Question 1

The $\chi ^ { 2 }$-test for independence is a useful tool for establishing a causal relationship between two factors.

Accepted Answer

A) True 
 B)False  False

Question 2

Describe probabilities of the k outcomes of the multinomial experiment trials.

Accepted Answer

The probabilities of the k outcomes remain the same from trial to trial and sum to 1.

Question 3

What are characteristics of the trials in a multinomial experiment?

Accepted Answer

The trials are independent; there are k possible outcomes to each trial; the
experiment consists of n identical trials.

Question 4

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: ʺThe government needs to be able to scan
Internet messages and user communications to prevent fraud and other crimes.ʺ The possible
Responses were ʺagree stronglyʺ, ʺagree somewhatʺ, ʺdisagree somewhatʺ, and ʺdisagree stronglyʺ.
The number of Internet users in each category is summarized in the table. $\begin{array}{l|l}
\text { RESPONSE } & \text { NUMBER } \
\hline \text { Agree Strongly } & 60 \
\text { Agree Somewhat } & 110 \
\text { Disagree Somewhat } & 80 \
\text { Disagree Strongly } & 50
\end{array}$

In order to determine whether the true proportions of Internet users in each response category differ, a one-way chi-square analysis should be conducted. As part of that analysis, a $95 \%$ confidence interval for the multinomial probability associated with the "Agree Strongly" response was desired. Which of the following confidence intervals should be used?

Accepted Answer

A)  $( 0.141,0.259 )$ 
B)  $( 0.162,0.238 )$ 
C)  $( 0.155,0.245 )$ 
D)  $( 0.145,0.255 )$ 
A)  $( 0.141,0.259 )$ 
B)  $( 0.162,0.238 )$ 
C)  $( 0.155,0.245 )$ 
D)  $( 0.145,0.255 )$

Question 5

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: ʺThe government needs to be able to scan
Internet messages and user communications to prevent fraud and other crimes.ʺ The possible
Responses were ʺagree stronglyʺ, ʺagree somewhatʺ, ʺdisagree somewhatʺ, and ʺdisagree stronglyʺ.
The number of Internet users in each category is summarized in the table. $\begin{array}{l|l}
\text { RESPONSE } & \text { NUMBER } \
\hline \text { Agree Strongly } & 60 \
\text { Agree Somewhat } & 110 \
\text { Disagree Somewhat } & 80 \
\text { Disagree Strongly } & 50
\end{array}$

In order to determine whether the true proportions of Internet users in each response category differ, a one-way chi-square analysis should be conducted. Calculate the value of the test statistic for the desired analysis.

Accepted Answer

A)  $\chi ^ { 2 } = 28.0$ 
B)  $\chi ^ { 2 } = 22.54$ 
C)  $x ^ { 2 } = 75$ 
D)  $\chi ^ { 2 } = 0.25$ 
A)  $\chi ^ { 2 } = 28.0$ 
B)  $\chi ^ { 2 } = 22.54$ 
C)  $x ^ { 2 } = 75$ 
D)  $\chi ^ { 2 } = 0.25$

Question 6

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: ʺThe government needs to be able to scan
Internet messages and user communications to prevent fraud and other crimes.ʺ The possible
Responses were ʺagree stronglyʺ, ʺagree somewhatʺ, ʺdisagree somewhatʺ, and ʺdisagree stronglyʺ.
The number of Internet users in each category is summarized in the table. $\begin{array}{l|l}
\text { RESPONSE } & \text { NUMBER } \
\hline \text { Agree Strongly } & 60 \
\text { Agree Somewhat } & 110 \
\text { Disagree Somewhat } & 80 \
\text { Disagree Strongly } & 50
\end{array}$

Specify the null hypothesis for testing whether the true proportions of Internet users in each response category are equal.

