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Chapter 12: Q. 12.22 (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 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: \(0.5, 0.3, 0.2\)

Observed frequencies: \(45, 39, 16\)

Significance level: \(=0.01\)

Short Answer

Expert verified

The variable has given distribution.

Step by step solution

01

Step 1. Given information

Distribution: \(0.5, 0.3, 0.2\)

Observed frequencies: \(45, 39, 16\)

Significance level: \(=0.01\)

02

Step 2. Calculation

Consider goodness of fit test

The sample size

\(n=45+39+16\)

\(=100\)

Observed frequencies

\(45\)

\(39\)

\(16\)

Relative frequencies


\(0.5\)

\(0.3\)

\(0.2\)

Expected frequencies

\(50\)

\(30\)

\(20\)

\(\frac{(obs-exp)^{2}}{exp}

\(0.5\)

\(2.9\)

\(0.8\)

\(\chi^{2}=\sum \frac{(obs-exp)^{2}}{exp}=4.2\)

The degrees of freedom of given data

\(=k-1\)

\(=3-1\)

\(=2\)

The critical value, with \(2df\) is \(\chi ^{2}_{0.01}=9.210\)

\(\chi ^{2}=4.2<\chi ^{2}_{0.05}=9.210\)

Because, it does not come in rejection region. So we accept the Null hypothesis.

Hence, the variable has given specified region.

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Most popular questions from this chapter

In this exercise, you are to consider two variables, xand ydefined on a hypothetical population. Following are the conditional distributions of the variable ycorresponding to each value of the variablex.
a. Are the variables xandy associated? Explain your answer.
b. Determine the marginal distribution ofy.
c. Can you determine the marginal distribution of x? Explain your answer.

Job Satisfaction. A CNN/USA TODAY poll conducted by Gallul asked a sample of employed Americans the following question: "Which do you enjoy more, the hours when you are on your job, or the hours when you are not on your job?" The responses to this question were cross-tabulated against several characteristics, among which were gender, age, type of community, educational attainment, income, and type of employer. The data are provided on the WeissStats site. In each of Exercises 12.87-12.92, use the technology of your choice to decide, at the 5% significance level, whether an association exists between the specified pair of variables.

Gender and response

State the null and alternative hypotheses for a chi-square homogeneity test
a. without using the terms homogeneous and nonhomogeneous.
b. using the terms homogeneous and nonhomogeneous

In each case, decide whether Assumptions 1and 2for using chi-square goodness-of-fit test are satisfied.

Sample size: n=100.

Relative frequencies:0.65,0.30,0.05.

In each of the given Exercises, we have presented a contingency table that gives a cross-classification of a random sample of values for two variables, x, and y, of a population. For each exercise, perform the following tasks.

a. Find the expected frequencies. Note: You will first need to compute the row totals, column totals, and grand total.

b. Determine the value of the chi-square statistic.

c. Decide at the 5% significance level whether the data provide sufficient evidence to conclude that the two variables are associated.
See all solutions

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