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Chapter 9: Problem 1

Briefly explain the meaning of each of the following terms. a. Null hypothesis b. Alternative hypothesis c. Critical point(s) d. Significance level e. Nonrejection region f. Rejection region \(\mathrm{g}\). Tails of a test h. Two types of errors

Short Answer

Expert verified
a. Null hypothesis: Statement we assume to be true.\nb. Alternative hypothesis: Counter statement to null hypothesis.\nc. Critical point(s): Values separating rejection and nonrejection regions.\nd. Significance level: Risk of making a Type I error.\ne. Nonrejection region: Range of values for which we accept the null hypothesis.\nf. Rejection region: Range of values for which we reject the null hypothesis.\ng. Tails of a test: Relates to rejection region(s) in the distribution.\nh. Two types of errors: Type I error (erroneously reject null), Type II error (fail to reject false null).

Step by step solution

01

Define Null Hypothesis

The null hypothesis, often denoted as \(H_0\), is a statement about a population parameter that we assume to be true until we have enough evidence to suggest otherwise.
02

Define Alternative Hypothesis

The alternative hypothesis, often denoted as \(H_a\) or \(H_1\), is the statement we believe might be true if the null hypothesis is false. It is basically the complement of the null hypothesis.
03

Define Critical point(s)

Critical points are values that separate the rejection region(s) from the nonrejection region in a testing distribution. At these points, if the calculated test statistic equals or exceeds the critical value, we reject the null hypothesis.
04

Define Significance Level

The significance level, often denoted as \(\alpha\) (alpha), is the probability of rejecting the null hypothesis when it is true. It represents the risk of making a Type I error.
05

Define Nonrejection Region

The nonrejection (or acceptance) region is the range of values for which we retain (do not reject) the null hypothesis.
06

Define Rejection Region

The rejection region is the range of values for which we reject the null hypothesis.
07

Define Tails of a Test

Tails of a test relate to the rejection region(s) in the testing distribution. A test can be one-tailed (rejection region is in one tail) or two-tailed (rejection regions are in both tails).
08

Define Types of Errors

There are two types of errors in hypothesis testing. Type I error occurs when we reject the null hypothesis when it's true. Type II error occurs when we fail to reject the null hypothesis when it's false.

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Most popular questions from this chapter

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 \(\alpha=.10\) ?

Records in a three-county area show that in the last few years, Girl Scouts sell an average of \(47.93\) boxes of cookies per year, with a population standard deviation of \(8.45\) boxes per year. Fifty randomly selected Girl Scouts from the region sold an average of \(46.54\) boxes this year. Scout leaders are concerned that the demand for Girl Scout cookies may have decreased. a. Test at the \(10 \%\) significance level whether the average number of boxes of cookies sold by all Girl Scouts in the three-county area is lower than the historical average. b. What will your decision be in part a if the probability of a Type I error is zero? Explain.

The Environmental Protection Agency (EPA) recommends that the sodium content in public water supplies should be no more than \(20 \mathrm{mg}\) per liter (http://www.disabledworld.com/artman/publish/ sodiumwatersupply.shtml). Forty samples were taken from a large reservoir, and the amount of sodium in each sample was measured. The sample average was \(23.5 \mathrm{mg}\) per liter. Assume that the population standard deviation is \(5.6 \mathrm{mg}\) per liter. The EPA is interested in knowing whether the average sodium content for the entire reservoir exceeds the recommended level. If so, the communities served by the reservoir will have to be made aware of the violation. a. Find the \(p\) -value for the test of hypothesis. Based on this \(p\) -value, would the communities need to be informed of an excessive average sodium level if the maximum probability of a Type I error is to be \(.05 ?\) What if the maximum probability of a Type I error is to be \(.01 ?\) b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.05 .\) Would the communities need to be notified? What if \(\alpha=.01 ?\) What if \(\alpha\) is zero?

The manager of a service station claims that the mean amount spent on gas by its customers is \(\$ 15.90\) per visit. You want to test if the mean amount spent on gas at this station is different from \(\$ 15.90\) per visit. Briefly explain how you would conduct this test when \(\sigma\) is not known.

Consider the null hypothesis \(H_{0}: \mu=100\). Suppose that a random sample of 35 observations is taken from this population to perform this test. Using a significance level of \(.01\), show the rejection and nonrejection regions and find the critical value(s) of \(t\) when the alternative hypothesis is as follows. a. \(H_{1}: \mu \neq 100\) b. \(H_{1}: \mu>100\) c. \(H_{1}: \mu<100\)
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