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Chapter 9: Problem 3

Explain how the tails of a test depend on the sign in the alternative hypothesis. Describe the signs in the null and alternative hypotheses for a two-tailed, a left-tailed, and a right-tailed test, respectively.

Short Answer

Expert verified
Two-tailed test holds a hypothesis of \(H_0: \mu = \mu_0\) and \(H_a: \mu \neq \mu_0\). Left-tailed test holds \(H_0: \mu \geq \mu_0\) and \(H_a: \mu < \mu_0\). Right-tailed test holds \(H_0: \mu \leq \mu_0\) and \(H_a: \mu > \mu_0\). The sign in the alternative hypothesis determines the type (direction) of the tail in the test. Two-tailed tests look for departures on either side of the hypothesized parameter value while one-tailed tests look specifically for increases (right-tailed) or decreases (left-tailed) from the current state.

Step by step solution

01

Two-tailed test

Two-tailed tests are used when the alternative hypothesis does not specify the direction of the effect. Thus, for a two-tailed test, the null hypothesis \(H_0\) will usually have an equals sign, e.g. \(H_0: \mu = \mu_0\), and the alternative hypothesis \(H_1\) or \(H_a\) will typically involve a not equals sign, e.g. \(H_a: \mu \neq \mu_0\). This means we're interested in departures on either side of the hypothesized parameter value.
02

Left-tailed test

Left-tailed tests are used when the direction of the effect specified in the alternative hypothesis is 'less than'. Thus, for a left-tailed test, the null hypothesis \(H_0\) will usually be \(H_0: \mu \geq \mu_0\), and the alternative hypothesis \(H_a\) will be \(H_a: \mu < \mu_0\). Envision this as the test looking for a decrease from the current state.
03

Right-tailed test

Right-tailed tests are used when the direction of the effect specified in the alternative hypothesis is 'greater than'. Thus, for a right-tailed test, the null hypothesis \(H_0\) will usually be \(H_0: \mu \leq \mu_0\), and the alternative hypothesis \(H_a\) will be \(H_a: \mu > \mu_0\). In essence, this test is looking for an increase from the current state.

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Most popular questions from this chapter

Briefly explain the procedure used to calculate the \(p\) -value for a two- tailed and for a one-tailed test. respectively.

Consider the null hypothesis \(H_{0}: p=.25 .\) Suppose a random sample of 400 observations is taken to perform this test about the population proportion. Using \(\alpha=.01\), show the rejection and nonrejection regions and find the critical value(s) of \(z\) for a a. left-tailed test b. two-tailed test c. right-tailed test

A tool manufacturing company claims that its top-of-the-line machine that is used to manufacture bolts produces an average of 88 or more bolts per hour. A company that is interested in buying this machine wants to check this claim. Suppose you are asked to conduct this test. Briefly explain how you would do so when \(\sigma\) is not known.

For each of the following significance levels, what is the probability of making a Type I error? \(\begin{array}{lll}\text { a. } \alpha=.10 & \text { b. } \alpha=.02 & \text { c. } \alpha=.005\end{array}\)

According to a New York Times/CBS News poll conducted during June \(24-28,2011,55 \%\) of the American adults polled said that owning a home is a very important part of the American Dream (The New York Times, June 30,2011 ). Suppose this result was true for the population of all American adults in \(2011 .\) In a recent poll of 1800 American adults, \(61 \%\) said that owning a home is a very important part of the American Dream. Perform a hypothesis test to determine whether it is reasonable to conclude that the percentage of all American adults who currently hold this opinion is higher than \(55 \%\). Use a \(2 \%\) significance level, and use both the \(p\) -value and the critical-value approaches.
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