rejected

If we fail to reject $H_{0}$, it indicates there is not enough evidence to support the claim that the drug is unsafe. Therefore, the test does not provide a basis to consider the drug unsafe, suggesting it might be safe.
05

Definition and Context of Type II Error

A Type II error happens when the null hypothesis is false, but we fail to reject it. In this case, a Type II error means the test fails to conclude that the drug is not safe when, in reality, it is unsafe.

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Key Concepts

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

Type I error
A Type I error occurs when we mistakenly reject a true null hypothesis. Imagine you are concluding a test for a new drug trial. You decide the drug is unsafe contrary to its actual safety. This erroneous conclusion is a Type I error. Here, the null hypothesis (which claims the drug is safe) is true, yet you reject it.

This error type is often termed a "false positive". It can lead to unnecessary alarm or discontinuation of beneficial treatments.
  • Avoiding Type I errors is crucial, hence the need for stringent significance levels in tests.
  • Significance level, denoted by alpha (b03b1), represents the probability of committing a Type I error.
Understanding this concept helps in designing experiments with proper safeguards against erroneous rejections.
Type II error
On the other hand, a Type II error happens when we fail to reject a false null hypothesis. In the context of our drug test, it means concluding the drug is safe when, in reality, it isn't. Essentially, the test misses a real issue, thus isn't catching a "false negative".

Avoiding Type II errors is vital because we might overlook potential risks associated with a drug.
  • The probability of making a Type II error is represented by beta (b2).
  • Power of a test (1 - beta) indicates the test's efficacy in correctly rejecting a false null hypothesis.
Balancing the risk of Type I and Type II errors is a key aspect of effective hypothesis testing.
null hypothesis
The null hypothesis acts as a starting point in hypothesis testing, presenting an assumption to be tested. In our case, the null hypothesis (b0H_{0}) suggests that the new drug is safe.

This claim serves as the default statement assumed to be true until evidence shows otherwise.
  • It's essential in testing as it provides a basis against which the evidence is compared.
  • Failing to reject the null hypothesis doesn't necessarily prove it's true; it only means the current test lacks evidence to disprove it.
Understanding the role of the null hypothesis clarifies why testing either supports or doesn't offer enough evidence to topple its stance.
alternative hypothesis
The alternative hypothesis (b0H_{a}) is a statement directly opposing the null hypothesis. In the scenario of our drug trial, it claims that the drug is not safe.

This hypothesis is crucial as it embodies the assertion to be proved through evidence.
  • When data backs the alternative hypothesis enough to reject the null hypothesis, it's an indication that the drug might indeed be unsafe.
  • The alternative hypothesis acts as the beacon for researchers, guiding towards what they aim to demonstrate through their study.
Through understanding both null and alternative hypotheses, one grasps the investigative framework within hypothesis testing.

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Most popular questions from this chapter

\(z\) test statistic To test \(\mathrm{H}_{0}: p=0.50\) that a population proportion equals 0.50 , the test statistic is a \(z\) -score that measures the number of standard errors between the sample proportion and the \(\mathrm{H}_{0}\) value of \(0.50 .\) If \(z=3.6,\) do the data support the null hypothesis, or do they give strong evidence against it? Explain.

A customer of a car workshop claimed that majority of customers were not satisfied with the services provided. In order to test this claim, officials in charge of the workshop delegated a third-party statistical company to administrate a satisfaction survey of its current customers. State the parameter of interest and the hypotheses for a significance test for testing this claim, where the alternative hypothesis will reflect the customer's claim.

A marketing study conducts 60 significance tests about means and proportions for several groups. Of them, 3 tests are statistically significant at the 0.05 level. The study's final report stresses only the tests with significant results, not mentioning the other 57 tests. What is misleading about this?

Practice mechanics of a \(t\) test A study has a random sample of 20 subjects. The test statistic for testing \(\mathrm{H}_{0}: \mu=100\) is \(t=2.40 .\) Find the approximate P-value for the alternative, (a) \(\mathrm{H}_{a}: \mu \neq 100\) (b) \(\mathrm{H}_{a}: \mu>100,\) and \((\mathrm{c}) \mathrm{H}_{a}: \mu<100\). (Among others, you can use the \(t\) Distribution web app to find the answer.)

A study (J Integr Med. \(2016 ; 14(2): 121-127)\) was conducted between May 2014 and April 2015 to assess the knowledge, attitude, and use of CIH strategies among nurses in Iran. In this study, 157 nurses from two urban hospitals of Zabol University of Medical Sciences in southeast Iran took part and their responses were analyzed. Most nurses \((n=95,60.5 \%)\) had some knowledge about the strategies. However, a majority \((n=90,57.3 \%)\) of the nurses never applied CIH methods. Does this suggest that nurses who never applied CIH methods would constitute a majority of the population, or are the results consistent with random variation? Answer by: a. Identifying the relevant variable and parameter. (Hint: The variable is categorical with two categories. The parameter is a population proportion for one of the categories.) b. Stating hypotheses for a large-sample two-sided test and checking that sample size guidelines are satisfied for that test. c. Finding the test statistic value. d. Finding and interpreting a P-value and stating the conclusion in context.
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