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Chapter 11: Problem 18

For a projected one-tailed test, lower tail critical, at the .05 level of significance, construct two rough graphs. Each graph should show the sector in the true sampling distribution that produces a type II error and the sector that produces a correct decision. One graph should reflect the case when \(H_{0}\) really is false because the true population mean is slightly less than the hypothesized population mean, and the other graph should reflect the case when \(H_{0}\) really is false because the true population mean is appreciably less than the hypothesized population mean. (Hint: First, identify the decision rule for the hypothesized population mean, and then draw the true sampling distribution for each case.)

Short Answer

Expert verified
Two graphs are constructed to visualize the sectors in the sampling distribution that produce Type II errors and correct decisions for a one-tail test at the 0.05 level of significance. In both cases, the true population mean is less than the hypothesized mean, but to varying degrees. The critical value (decision rule) for rejecting the null hypothesis is -1.645. The areas representing correct decisions and Type II errors differ based on how much the true mean deviates from the hypothesized mean.

Step by step solution

01

Understand the Hypotheses

Start by understanding that the null hypothesis, \(H_{0}\), is the status quo, typically indicating that no effect or change is present. On the contrary, the alternative hypothesis indicates some degree of effect or change. In this exercise, we're considering a one-tailed test where we're only interested in whether the true population mean is less than the hypothesized mean, not if it's higher or equal.
02

Identify Decision Rule

The decision rule for this exercise would be: Reject \(H_{0}\) if the sample mean is less than the critical value. The critical value for a one-tail test at the 0.05 significance level is about -1.645 in the z-distribution. If the test statistic falls in the critical region (is smaller than -1.645), we reject the null and accept the alternate hypothesis that the true mean is less than hypothesized.
03

Create Graphs

Two graphs need to be created:1) True Mean is Slightly Less than Hypothesized Mean: Draw the sampling distribution with its centre slightly to the left of the hypothesized mean. Mark the critical value (-1.645). The area to the left of the critical value represents the correct decision to reject \(H_{0}\) (denoted in one colour). The area to the right of the critical value, but still under the true mean distribution, represents Type II errors, where \(H_{0}\) is not rejected when it should be (denoted in a different colour).2) True Mean is Significantly Less than Hypothesized Mean: Follow the same process as the first graph, but draw the centre of the true mean sampling distribution substantially to the left of the hypothesized mean. Note that the area of Type II error (not rejecting \(H_{0}\) when it should be rejected) will be smaller compared to the first graph, as the true mean is further away from the hypothesized mean.

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Most popular questions from this chapter

How should a projected hypothesis test be modified if you're particularly concerned about (a) the type I error? (b) the type Il error?

For each of the following situations, indicate whether \(H_{0}\) should be retained or rejected. Given a one-tailed test, lower tail critical with \(\alpha=.01,\) and (a) \(z=-2.34\) (b) \(z=-5.13\) (c) \(z=4.04\) Given a one-tailed test, upper tail critical with \(\alpha=.05,\) and (d) \(z=2.00\) (e) \(z=-1.80\) (f) \(z=1.61\)

Progress Gheck \(^{\star} 11.3\) Specify the decision rule for each of the following situations (referring to Table 11.1 to find critical \(z\) values): (a) a two-tailed test with \(\alpha=.05\) (b) a one-tailed test, upper tail critical, with \(\alpha=.01\) (c) a one-tailed test, lower tail critical, with \(\alpha=.05\) (d) a two-tailed test with \(\alpha=.01\)

Each of the following statements could represent the point of departure for a hypothesis test. Given only the information in each statement, would you use a two-tailed (or nondirectional) test, a one-tailed (or directional) test with the lower tail critical, or a one-tailed (or directional) test with the upper tail critical? Indicate your decision by specifying the appropriate \(H_{0}\) and \(H_{1}\). Furthermore, whenever you conclude that the test is one-tailed, indicate the precise word (or words) in the statement that justifies the one- tailed test. (a) An investigator wishes to determine whether, for a sample of drug addicts, the mean score on the depression scale of a personality test differs from a score of \(60,\) which, according to the test documentation, represents the mean score for the general population. (b) To increase rainfall, extensive cloud-seeding experiments are to be conducted, and the results are to be compared with a baseline figure of 0.54 inch of rainfall (for comparable periods when cloud seeding was not done). (c) Public health statistics indicate, we will assume, that American males gain an average of 23 Ibs during the 20-year period after age 40. An ambitious weight-reduction program, spanning 20 years, is being tested with a sample of 40 -year-old men. (d) When untreated during their lifetimes, cancer-susceptible mice have an average life span of 134 days. To determine the effects of a potentially life- prolonging (and cancer-retarding) drug, the average life span is determined for a group of mice that receives this drug.
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