Accepted Answer

A)  $\mathrm { H } _ { 0 } : \mathrm { p } _ { 1 } = \mathrm { p } _ { 2 } = \mathrm { p } _ { 3 } = \mathrm { p } _ { 4 } = 0.25$, where $\mathrm { p } _ { 1 }$ represents the proportion of Internet users in one of the four response categories 
B)  $\mathrm { H } _ { 0 }$ : Internet users and Government scanning are independent 
C)  $\mathrm { H } _ { 0 } : \mathrm { p } 1 = 60 , \mathrm { P } _ { 2 } = 110 , \mathrm { P } 3 = 80 , \mathrm { p } 4 = 50$, where $\mathrm { p } _ { \mathrm { i } }$ represents the proportion of Internet users in one of the four response categories 
D)  $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 }$, where $\mu _ { \mathrm { i } }$ represents the average number of Internet users in one of the found response categories 
A)  $\mathrm { H } _ { 0 } : \mathrm { p } _ { 1 } = \mathrm { p } _ { 2 } = \mathrm { p } _ { 3 } = \mathrm { p } _ { 4 } = 0.25$, where $\mathrm { p } _ { 1 }$ represents the proportion of Internet users in one of the four response categories 
B)  $\mathrm { H } _ { 0 }$ : Internet users and Government scanning are independent 
C)  $\mathrm { H } _ { 0 } : \mathrm { p } 1 = 60 , \mathrm { P } _ { 2 } = 110 , \mathrm { P } 3 = 80 , \mathrm { p } 4 = 50$, where $\mathrm { p } _ { \mathrm { i } }$ represents the proportion of Internet users in one of the four response categories 
D)  $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 }$, where $\mu _ { \mathrm { i } }$ represents the average number of Internet users in one of the found response categories

Question 7

A teacher finds that final grades in the statistics department are distributed as: A, 25\%; B, 25\%; C, $40 \%$; D, $5 \% ; F , 5 \%$. At the end of a randomly selected semester, the following grades were recorded. Find the rejection region used to determine if the grade distribution for the department is different than expected. Use $\alpha = 0.01$.
$\begin{array}{l|lllll}
\text { Grade } & \text { A } & \text { B } & \text { C } & \text { D } & \text { F } \
\hline \text { Number } & 36 & 42 & 60 & 8 & 14
\end{array}$

Accepted Answer

A)  $\chi ^ { 2 } > 13.277$ 
B)  $\chi ^ { 2 } > 9.488$ 
C)  $\chi ^ { 2 } > 7.779$ 
D)  $\chi ^ { 2 } > 11.143$ 
A)  $\chi ^ { 2 } > 13.277$ 
B)  $\chi ^ { 2 } > 9.488$ 
C)  $\chi ^ { 2 } > 7.779$ 
D)  $\chi ^ { 2 } > 11.143$

Question 8

A coffeehouse wishes to see if customers have any preference among 5 different brands of coffee. A sample of 200 customers provided the data below. Find the rejection region used to test the claim that the probabilities show no preference. Use $\alpha = 0.01$.
$\begin{array}{l|cllll}
\text { Brand } & 1 & 2 & 3 & 4 & 5 \
\hline \text { Customers } & 32 & 65 & 18 & 55 & 30
\end{array}$

Accepted Answer

A)  $\chi ^ { 2 } > 14.860$ 
B)  $\chi ^ { 2 } > 11.143$ 
C)  $\chi ^ { 2 } > 13.277$ 
D)  $\chi ^ { 2 } > 9.488$ 
A)  $\chi ^ { 2 } > 14.860$ 
B)  $\chi ^ { 2 } > 11.143$ 
C)  $\chi ^ { 2 } > 13.277$ 
D)  $\chi ^ { 2 } > 9.488$

Question 9

Use the appropriate table to find the following chi-square value: $\chi _ { .05 } ^ { 2 }$ for $\mathrm { df } = 3$.

