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Researchers at the University of Washington and Harvard University analyzed records of breast cancer screening and diagnostic evaluations ("Mammogram Cancer Scares More Frequent than Thought," USA Today, April 16, 1998). Discussing the benefits and downsides of the screening process, the article states that, although the rate of false-positives is higher than previously thought, if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise. Suppose that such a screening test is used to decide between a null hypothesis of \(H_{0}:\) no cancer is present and an alternative hypothesis of \(H_{a}:\) cancer is present. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. Would a false-positive (thinking that cancer is present when in fact it is not) be a Type I error or a Type II error? b. Describe a Type I error in the context of this problem, and discuss the consequences of making a Type I error. c. Describe a Type II error in the context of this problem, and discuss the consequences of making a Type II error. d. What aspect of the relationship between the probability of Type I and Type II errors is being described by the statement in the article that if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise?

Short Answer

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a. A false-positive is a Type I error. b. A Type I error in this context is diagnosing a patient with cancer when they don't have it, leading to unnecessary stress and treatments. c. A Type II error is failing to diagnose a patient who has cancer leading to late treatment. d. The statement from the article illustrates the trade-off between the probabilities of Type I and Type II errors i.e. decreasing false-positives (Type I errors) may increase missed diagnoses (Type II errors).

Step by step solution

01

Identification of Type of Error

A false-positive error means thinking that cancer is present when in fact it is not. This corresponds to rejecting the null hypothesis of 'no cancer is present' when it is actually true, hence a false-positive is a Type I error.
02

Describe Type I Error and Its Consequences

A Type I error in this context is when the screening test indicates that a patient has cancer (rejecting the null hypothesis), when in fact, they do not have cancer (i.e., the null hypothesis is true). The consequence of a Type I error in this instance would be the unnecessary stress and possibly additional invasive tests and treatments for a person who does not have cancer.
03

Describe Type II Error and Its Consequences

A Type II error, on the other hand, would occur if the test indicates that a patient does not have cancer (not rejecting the null hypothesis), when in reality, they do have cancer (i.e., the null hypothesis is false). The consequence of a Type II error could be far more dangerous, as it could potentially lead to a late diagnosis, resulting in a delay in necessary treatments.
04

Discussing Probability of Error Types Trade-Off

The statement from the article suggests that if doctors become less aggressive in following up on suspicious tests, the frequency of Type I errors (false-positives) would decrease; however, the frequency of Type II errors (missed cancer diagnoses) would increase. This statement is illustrating the trade-off between the probability of Type I and Type II errors; as steps are taken to decrease one type of error, it may lead to an increase in the other type of error.

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Key Concepts

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

Type I Error
In the context of hypothesis testing, a Type I error occurs when the null hypothesis is rejected even though it is true. For medical tests, like breast cancer screenings, a Type I error is also known as a false-positive.
This means the test result suggests a problem (cancer) that doesn't actually exist. In this scenario, the patient is told they might have cancer, but in reality, they do not.
Such an error can cause unnecessary fear and anxiety, leading to further testing, which can often be invasive or stressful.
  • Patients might undergo unnecessary treatment.
  • This could lead to avoidable expenses and physical discomfort.
Understanding the impacts of a Type I error is crucial in developing testing methods and communicating results with patients.
Type II Error
A Type II error happens when the test fails to reject the null hypothesis when the alternative hypothesis is true. In simpler terms, a Type II error is a false-negative.
In the breast cancer screening example, this error means the test fails to detect cancer when it is indeed present in the patient.
Missing a diagnosis could have severe consequences because the patient does not receive timely treatment, potentially worsening their health condition.
  • Cancer remains untreated, which can affect prognosis.
  • There may be a delay in the patient receiving necessary care or follow-up.
The importance of being aware of the potential for a Type II error is clear: it underscores the need for accuracy and reliability in medical testing.
False Positives
False positives are test results that indicate the presence of a condition, such as cancer, that is not actually present. In statistical terms, false positives are considered Type I errors.
They are particularly concerning in medical diagnostics as they can lead to emotional distress and potentially unnecessary medical procedures.
False positives speak to the balance that must be found between sensitivity (the ability to detect true positives) and specificity (the ability to avoid false positives) in a test.
  • Sensitive tests maximize true positive rates but might increase false positives.
  • False positives can drain resources if many healthy individuals undergo further unnecessary testing.
Healthcare providers strive to minimize false positives through refined testing strategies and communication with patients.
Trade-Off in Error Types
The relationship between Type I and Type II errors involves a delicate balance. Reducing the likelihood of one error type often increases the likelihood of another.
This is captured in the medical context by the statement: less aggressive follow-ups might reduce false positives but increase missed diagnoses.
  • Investing in highly sensitive tests can increase Type I errors.
  • Alternatively, highly specific tests may increase Type II errors.
Understanding this trade-off helps in designing tests that are not only accurate but also aligned with the health priorities and concerns of patients and healthcare systems.

