Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 12: Q. 12.18 (page 491)

In each of Exercises 12.18-12.23, we have provided a distribution and the observed frequencies of the values of a variable from a simple random sample of a population. In each case, use the chi-square goodness-of-fit test to decide, at the specified significance level, whether the distribution of the variable differs from the given distribution.

Distribution: 0.2, 0.4, 0.3, 0.1

Observed frequencies: 39, 78, 64, 19

Significance level = 0.05

Short Answer

Expert verified

The test hypotheses,

H0: The variable has the given specified distribution

H1: The variable differ from the given distribution

wo do not reject the null hypothesis,H0

∴DonotrejectH0,H0Variable has distribution given in the problem .

Therefore, the variable has the given specified distribution.

Step by step solution

01

Step 1. Given 

The sample size is n= 39+78+64+19=200.

Level of significance ,α=0.05

02

Step 2. Calculation the goodness of fit .

Now, we want to perform the hypothesis test

The test hypotheses,

H0: The variable has the given specified distribution

H1: The variable differ from the given distribution

Calculating the Goodness of fit :

Observed
frequencies		Relative
frequencies		Expected
frequencies		Obs - Exp(Obs-Exp)2Exp
390.240-10.025
780.480-20.050
640.36040.267
190.120-10.050




∑(Obs-Exp)2Exp=0.392

∑(Obs-Exp)2Exp=0.392

The degrees of freedom of the given data is, k-1

=4-1 = 3

Critical value is X20.05for 3df is 9.815

The value of the test statistic is, X2=0.392

X2=0.392<X20.05= 7.815, because it does not fall in the rejection region So, wo do not reject the null hypothesis,H0

∴DonotrejectH0,H0Variable has distribution given in the problem .

Therefore, the variable has the given specified distribution.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The t-table has entries for areas of 0.10,0.05,0.025,0.01and0.005. In contrast, the χ2-table has entries for those areas and for 0.995,0.99,0.975,0.95and 0.90.Explain why the t-values corresponding to these additional areas can be obtained from the existing t-table but must be provided explicitly in the χ2-table.

In each of Exercises 12.24-12.33, apply the chi-square goodness-of-fit test, using either the critical-value approach or theP-value approach, to perform the required hypothesis test.
An American roulette wheel contains 18red numbers, 18black numbers, and 2green numbers. The following table shows the frequency with which the ball landed on each color in 200trials.

At the 5%significance level, do the data suggest that the wheel is out of balance?

In each of exeercises 12.57-12.59, use the technology of your choice to solve the specified problems.

The U.S. Congress, Joint Committee on Printing. provides information on the composition of Congress in Congressional Directory. On the WeissStats CD, we present data on party and class for the senators in the 113th Congress.
a. Group the bivariate data for these two variables into a contingency table.
a. Group the bivariate data for these two variables into a contingency table.
b. Determine the conditional distribution of party within each class and the marginal distribution of party.
c. Determine the conditional distribution of class within each party and the marginal distribution of class.
d. Are the variables "party" and "class" for U.S. senators in the 111th Congress associated? Explain your answer.

Ancestry and Region. The U.S. Census Bureau collects information on the U.S. population by ancestry and region of residenc and publishes the results in American Community Survey. Accordin; to that document, 18 % of the population resides in the Northeast.

a. If ancestry and region of residence are not associated, wha percentage of Americans of Irish ancestry would reside in the Northeast?

b. There are roughly 37 million Americans of Irish ancestry. If ancestry and region of residence are not associated, how many Americans of Irish ancestry would reside in the Northeast?

c. There are, in fact, 9.25 million Americans of Irish ancestry who reside in the Northeast. Given this information and your answer to part (b), what can you conclude?

The chi-square goodness-of-fit test provides a method for performing a hypothesis test about the distribution of a variable that has c possible values. If the number of possible values is 2, that is, c=2, the chi-square goodness-of-fit test is equivalent to a procedure that you studied earlier.

a) Which procedure is that? Explain your answer.

b) Suppose that you want to perform a hypothesis test to decide whether the proportion of a population that has a specified attribute is different from p_0. Discuss the method for performing such a test if you use (1) the one-proportion z-test (page 463) or (2) the chi-square goodness-of-fit test.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free