Accepted Answer

A)  $9.488$ 
B)  $0.352$ 
C)  $7.815$ 
D)  $5.991$ 
A)  $9.488$ 
B)  $0.352$ 
C)  $7.815$ 
D)  $5.991$

Question 10

The contingency table below shows the results of a random sample of 200 state
representatives that was conducted to see whether their opinions on a bill are related to
their party affiliations. $$\begin{array}{l}
\begin{array} { l | c c c } 
& { \text { Opinion } } \
\hline \text { Party } & \text { Approve } & \text { Disapprove } & \text { No Opinion } \
\text { Republican } & 42 & 20 & 14 \
\text { Democrat } & 50 & 24 & 18 \
\text { Independent } & 10 & 16 & 6
\end{array}\
\text { Test the claim of independence. Use } \alpha = .05 \text {. }
\end{array}$$

Accepted Answer

The rejection region is @#LAT-DLM&; chi-

Question 11

Find the rejection region for a one-dimensional chi-square test of a null hypothesis concerning $p _ { 1 } , p _ { 2 } , \ldots p _ { \mathrm { k } }$ if $k = 4$ and $\alpha = .01$.

Accepted Answer

A)  $\chi ^ { 2 } > 11.345$ 
B)  $\chi ^ { 2 } > 0.115$ 
C)  $\chi ^ { 2 } > 13.277$ 
D)  $\chi ^ { 2 } > 15.086$ 
A)  $\chi ^ { 2 } > 11.345$ 
B)  $\chi ^ { 2 } > 0.115$ 
C)  $\chi ^ { 2 } > 13.277$ 
D)  $\chi ^ { 2 } > 15.086$

Question 12

The contingency table below shows the results of a random sample of 200 state representatives that was conducted to see whether their opinions on a bill are related to their party affiliation. $\begin{array}{l|ccc} 
&{\text { Opinion }} \
\hline \text { Party } & \text { Approve } & \text { Disapprove } & \text { No Opinion } \
\text { Republican } & 42 & 20 & 14 \
\text { Democrat } & 50 & 24 & 18 \
\text { Independent } & 10 & 16 & 6
\end{array}$

Find the chi-square test statistic $\chi ^ { 2 }$ used to test the claim of independence.

Accepted Answer

A)  $9.483$ 
B)  $11.765$ 
C)  $7.662$ 
D)  $8.030$ 
A)  $9.483$ 
B)  $11.765$ 
C)  $7.662$ 
D)  $8.030$

Question 13

The contingency table below shows the results of a random sample of 200 state representatives that was conducted to see whether their opinions on a bill are related to their party affiliations.
Assuming the row and column classifications are independent, find an estimate for the expected cell count $E _ { 22 }$.

Accepted Answer

A)  $46.92$ 
B)  $22.8$ 
C)  $17.48$ 
D)  $27.6$ 
A)  $46.92$ 
B)  $22.8$ 
C)  $17.48$ 
D)  $27.6$

Question 14

A sports researcher is interested in determining if there is a relationship between the number of home team and visiting team wins and different sports. A random sample of 526 games is selected and the results are given below. Find the rejection region used to test the claim that the number of home team and visiting team wins is independent of the sport. Use $\alpha = 0.01$.
$\begin{array}{l|cccc} 
& \text { Football } & \text { Basketball } & \text { Soccer } & \text { Baseball } \
\hline \text { Home team wins } & 39 & 156 & 25 & 83 \
\text { Visiting team wins } & 31 & 98 & 19 & 75
\end{array}$

Accepted Answer

A)  $\chi ^ { 2 } > 12.838$ 
B)  $\chi ^ { 2 } > 9.348$ 
C)  $x ^ { 2 } > 7.815$ 
D)  $\chi ^ { 2 } > 11.345$ 
A)  $\chi ^ { 2 } > 12.838$ 
B)  $\chi ^ { 2 } > 9.348$ 
C)  $x ^ { 2 } > 7.815$ 
D)  $\chi ^ { 2 } > 11.345$

