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Chapter 12: Q. 12.28 (page 492)

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?

Short Answer

Expert verified

The data does not provide enough evidence to indicate that the wheel is out of balance.

Step by step solution

01

Step 1. Given Information

A table with the distribution of trials:

02

Step 2. Hypotheses and calculating goodness of fit

The null and alternate hypotheses are:
H0:The data does not provide enough evidence to indicate that the wheel is out of balance.
Ha:The data provides enough evidence to indicate that the wheel is out of balance.
Calculating the goodness of fit

蠂2=鈭�(O-E)2E
=1.0622
The degrees of freedom of the given data is
df=k-1
=3-1=2
The critical value is 蠂0.0522dfis 5.991
The test statistic is 蠂2=1.0622
蠂2=1.0622<蠂0.052=5.991as it does not fall in the rejection region.

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Most popular questions from this chapter

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

The Scottish Executive, Analytical Services Division Transport Statistics, compiles information on motorcycle accidents in Scotland. During one year, data on the number of motorcycle accidents, by day of the week and type of road (built-up or non built-up), are as presented on the WeissStats CD.
a. Group the bivariate data for these two variables into a contingency table.
b. Determine the conditional distribution of day of the week within each type-of-road category and the marginal distribution of day of the week.
c. Determine the conditional distribution of type of road within each day of the week and the marginal distribution of type of road.
d. Does an association exist between the variables "day of the week" and "type of road" for these motorcycle accidents? Explain your answer.

Primary Heating Fuel. According to Current Housing Reports, published by the U.S. Census Bureau, the primary heating fuel for all occupied housing units is distributed as follows.

Suppose that you want to determine whether the distribution of primary heating fuel for occupied housing units built after 2010 differs from that of all occupied housing units. To decide, you take a random sample of housing units built after 2010 and obtain a frequency distribution of their primary heating fuel.

a. Identify the population and variable under consideration here.

b. For each of the following sample sizes, determine whether conducting a chi-square goodness-of-fit test is appropriate and explain

your answers: 200; 300; 400.

c. Strictly speaking, what is the smallest sample size for which conducting a chi-square goodness-of-fit test is appropriate?

Education of Prisoners. Refer to Exercise 12.49.

a. Find the conditional distribution of educational attainment with in each type of prison facility.

b. Does an association exist between educational attainment and of prison facility for prisoners? Explain your answer.

c. Determine the marginal distribution of educational attainment for prisoners.

d. Construct a segmented bar graph for the conditional distributions of educational attainment and marginal distribution of educational attainment that you obtained in parts (a) and (c), respectively. Interpret the graph in light of your answer to part (b).

e. Without doing any further calculations, respond true or false to the following statement and explain your answer: "The conditional distributions of facility type within educational attainment categories are identical."

f. Determine the marginal distribution of facility type and the conditional distributions of facility type within educational attainment

categories.

g. Find the percentage of prisoners who are in federal facilities.

h. Find the percentage of prisoners with at most an 8th-gradetion who are in federal facilities.

i. Find the percentage of prisoners in federal facilities who have at most an 8th-grade education.

Identify three ways in which the total number of observations of bivariate data can be obtained from the frequencies in a contingency table.

Although the chi-square homogeneity test and the chi-square independence test use essentially the same basic procedure, the context of the two tests are quite different. Explain the difference.
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