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Chapter 9: Problem 12

What does the level of significance represent in a test of hypothesis? Explain.

Short Answer

Expert verified
The level of significance, denoted as \(\alpha\), in a hypothesis test is the probability that the test incorrectly rejects the null hypothesis when it is true, or a false positive. It is a pre-set threshold by the researcher that if the p-value (the evidence against the null hypothesis) is below \(\alpha\), the null hypothesis is rejected.

Step by step solution

01

Understanding Hypothesis Testing

Before discussing the level of significance, it's crucial to grasp what a hypothesis test is. This is a statistical method used to make decisions or reach conclusions about populations based on sample data.
02

Introduction to Level of Significance

The level of significance, often denoted as \(\alpha\), is a value in a hypothesis test that helps to manage the risk of concluding a false positive. In other words, it's the probability of rejecting the null hypothesis when it's true.
03

Application of the Level of Significance

The level of significance is set by the researcher before conducting the test. A common \(\alpha\) value is 0.05. This means that we'd be willing to accept a 5% chance of wrongly rejecting the null hypothesis. If the p-value obtained from the test is less than \(\alpha\) (0.05), we reject the null hypothesis because this lesser value indicates stronger evidence against the null hypothesis.
04

Understanding the Implication

A lower \(\alpha\) value indicates a lower willingness to accept a false positive. It reflects the strictness of the researcher. Remember, the level of significance is an arbitrary set standard, and it can vary depending on the nature and field of the study.

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Most popular questions from this chapter

Thirty percent of all people who are inoculated with the current vaccine used to prevent a disease contract the disease within a year. The developer of a new vaccine that is intended to prevent this disease wishes to test for significant evidence that the new vaccine is more effective. a. Determine the appropriate null and alternative hypotheses. b. The developer decides to study 100 randomly selected people by inoculating them with the new vaccine. If 84 or more of them do not contract the disease within a year, the developer will conclude that the new vaccine is superior to the old one. What significance level is the developer using for the test? c. Suppose 20 people inoculated with the new vaccine are studied and the new vaccine is concluded to be better than the old one if fewer than 3 people contract the disease within a year. What is the significance level of the test?

According to the Magazine Publishers of America (www.magazine.org), the average visit at the magazines' Web sites during the fourth quarter of 2007 lasted \(4.145\) minutes. Forty-six randomly selected visits to magazine's Web sites during the fourth quarter of 2009 produced a sample mean visit of \(4.458\) minutes, with a standard deviation of \(1.14\) minutes. Using the \(10 \%\) significance level and the critical value approach, can you conclude that the length of an average visit to these Web sites during the fourth quarter of 2009 was longer than \(4.145\) minutes? Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value range and \(\alpha=.10\) ?

Briefly explain the conditions that must hold true to use the \(t\) distribution to make a test of hypothesis about the population mean.

A real estate agent claims that the mean living area of all single-family homes in his county is at most 2400 square feet. A random sample of 50 such homes selected from this county produced the mean living area of 2540 square feet and a standard deviation of 472 square feet. a. Using \(\alpha=.05\), can you conclude that the real estate agent's claim is true? What will your conclusion be if \(\alpha=.01 ?\)

Make the following tests of hypotheses. a. \(H_{0}: \mu=80, \quad H_{1}: \mu \neq 80, \quad n=33, \quad \bar{x}=76.5, \quad \sigma=15, \quad \alpha=.10\) b. \(H_{0}: \mu=32, \quad H_{1}: \mu<32, \quad n=75, \quad \bar{x}=26.5, \quad \sigma=7.4, \quad \alpha=.01\) c. \(H_{0}: \mu=55, \quad H_{1}: \mu>55, \quad n=40, \quad \bar{x}=60.5, \quad \sigma=4, \quad \alpha=.05\)
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