Question 15

A random sample of 160 car accidents are selected and categorized by the age of the driver determined to be at fault. The results are listed below. The age distribution of drivers for the given categories is $18 \%$ for the under 26 group, $39 \%$ for the $26 - 45$ group, $31 \%$ for the $46 - 65$ group, and $12 \%$ for the group over 65 . Test the claim that all ages have crash rates proportional to their number of drivers. Use $\alpha = 0.05$.
$\begin{array}{l|cccc}
\text { Age } & \text { Under 26 } & 26-45 & 46-65 & \text { Over } 65 \
\hline \text { Drivers } & 66 & 39 & 25 & 30
\end{array}$

Accepted Answer

The rejection region is @#LAT-DLM&; chi-

Question 16

$$\text { The sampling distribution for } \chi ^ { 2 } \text { works well when expected counts are very small. }$$

Accepted Answer

A) True 
 B)False

Question 17

In a test of independence, it is safe to conclude that the events are independent when the value of $\chi ^ { 2 }$ is very small.

Accepted Answer

A) True 
 B)False

Question 18

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: ʺThe government needs to be able to scan
Internet messages and user communications to prevent fraud and other crimes.ʺ The possible
Responses were ʺagree stronglyʺ, ʺagree somewhatʺ, ʺdisagree somewhatʺ, and ʺdisagree stronglyʺ.
The number of Internet users in each category is summarized in the table. $\begin{array}{l|l}
\text { RESPONSE } & \text { NUMBER } \
\hline \text { Agree Strongly } & 60 \
\text { Agree Somewhat } & 110 \
\text { Disagree Somewhat } & 80 \
\text { Disagree Strongly } & 50
\end{array}$

In order to determine whether the true proportions of Internet users in each response category differ, a one-way chi-square analysis should be conducted. When calculating the test statistic, what values for the expected counts should be used in the calculation?

Accepted Answer

A)  $\mathrm { E } _ { 1 } = 0.25 , \mathrm { E } _ { 2 } = 0.25 , \mathrm { E } _ { 3 } = 0.25 , \mathrm { E } _ { 4 } = 0.25$ 
B)  $\mathrm { E } _ { 1 } = 75 , \mathrm { E } _ { 2 } = 75 , \mathrm { E } _ { 3 } = 75 , \mathrm { E } _ { 4 } = 75$ 
C)  $\mathrm { E } _ { 1 } = 60 , \mathrm { E } _ { 2 } = 110 , \mathrm { E } _ { 3 } = 80 , \mathrm { E } _ { 4 } = 50$ 
D)  $\mathrm { E } _ { 1 } = 300 , \mathrm { E } _ { 2 } = 300 , \mathrm { E } _ { 3 } = 300 , \mathrm { E } _ { 4 } = 300$ 
A)  $\mathrm { E } _ { 1 } = 0.25 , \mathrm { E } _ { 2 } = 0.25 , \mathrm { E } _ { 3 } = 0.25 , \mathrm { E } _ { 4 } = 0.25$ 
B)  $\mathrm { E } _ { 1 } = 75 , \mathrm { E } _ { 2 } = 75 , \mathrm { E } _ { 3 } = 75 , \mathrm { E } _ { 4 } = 75$ 
C)  $\mathrm { E } _ { 1 } = 60 , \mathrm { E } _ { 2 } = 110 , \mathrm { E } _ { 3 } = 80 , \mathrm { E } _ { 4 } = 50$ 
D)  $\mathrm { E } _ { 1 } = 300 , \mathrm { E } _ { 2 } = 300 , \mathrm { E } _ { 3 } = 300 , \mathrm { E } _ { 4 } = 300$

Question 19

Test the null hypothesis of independence of the two classifications, $\mathrm { A }$ and $\mathrm { B }$, of the $3 \times 3$ contingency table shown below. Test using $\alpha = .025$.

