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Chapter 8: Q82E (page 501)

a.Consider testing \(H0\) : \(\mu \)=80. Under what conditions should you use the t-distribution to conduct the test?

Short Answer

Expert verified

a.Two conditions to conduct the test for t-distribution

Step by step solution

01

Given information

Testing hypothesis are as follows

\(H0\): \(\mu \)=80.

\(H\alpha :\mu \ne 80\)

02

Explaining the t-distribution

When the population standard deviation is unknown and the data are from a normally distributed population, the t-distribution defines the standardized distance between the sample and the population mean.

03

Conditions to use the t-distribution to conduct the test

Two requirements have got to be met before the t-distribution is employed. The first is the sampling distribution's normality. Such that the x-bar uses a traditional distribution. It may accomplish in one of two ways. It would be best to grasp that the individual observations follow a standard distribution, or like to own a large sample size (more than 30), so it can depend upon the central limit theorem. After calculating the population variance with the sample variance, we now have the t-distribution.

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Most popular questions from this chapter

The gender diversity of a large corporation鈥檚 board of directors was studied in Accounting & Finance (December 2015). In particular, the researchers wanted to know whether firms with a nominating committee would appoint more female directors than firms without a nominating committee. One of the key variables measured at each corporation was the percentage of female board directors. In a sample of 491firms with a nominating committee, the mean percentage was 7.5%; in an independent sample of 501firms without a nominating committee, the mean percentage was role="math" localid="1652702402701" 4.3% .

a. To answer the research question, the researchers compared the mean percentage of female board directors at firms with a nominating committee with the corresponding percentage at firms without a nominating committee using an independent samples test. Set up the null and alternative hypotheses for this test.

b. The test statistic was reported as z=5.1 with a corresponding p-value of 0.0001. Interpret this result if 伪=0.05.

c. Do the population percentages for each type of firm need to be normally distributed for the inference, part b, to be valid? Why or why not?

d. To assess the practical significance of the test, part b, construct a 95% confidence interval for the difference between the true mean percentages at firms with and without a nominating committee. Interpret the result.

A random sample of n observations is selected from a normal population to test the null hypothesis that 蟽2=25. Specify the rejection region for each of the following combinations of Ha,伪and n.

a.Ha:蟽2鈮�25;伪=0.5;n=16

b.Ha:蟽2>25;伪=.10;n=15

c.Ha:蟽2>25;伪=.01;n=23

d. Ha:蟽2<25;伪=.01;n=13

e. Ha:蟽2鈮�25;伪=.10;n=7

f. Ha:蟽2<25;伪=.05;n=25

Question: Independent random samples from approximately normal populations produced the results shown below.

Sample 1

Sample 2

52 33 42 4441 50 44 5145 38 37 4044 50 43

52 43 47 5662 53 61 5056 52 53 6050 48 60 55

a. Do the data provide sufficient evidence to conclude that (渭1-渭2)>10? Test using伪=0.1.

b. Construct a confidence interval for (渭1-渭2). Interpret your result.

Last name and acquisition timing. Refer to the Journal of Consumer Research (August 2011) study of the last name effect in acquisition timing, Exercise 8.13 (p. 466). Recall that the mean response times (in minutes) to acquire free tickets were compared for two groups of MBA students鈥� those students with last names beginning with one of the first nine letters of the alphabet and those with last names beginning with one of the last nine letters of the alphabet. How many MBA students from each group would need to be selected to estimate the difference in mean times to within 2 minutes of its true value with 95% confidence? (Assume equal sample sizes were selected for each group and that the response time standard deviation for both groups is 蟽鈮� 9 minutes.)

Question: Performance ratings of government agencies. The U.S. Office of Management and Budget (OMB) requires government agencies to produce annual performance and accounting reports (PARS) each year. A research team at George Mason University evaluated the quality of the PARS for 24 government agencies (The Public Manager, Summer 2008), where evaluation scores ranged from 12 (lowest) to 60 (highest). The accompanying file contains evaluation scores for all 24 agencies for two consecutive years. (See Exercise 2.131, p. 132.) Data for a random sample of five of these agencies are shown in the accompanying table. Suppose you want to conduct a paired difference test to determine whether the true mean evaluation score of government agencies in year 2 exceeds the true mean evaluation score in year 1.

Source: J. Ellig and H. Wray, 鈥淢easuring Performance Reporting Quality,鈥� The Public Manager, Vol. 37, No. 2, Summer 2008 (p. 66). Copyright 漏 2008 by Jerry Ellig. Used by permission of Jerry Ellig.

a. Explain why the data should be analyzedusing a paired difference test.

b. Compute the difference between the year 2 score and the year 1 score for each sampled agency.

c. Find the mean and standard deviation of the differences, part

b. Use the summary statistics, part c, to find the test statistic.

e. Give the rejection region for the test using a = .10.

f. Make the appropriate conclusion in the words of the problem.
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