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Chapter 8: Q77E (page 500)

Specify the appropriate rejection region for testing \({H_0}:\sigma _1^2 = \sigma _2^2\) in each of the following situations:

  1. \({H_a}:\sigma _1^2 > \sigma _2^2\,\alpha = .05,\,{n_1} = 25,{n_2} = 20\)

Short Answer

Expert verified
  1. The rejection region for right tailed test is given by \(F > 2.1141\)

Step by step solution

01

Given Information

The hypothesis are given by

\(\begin{aligned}{H_0}:\sigma _1^2 = \sigma _2^2\\{H_a}:\sigma _1^2 > \sigma _2^2\end{aligned}\)

The level of significance is 0.05

02

Compute degrees of freedom

\(\begin{aligned}{v_1} &= {n_1} - 1\\ &= 25 - 1\\ &= 24\\{v_2} &= {n_2} - 1\\ &= 20 - 1\\ &= 19\end{aligned}\)

03

Test statistic

The test statistic is computed as

\(F = \frac{{s_1^2}}{{s_2^2}}\)

04

Critical value

For,

\(\alpha = 0.05\,and\,{v_1} = 24\;and\,{v_2} = 19\)

The critical value is

\({F^ \bullet } = {F_{0.05}}\left( {24,19} \right) = 2.1141\)

05

Decision-rule

The rejection region for right tailed test is given by

\(\begin{aligned}F > {F^ \bullet }\\F > 2.1141\end{aligned}\)

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Most popular questions from this chapter

Find the numerical value of

a.6! b.(109)c. (101)d.(63)e.0!

Question: Two independent random samples have been selected—100 observations from population 1 and 100 from population 2. Sample means x¯1=26.6,x¯2= 15.5 were obtained. From previous experience with these populations, it is known that the variances areσ12=9andσ22=16 .

a. Find σ(x¯1-x¯2).

b. Sketch the approximate sampling distribution for (x¯1-x¯2), assuming (μ1-μ2)=10.

c. Locate the observed value of (x¯1-x¯2)the graph you drew in part

b. Does it appear that this value contradicts the null hypothesis H0:(μ1-μ2)=10?

d. Use the z-table to determine the rejection region for the test againstH0:(μ1-μ2)≠10. Useα=0.5.

e. Conduct the hypothesis test of part d and interpret your result.

f. Construct a confidence interval for μ1-μ2. Interpret the interval.

g. Which inference provides more information about the value of μ1-μ2— the test of hypothesis in part e or the confidence interval in part f?

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a. Specify the parameter of interest to the experimenters.

b. Set up the null and indispensable suppositions for testing whether the proportion of directors willing to offer windrowing services differs in Missouri and Illinois.

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d. provide the rejection region for the test using a = .01.

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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, “Measuring Performance Reporting Quality,” The Public Manager, Vol. 37, No. 2, Summer 2008 (p. 66). Copyright © 2008 by Jerry Ellig. Used by permission of Jerry Ellig.

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