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Chapter 10: Problem 17

One type of error in a hypothesis test is rejecting the null hypothesis when it is true. What is the other type of error that might occur when a hypothesis test is carried out?

Short Answer

Expert verified
The other type of error that might occur when a hypothesis test is carried out is a Type II error. This is when we fail to reject the null hypothesis when it is actually false (a false negative).

Step by step solution

01

Understanding Type II error

Type II error, in the context of hypothesis testing, is the error that occurs when the null hypothesis is not rejected when it actually should be rejected. This is also known as a false negative. In other words, the second type of error occurs when failing to detect a difference or effect that is actually present.
02

Type II Error Definition

Therefore, the other type of error that might occur when a hypothesis test is carried out is when we fail to reject the null hypothesis when it is false. This error is known as a Type II error or a false negative.

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Most popular questions from this chapter

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 falsepositives 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. Recall 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. What aspect of the relationship between the probability of a Type I error and the probability of a Type II error is being described here?

In a hypothesis test, what does it mean to say that the null hypothesis was not rejected?

The article "The Benefits of Facebook Friends: Social Capital and College Students' Use of Online Social Network Sites" (Journal of Computer-Mediated Communication [2007]: \(1143-1168\) ) describes a study of \(n=286\) undergraduate students at Michigan State University. Suppose that it is reasonable to regard this sample as a random sample of undergraduates at Michigan State. You want to use the survey data to decide if there is evidence that more than \(75 \%\) of the students at this university have a Facebook page that includes a photo of themselves. Let \(p\) denote the proportion of all Michigan State undergraduates who have such a page. (Hint: See Example 10.10\()\) a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for random samples of size 286 if the null hypothesis \(H_{0}: p=0.75\) is true. b. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.83\) for a sample of size 286 if the null hypothesis \(H_{0}: p=0.75\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.79\) for a sample of size 286 if the null hypothesis \(H_{0}: p=0.75\) were true? Explain why or why not. d. The actual sample proportion observed in the study was \(\hat{p}=0.80 .\) Based on this sample proportion, is there convincing evidence that the null hypothesis \(H_{0}: p=\) 0.75 is not true, or is \(\hat{p}\) consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation.

Step 5 of the five-step process for hypothesis testing is communication of results. What is involved in completing this step?

In a hypothesis test, what does it mean to say that the null hypothesis was rejected?
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