Question 1

A survey of students at a college was asked if they lived at home with their parents, rented an apartment, or
owned their own home. The results are shown in the table below sorted by gender. At α = 0.05, test the claim
that living accommodations are independent of the gender of the student. $$\begin{array} { r | c c c } 
& \text { Live with Parent } & \text { Rent Apartment } & \text { Own Home } \
\hline \text { Male } & 20 & 26 & 19 \
\text { Female } & 18 & 28 & 30
\end{array}$$

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The answer of A survey of students at a college...

Question 2

According to Benford's Law, a variety of different data sets include numbers with leading (first) digits that
follow the distribution shown in the table below. Test for goodness-of-fit with Benford's Law. $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | } 
\hline \text { Leading Digit } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \
\hline \begin{array} { l } 
\text { Benford's law: } \
\text { distribution of } \
\text { leading digits }
\end{array} & 30.1 \% & 17.6 \% & 12.5 \% & 9.7 \% & 7.9 \% & 6.7 \% & 5.8 \% & 5.1 \% & 4.6 \% \
\hline
\end{array}$$ When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the
amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 18, 0, 79, 476,
180, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the
observed frequencies are substantially different from the frequencies expected with Benford's Law, the check
amounts appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit with Benford's
Law. Does it appear that the checks are the result of fraud?

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Question 3

Discuss the three characteristics of a chi-square distribution.

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Question 4

Describe the null hypothesis for the test of independence. List the assumptions for the $x ^ { 2 }$ test of independence.

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Question 5

Describe a goodness-of-fit test. What assumptions are made when using a goodness-of-fit test?

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Question 6

Use a 0.01 significance level to test the claim that the proportion of men who plan to vote in the next election is
the same as the proportion of women who plan to vote. 300 men and 300 women were randomly selected and
asked whether they planned to vote in the next election. The results are shown below. $$\begin{array} { r | c c } 
& \text { Men } & \text { Women } \
\hline \text { Plan to vote } & 170 & 185 \
\text { Do not plan to vote } & 130 & 115
\end{array}$$

Accepted Answer

The answer of Use a 0.01 significance level to test...

Question 7

Explain the computation of expected values for contingency tables in terms of probabilities. Refer to the
assumptions of the null hypothesis as part of your explanation. You might give a brief example to illustrate.

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Question 8

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and
chiropractor) to relieve back pain. If we test the claim at a 5% level of significance that success is independent of
the method used, technology provides a P-value of 0.0655. What does the P-value tell us about the claim?

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The answer of A table summarizes the success and failures...

Question 9

A researcher wishes to test the effectiveness of a flu vaccination. 150 people are vaccinated, 180 people are
vaccinated with a placebo, and 100 people are not vaccinated. The number in each group who later caught the
flu was recorded. The results are shown below. $$\begin{array} { r | c c c } 
& \text { Vaccinated } & \text { Placebo } & \text { Control } \
\hline \text { Caught the flu } & 8 & 19 & 21 \
\text { Did not catch the flu } & 142 & 161 & 79
\end{array}$$

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Question 10

Using the data below and a 0.05 significance level, test the claim that the responses occur with percentages of
15%, 20%, 25%, 25%, and 15% respectively. $$\begin{array} { r | c c c c c } 
\text { Response } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { E } \
\hline \text { Frequency } & 12 & 15 & 16 & 18 & 19
\end{array}$$

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The answer of Using the data below and a 0.05...

Question 11

Use a $x ^ { 2 }$ test to test the claim that in the given contingency table, the row variable and the column variable are
independent. Responses to a survey question are broken down according to employment status and the sample
results are given below. At the 0.10 significance level, test the claim that response and employment status are
independent. $$\begin{array} { r | c c c } 
& \text { Yes } & \text { No } & \text { Undecided } \
\hline \text { Employed } & 30 & 15 & 5 \
\text { Unemployed } & 20 & 25 & 10
\end{array}$$

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Question 12

Describe the test of homogeneity. What characteristic distinguishes a test of homogeneity from a test of
independence?