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Most popular questions from this chapter

Let \(\mu\) denote the mean diameter for bearings of a certain type. A test of \(H_{0}: \mu=0.5\) versus \(H_{a}: \mu \neq 0.5\) will be based on a sample of \(n\) bearings. The diameter distribution is believed to be normal. Determine the value of \(\beta\) in each of the following cases: a. \(\quad n=15, \alpha=.05, \sigma=0.02, \mu=0.52\) b. \(n=15, \alpha=.05, \sigma=0.02, \mu=0.48\) c. \(\quad n=15, \alpha=.01, \sigma=0.02, \mu=0.52\) d. \(\quad n=15, \alpha=.05, \sigma=0.02, \mu=0.54\) e. \(n=15, \alpha=.05, \sigma=0.04, \mu=0.54\) f. \(\quad n=20, \alpha=.05, \sigma=0.04, \mu=0.54\) g. Is the way in which \(\beta\) changes as \(n, \alpha, \sigma,\) and \(\mu\) vary consistent with your intuition? Explain.

Use the definition of the \(P\) -value to explain the following: a. Why \(H_{0}\) would be rejected if \(P\) -value \(=.0003\) b. Why \(H_{0}\) would not be rejected if \(P\) -value \(=.350\)

In a survey conducted by CareerBuilder.com, employers were asked if they had ever sent an employee home because they were dressed inappropriately (June 17 . 2008 , www.careerbuilder.com). A total of 2765 employers responded to the survey, with 968 saying that they had sent an employee home for inappropriate attire. In a press release, CareerBuilder makes the claim that more than one- third of employers have sent an employee home to change clothes. Do the sample data provide convincing evidence in support of this claim? Test the relevant hypotheses using \(\alpha=.05 .\) For purposes of this exercise, assume that it is reasonable to regard the sample as representative of employers in the United States.

An automobile manufacturer is considering using robots for part of its assembly process. Converting to robots is an expensive process, so it will be undertaken only if there is strong evidence that the proportion of defective installations is lower for the robots than for human assemblers. Let \(p\) denote the proportion of defective installations for the robots. It is known that human assemblers have a defect proportion of .02 . a. Which of the following pairs of hypotheses should the manufacturer test: \(H_{0}: p=.02\) versus \(H_{a}: p<.02\) or \(H_{0}: p=.02\) versus \(H_{a}: p>.02\) Explain your answer. b. In the context of this exercise, describe Type I and Type II errors. c. Would you prefer a test with \(\alpha=.01\) or \(\alpha=.1 ?\) Explain your reasoning.

The international polling organization Ipsos reported data from a survey of 2000 randomly selected Canadians who carry debit cards (Canadian Account Habits Survey, July 24, 2006 ). Participants in this survey were asked what they considered the minimum purchase amount for which it would be acceptable to use a debit card. Suppose that the sample mean and standard deviation were \(\$ 9.15\) and \(\$ 7.60\), respectively. (These values are consistent with a histogram of the sample data that appears in the report.) Do these data provide convincing evidence that the mean minimum purchase amount for which Canadians consider the use of a debit card to be appropriate is less than \(\$ 10\) ? Carry out a hypothesis test with a significance level of .01 .

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