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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

A paint manufacturing company claims that the mean drying time for its paints is not longer than 45 minutes. A random sample of 20 gallons of paints selected from the production line of this company showed that the mean drying time for this sample is \(49.50\) minutes with a standard deviation of 3 minutes. Assume that the drying times for these paints have a normal distribution. a. Using a \(1 \%\) significance level, would you conclude that the company's claim is true? b. What is the Type I error in this exercise? Explain in words. What is the probability of making such an error?

The police that patrol a heavily traveled highway claim that the average driver exceeds the 65 miles per hour speed limit by more than 10 miles per hour. Seventy-two randomly selected cars were clocked by airplane radar. The average speed was \(77.40\) miles per hour, and the standard deviation of the speeds was \(5.90\) miles per hour. Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value range and \(\alpha=.02\) ?

Consider the null hypothesis \(H_{0}: \mu=12.80 .\) A random sample of 58 observations is taken from this population to perform this test. Using \(\alpha=.05\), show the rejection and nonrejection regions on the sampling distribution curve of the sample mean and find the critical value(s) of \(t\) for the following. a. a right-tailed test \(\underline{\text { b. a left-tailed test }}\) c. a two-tailed test

By rejecting the null hypothesis in a test of hypothesis example, are you stating that the alternative hypothesis is true?

Consider the null hypothesis \(H_{0}: \mu=5 .\) A random sample of 140 observations is taken from a population with \(\sigma=17\). Using \(\alpha=.05\), show the rejection and nonrejection regions on the sampling distribution curve of the sample mean and find the critical value(s) of \(z\) for the following. \(\begin{array}{lll}\text { a. a right-tailed test } & \text { b. a left-tailed test } & \text { c. a two-tailed test }\end{array}\)
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