Accepted Answer

Since @#LAT-DLM&, we reject the null hyp

Question 20

A multinomial experiment with k = 3 cells and n =30 has been conducted and the results
are shown in the table.

Explain why the sample size is not large enough to test whether $p _ { 1 } = .55 , p _ { 2 } = .40$, and $p _ { 3 } = .05$.

Accepted Answer

E(n3)= .05(30)= 1.5

The $\chi ^ { 2 }$ -test for independence is a useful tool for establishing a causal relationship between two factors.

Describe probabilities of the k outcomes of the multinomial experiment trials.

What are characteristics of the trials in a multinomial experiment?

Use the appropriate table to find the following chi-square value: $\chi _ { .05 } ^ { 2 }$ for $\mathrm { df } = 3$ .

Find the rejection region for a one-dimensional chi-square test of a null hypothesis concerning $p _ { 1 } , p _ { 2 } , \ldots p _ { \mathrm { k } }$ if $k = 4$ and $\alpha = .01$ .

$\text { The sampling distribution for } \chi ^ { 2 } \text { works well when expected counts are very small. }$

In a test of independence, it is safe to conclude that the events are independent when the value of $\chi ^ { 2 }$ is very small.

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Discrete Random Variables

Continuous Random Variables

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single

Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Analysis of Variance: Comparing More Than Two Means

Simple Linear Regression

Multiple Regression and Model Building

Nonparametric Statistics Available Online

Filters

Exam 13: Categorical Data Analysis

The χ2\chi ^ { 2 }χ2 -test for independence is a useful tool for establishing a causal relationship between two factors.

Describe probabilities of the k outcomes of the multinomial experiment trials.

What are characteristics of the trials in a multinomial experiment?

Use the appropriate table to find the following chi-square value: χ.052\chi _ { .05 } ^ { 2 }χ.052​ for df=3\mathrm { df } = 3df=3 .

Find the rejection region for a one-dimensional chi-square test of a null hypothesis concerning p1,p2,…pkp _ { 1 } , p _ { 2 } , \ldots p _ { \mathrm { k } }p1​,p2​,…pk​ if k=4k = 4k=4 and α=.01\alpha = .01α=.01 .

The sampling distribution for χ2 works well when expected counts are very small. \text { The sampling distribution for } \chi ^ { 2 } \text { works well when expected counts are very small. } The sampling distribution for χ2 works well when expected counts are very small.

In a test of independence, it is safe to conclude that the events are independent when the value of χ2\chi ^ { 2 }χ2 is very small.

Test the null hypothesis of independence of the two classifications, A\mathrm { A }A and B\mathrm { B }B , of the 3×33 \times 33×3 contingency table shown below. Test using α=.025\alpha = .025α=.025 .

A multinomial experiment with k = 3 cells and n =30 has been conducted and the results are shown in the table. Explain why the sample size is not large enough to test whether p1=.55,p2=.40p _ { 1 } = .55 , p _ { 2 } = .40p1​=.55,p2​=.40 , and p3=.05p _ { 3 } = .05p3​=.05 .

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Discrete Random Variables

Continuous Random Variables

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single

Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Analysis of Variance: Comparing More Than Two Means

Simple Linear Regression

Multiple Regression and Model Building

Nonparametric Statistics Available Online

Filters

The $\chi ^ { 2 }$ -test for independence is a useful tool for establishing a causal relationship between two factors.

Use the appropriate table to find the following chi-square value: $\chi _ { .05 } ^ { 2 }$ for $\mathrm { df } = 3$ .

Find the rejection region for a one-dimensional chi-square test of a null hypothesis concerning $p _ { 1 } , p _ { 2 } , \ldots p _ { \mathrm { k } }$ if $k = 4$ and $\alpha = .01$ .

$\text { The sampling distribution for } \chi ^ { 2 } \text { works well when expected counts are very small. }$

In a test of independence, it is safe to conclude that the events are independent when the value of $\chi ^ { 2 }$ is very small.