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

Accepted Answer

$H _ { 0 }$ : Living accommodations and gender are independent.
$H _ { 1 }$ : Living accommodations and gender are dependent.
Test statistic: $\chi ^ { 2 } = 0.4024 . P$-value $= 0.8177$. Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that type of
living accommodation and gender are independent.

Question 2

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

Accepted Answer

$H _ { 0 }$ : The die is fair (all numbers occur with equal frequency).
$H _ { 1 }$ : The die is not fair.
Test statistic: $\chi ^ { 2 } = 21.75$. Critical value: $\chi ^ { 2 } = 11.070$. Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the die is fair.

Question 3

The following table shows the number of employees who called in sick at a business for
different days of a particular week. $$\begin{array} { r | c c c c c c c } 
\text { Day } & \text { Sun } & \text { Mon } & \text { Tues } & \text { Wed } & \text { Thurs } & \text { Fri } & \text { Sat } \
\hline \text { Number sick } & 8 & 12 & 7 & 11 & 9 & 11 & 12
\end{array}$$ 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

i) $H _ { 0 } : p _ { 0 } = p _ { 1 } = \ldots = p _ { 7 } = 1 / 7 . H _ { 1 } :$ At least one of these probabilities is different from the others. The value of the test statistic is $\chi 2 = 2.4$, which is less than the critical value of $\chi ^ { 2 } = 12.59$. We fail to reject the null hypothesis.
ii) The value of the test statistic is $\chi ^ { 2 } = 579.5$, which is greater than the critical value of $\chi ^ { 2 } = 12.59$. We reject the null hypothesis. An outlier has a significant effect on the $\chi ^ { 2 }$ test statistic.

Question 4

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

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

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

Use a $\chi ^ { 2 }$ 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).
$\begin{array}{r|cc} 
& \text { Drug } & \text { Placebo } \
\hline \text { Headaches } & 11 & 7 \
\text { No headaches } & 73 & 91
\end{array}$

Accepted Answer

The answer of Use a $\chi ^ { 2 }$...

Question 7

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. $$\begin{array} { r | c c c c } 
& \text { College A } & \text { College B } & \text { College C } & \text { College D } \
\hline \text { Smoke } & 17 & 26 & 11 & 34 \
\text { Don't Smoke } & 83 & 74 & 89 & 66
\end{array}$$ Use a 0.01 significance level to test the claim that the proportion of students smoking is the
same at all four colleges.

Accepted Answer

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

Question 8

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 9

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 10

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 11

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. $$\begin{array} { c | c c c } 
& \text { Red } & \text { Blue } & \text { White } \
\hline \text { Car has been in accident } & 28 & 33 & 36 \
\text { Car has not been in accident } & 23 & 22 & 30
\end{array}$$

Accepted Answer

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

Question 12

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 13

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 14

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 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.
$\begin{array}{c|ccc} 
& \text { Rock } & \text { Pop } & \text { Classical } \
\hline \mathbf{1 5 - 2 5} & 50 & 85 & 73 \
\mathbf{2 5 - 3 5} & 68 & 91 & 60 \
\text { 35-45 } & 90 & 74 & 77
\end{array}$
 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

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

Accepted Answer

The answer of Describe a goodness-of-fit test. What assumptions are...

Question 17

Use a $\chi ^ { 2 }$ 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.
$\begin{array}{r|cc} 
& \text { Correct } & \text { Incorrect } \
\hline \text { Math } & 27 & 53 \
\text { English } & 43 & 37
\end{array}$

Accepted Answer

The answer of Use a $\chi ^ { 2 }$...

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

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 20

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

Accepted Answer

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

Quiz 11: Chi-Square and Analysis of Variance

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?

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

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

Discuss the three characteristics of a chi-square distribution.

Describe the null hypothesis for the test of independence. List the assumptions for the χ2\chi ^ { 2 }χ2 test of independence.

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