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Chapter 9: Problem 5

Explain which of the following is a two-tailed test, a left-tailed test, or a right-tailed test. a. \(H_{0}: \mu=12, \quad H_{1}: \mu<12\) b. \(H_{0}: \mu \leq 85, \quad H_{1}: \mu>85\) c. \(H_{0}: \mu=33, \quad H_{1}: \mu \neq 33\)

Short Answer

Expert verified
a. Left-tailed test. b. Right-tailed test. c. Two-tailed test.

Step by step solution

01

Identify the Type of Test for Option a

For the pair of hypothesis \(H_{0}: \mu=12, \quad H_{1}: \mu<12\), it is observed that the alternative hypothesis \(H_{1}\) states that the population mean (\(\mu\)) is less than 12. Since the focus is on the 'less than' (<) direction, this is a left-tailed test.
02

Identify the Type of Test for Option b

For the pair of hypothesis \(H_{0}: \mu \leq 85, \quad H_{1}: \mu>85\), it is observed that the alternative hypothesis \(H_{1}\) states that the population mean (\(\mu\)) is greater than 85. Since the focus is on the 'greater than' (>) direction, this is a right-tailed test.
03

Identify the Type of Test for Option c

For the pair of hypothesis \(H_{0}: \mu=33, \quad H_{1}: \mu \neq 33\), it is observed that the alternative hypothesis \(H_{1}\) states that the population mean (\(\mu\)) is not equal to 33. As there is no specific direction indicated and it could either be less than or greater than 33, this is a two-tailed test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-Tailed Test
A two-tailed test is often used when we want to determine if a parameter is significantly different from a known value, without specifying a direction of difference.

In this case, the test takes into account both possibilities: the parameter being either less or greater than the specified value.

One can visualize this by imagining a normal distribution curve. The critical areas or 'tails' of interest in a two-tailed test are on both ends of the curve.
  • It is typically applied in situations where the research question is simply about the difference, regardless of direction.
  • For instance, with the hypothesis statement: \( H_{1}: \mu eq 33 \), we don't specify if \( \mu \) is greater or lesser, just that it is different.
  • This leads to testing for the effects that may show up on either side of the mean.
Whenever you suspect a change, but don't know in which direction, a two-tailed test is the go-to choice.
Left-Tailed Test
The left-tailed test is used to assess whether a parameter is significantly less than a hypothesized value.

It focuses on the left side or lower end of the distribution.

In a left-tailed test, the critical region is in the left 'tail' of the distribution curve.
  • For example, with the hypothesis \( H_{1}: \mu < 12 \), we are testing if the mean is significantly less than 12.
  • We concentrate on how extreme the data is on the lower end of the hypothesized mean.
  • Left-tailed tests are used when a hypothesis predicts a decrease or lesser value.
Remember, if your research question suggests a drop or decrease, this is the test you would usually choose.
Right-Tailed Test
A right-tailed test is employed when you are interested in finding out if a parameter is significantly greater than a specific value.

This test considers the right side of the distribution curve.

The critical area in this test exists at the right 'tail' of the distribution.
  • Consider the hypothesis \( H_{1}: \mu > 85 \), where it questions if the mean is greater than 85.
  • We're therefore looking at higher values that are significantly different.
  • Right-tailed tests are suitable when a hypothesis indicates growth or an increase.
When the expected outcome is an increase, the right-tailed test will lead your analysis in the correct direction.

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Most popular questions from this chapter

For each of the following examples of tests of hypothesis about \(\mu\), show the rejection and nonrejection regions on the \(t\) distribution curve. a. A two-tailed test with \(\alpha=.02\) and \(n=20\) b. A left-tailed test with \(\alpha=.01\) and \(n=16\) c. A right-tailed test with \(\alpha=.05\) and \(n=18\)

In an observational study at Turner Field in Atlanta, Georgia, \(43 \%\) of the men were observed not washing their hands after going to the bathroom (see Exercise \(7.80\) ). Suppose that in a random sample of 95 men who used the bathroom at Camden Yards in Baltimore, Maryland, 26 did not wash their hands. a. Using the critical-value approach and \(\alpha=.10\), test whether the percentage of all men at Camden Yards who use the bathroom and do not wash their hands is less than \(43 \%\). b. How do you explain the Type I error in part a? What is the probability of making this error in part a? c. Calculate the \(p\) -value for the test of part a. What is your conclusion if \(\alpha=.10 ?\)

Perform the following tests of hypotheses for data coming from a normal distribution. a. \(H_{0}: \mu=94.80, H_{1}: \mu<94.80, n=12, \quad \bar{x}=92.87, s=5.34, \quad \alpha=.10\) b. \(H_{0}: \mu=18.70, \quad H_{1}: \mu \neq 18.70, n=25, \quad \bar{x}=20.05, s=2.99, \quad \alpha=.05\) c. \(H_{0}: \mu=59, \quad H_{1}: \mu>59, \quad n=7, \quad \bar{x}=59.42, s=.418, \quad \alpha=.01\)

According to a 2008 survey by the Royal Society of Chemistry, \(30 \%\) of adults in Great Britain stated that they typically run the water for a period of 6 to 10 minutes while taking the shower (http://www.rsc.org/ AboutUs/News/PressReleases/2008/EuropeanShowerHabits.asp). Suppose that in a recent survey of 400 adults in Great Britain, 104 stated that they typically run the water for a period of 6 to 10 minutes when they take a shower. At the \(5 \%\) significance level, can you conclude that the proportion of all adults in Great Britain who typically run the water for a period of 6 to 10 minutes when they take a shower is less than 30 ? Use both the \(p\) -value and the critical value approaches.

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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