Question 1

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. $$\begin{array} { r | c c c } 
& \text { Vaccinated } & \text { Placebo } & \text { Control } \
\hline \text { Caught the flu } & 8 & 19 & 21 \
\text { Did not catch the flu } & 142 & 161 & 79
\end{array}$$ Use a 0.05 significance level to test the claim that the proportion of people catching the flu is the same in all three groups.

Accepted Answer

$H _ { 0 }$ : The proportion of people catching the flu is the same in all three groups.
$H _ { 1 } :$ At least one of the proportions are different.
Test statistic: $\chi ^ { 2 } = 14.965$. Critical value: $\chi ^ { 2 } = 5.991$.
Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the proportior people catching the flu is the same in all three groups.

Question 2

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

Accepted Answer

A goodness-of-fit test is used to test the hypothesis that an observed frequency distribution fits some
claimed distribution. The assumptions are 1) the sample data are randomly selected; 2) the sample data
consists of frequency counts for the different categories; and 3) for each of the categories, the expected
frequency is at least 5.

Question 3

Use a $x ^ { 2 }$ test to test the claim that in the given contingency table, the row variable and the column variable are
independent. Tests for adverse reactions to a new drug yielded the results given in the table. At the 0.05
significance level, test the claim that the treatment (drug or placebo) is independent of the reaction (whether or
not headaches were experienced). $$\begin{array} { r | c c } 
& \text { Drug } & \text { Placebo } \
\hline \text { Headaches } & 11 & 7 \
\text { No headaches } & 73 & 91
\end{array}$$

Accepted Answer

$H _ { 0 }$ : Treatment and reaction are independent.
$H _ { 1 }$ : Treatment and reaction are dependent.
Test statistic: $\chi ^ { 2 } = 1.798$. Critical value: $\chi ^ { 2 } = 3.841$.
Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that treatment and reaction are independent. nt

Question 4

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 $\alpha = 0.10$ 0, test the claim
that living accommodations are independent of the gender of the student. $$\begin{array} { r | c c c } 
& \text { Live with Parent } & \text { Rent Apartment } & \text { Own Home } \
\hline \text { Male } & 20 & 26 & 19 \
\text { Female } & 22 & 28 & 26
\end{array}$$

Accepted Answer

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

Question 5

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 6

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

Accepted Answer

The answer of Describe the test of homogeneity. What characteristic...

Question 7

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

Accepted Answer

The answer of The table in number 18 is called...

Question 8

Use the sample data below to test whether car color affects the likelihood of being in an accident. Use a
significance level of 0.01. $$\begin{array} { | c | c c c | } 
\hline & \text { Red } & \text { Blue } & \text { White } \
\hline \text { Car has been in accident } & 28 & 33 & 36 \
\text { Car has not been in accident } & 23 & 22 & 30 \
\hline
\end{array}$$

Accepted Answer

The answer of Use the sample data below to test...

Question 9

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?

Accepted Answer

The answer of According to Benford's Law, a variety of...

Question 10

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.

Accepted Answer

The answer of Explain the computation of expected values for...

Question 11

Perform the indicated goodness-of-fit test. A company manager wishes to test a union leader's claim that
absences occur on the different week days with the same frequencies. Test this claim at the 0.05 level of
significance if the following sample data have been compiled. $$\begin{array} { c | c c c c c } 
\text { Day } & \text { Mon } & \text { Tue } & \text { Wed } & \text { Thurs } & \text { Fri } \
\hline \text { Absences } & 37 & 15 & 12 & 23 & 43
\end{array}$$

Accepted Answer

The answer of Perform the indicated goodness-of-fit test. A company...

Question 12

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.

Accepted Answer

The answer of Among the four northwestern states, Washington has...

Question 13

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

Accepted Answer

The answer of A researcher wishes to test whether the...

Question 14

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

Accepted Answer

The answer of Perform the indicated goodness-of-fit test. You roll...

Question 15

Use a $\chi ^ { 2 }$ 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 $\chi ^ { 2 }$...

Question 16

Use a 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. $$\begin{array} { | r | c c | } 
\hline & \text { Correct } & \text { Incorrect } \
\hline \text { Math } & 27 & 53 \
\text { English } & 43 & 37 \
\hline
\end{array}$$

Accepted Answer

The answer of Use a 2 test to test the...

Question 17

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

Accepted Answer

The answer of Describe the null hypothesis for the test...

Question 18

Discuss the three characteristics of a chi-square distribution.

Accepted Answer

The answer of Discuss the three characteristics of a chi-square...

Question 19

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 0.0355. What does the P-value tell us about the claim?

Accepted Answer

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

Question 20

The following table shows the number of employees who called in sick at a business for different days of a
particular week. $$\begin{array} { | r | c c c c c c c | } 
\hline \text { Day } & \text { Sun } & \text { Mon } & \text { Tues } & \text { Wed } & \text { Thurs } & \text { Fri } & \text { Sat } \
\hline \text { Number sick } & 8 & 12 & 7 & 11 & 9 & 11 & 12 \
\hline
\end{array}$$ i) At the 0.05 level of significance, test the claim that sick days occur with equal frequency on the different days of the
week.
ii) Test the claim after changing the frequency for Saturday to 152. Describe the effect of this outlier on the test.

Accepted Answer

The answer of The following table shows the number of...

Quiz 11: Goodness-Of-Fit and Contingency Tables

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

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

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

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

Discuss the three characteristics of a chi-square distribution.

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