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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

What are the four possible outcomes for a test of hypothesis? Show these outcomes by writing a table. Briefly describe the Type I and Type II errors.

A soft-drink manufacturer claims that its 12 -ounce cans do not contain, on average, more than 30 calories. A random sample of 64 cans of this soft drink, which were checked for calories, contained a mean of 32 calories with a standard deviation of 3 calories. Does the sample information support the altemative hypothesis that the manufacturer's claim is false? Use a significance level of \(5 \%\). Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value and \(\alpha=.05 ?\)

Consider \(H_{0^{\circ}} \mu=80\) versus \(H_{1}: \mu \neq 80\) for a population that is normally distributed. a. A random sample of 25 observations taken from this population produced a sample mean of 77 and a standard deviation of 8 . Using \(\alpha=.01\), would you reject the null hypothesis? b. Another random sample of 25 observations taken from the same population produced a sample mean of 86 and a standard deviation of \(6 .\) Using \(\alpha=.01\), would you reject the null hypothesis?

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

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