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Question 13

A researcher wishes to test whether the proportion of college students who smoke is the same in four different
colleges. She randomly selects 100 students from each college and records the number that smoke. The results
are shown below. $$\begin{array} { r | c c c c } 
& \text { College A } & \text { College B } & \text { College C } & \text { College D } \
\hline \text { Smoke } & 17 & 26 & 11 & 34 \
\text { Don't Smoke } & 83 & 74 & 89 & 66
\end{array}$$ Use a 0.01 significance level to test the claim that the proportion of students smoking is the same at all four colleges.

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Question 14

Define categorical data and give an example.

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Question 15

In studying the responses to a multiple-choice test question, the following sample data were obtained. At the
0.05 significance level, test the claim that the responses occur with the same frequency. $$\begin{array} { r | c c c c c } 
\text { Response } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \
\hline \text { Frequency } & 12 & 15 & 16 & 18 & 19
\end{array}$$

Accepted Answer

The answer of In studying the responses to a multiple-choice...

Question 16

Use a $x ^ { 2 }$ test to test the claim that in the given contingency table, the row variable and the column variable are
independent. The table below shows the age and favorite type of music of 668 randomly selected people. $$\begin{array} { c | c c c } 
& \text { Rock } & \text { Pop } & \text { Classical } \
\hline \mathbf { 1 5 } - \mathbf { 2 5 } & 50 & 85 & 73 \
\mathbf { 2 5 } - \mathbf { 3 5 } & 68 & 91 & 60 \
\mathbf { 3 5 } - \mathbf { 4 5 } & 90 & 74 & 77
\end{array}$$ Use a 5 percent level of significance to test the null hypothesis that age and preferred music type are independent.

Accepted Answer

The answer of Use a $x ^ { 2 }$...

Question 17

Use a $x ^ { 2 }$ test to test the claim that in the given contingency table, the row variable and the column variable are
independent. 160 students who were majoring in either math or English were asked a test question, and the
researcher recorded whether they answered the question correctly. The sample results are given below. At the
0.10 significance level, test the claim that response and major are independent. $$\begin{array} { r | c c } 
& \text { Correct } & \text { Incorrect } \
\hline \text { Math } & 27 & 53 \
\text { English } & 43 & 37
\end{array}$$

Accepted Answer

The answer of Use a $x ^ { 2 }$...

Question 18

Perform the indicated goodness-of-fit test. Among the four northwestern states, Washington has 51% of the
total population, Oregon has 30%, Idaho has 11%, and Montana has 8%. A market researcher selects a sample of
1000 subjects, with 450 in Washington, 340 in Oregon, 150 in Idaho, and 60 in Montana. At the 0.05 significance
level, test the claim that the sample of 1000 subjects has a distribution that agrees with the distribution of state
populations.

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Question 19

The table in number 18 is called a two-way table. Why is the terminology of two-way table used?

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Question 20

Perform the indicated goodness-of-fit test. You roll a die 48 times with the following results. $$\begin{array} { l | c | c | c | c | c | c } 
\text { Number } & 1 & 2 & 3 & 4 & 5 & 6 \
\hline \text { Frequency } & 4 & 13 & 2 & 14 & 13 & 2
\end{array}$$ Use a significance level of 0.05 to test the claim that the die is fair.

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Quiz 11: Goodness-Of-Fit and Contingency Tables

Discuss the three characteristics of a chi-square distribution.

Describe the null hypothesis for the test of independence. List the assumptions for the x2x ^ { 2 }x2 test of independence.

Describe a goodness-of-fit test. What assumptions are made when using a goodness-of-fit test?

Explain the computation of expected values for contingency tables in terms of probabilities. Refer to the assumptions of the null hypothesis as part of your explanation. You might give a brief example to illustrate.

Describe the test of homogeneity. What characteristic distinguishes a test of homogeneity from a test of independence?

Define categorical data and give an example.

The table in number 18 is called a two-way table. Why is the terminology of two-way table used?

Describe the null hypothesis for the test of independence. List the assumptions for the $x ^ { 2 }$ test of independence.