Deck 13: Categorical Data Analysis

Question 1

Answer 1

A

Question 2

Many track runners believe that they have a better chance of winning if they start in the inside lane that is closest to the field. For the data below, the lane closest to the field is Lane 1, the next lane is Lane 2, and so on until the outermost lane, Lane 6. The table displays the starting positions for the winnners of 240 competitions. Calculate the chi-square test statistic $\chi ^ { 2 }$ used to test the claim that the probability of winning is the same regardless of starting position.. $$\begin{array} { l | c c c c c c } \text { Starting Position } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Number of Wins } & 32 & 45 & 50 & 36 & 33 & 44\end{array}$$

A) 6.750
B) 9.326
C) 12.592
D) 15.541

Answer 2

6.750

Question 3

Use the appropriate table to find the following probability: $\mathrm { P } \left( \chi ^ { 2 } < 13.277 \right) \text { for } d f = 4.$

A) 0.990
B) 0.010
C) 0.995
D) 0.005

Answer 3

0.990

Question 4

A multinomial experiment with k = 3 cells and n =100 has been conducted and the results are shown in the table. Construct a 99% confidence interval for the multinomial probability associated with cell 2. 3 Perform One-Way Chi Square Test

Question 5

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". 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?

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

Question 6

Many track runners believe that they have a better chance of winning if they start in the inside lane that is closest to the field. For the data below, the lane closest to the field is Lane 1, the next lane is Lane 2, and so on until the outermost lane, Lane 6. The table displays the starting positions for the winners of 240 competitions. Find the rejection region used to test the claim that the probability of winning is the same regardless of starting position. Use α = 0.05. The results are based on 240 wins.

Question 7

Question 8

An adverse drug effect (ADE) is an unintended injury caused by prescribed medication. The table summarizes the proximal cause of 95 ADEs that resulted from a dosing error at a Boston hospital. 2 Construct Confidence Interval

Question 9

Many track runners believe that they have a better chance of winning if they start in the inside lane that is closest to the field. For the data below, the lane closest to the field is Lane 1, the next lane is Lane 2, and so on until the outermost lane, Lane 6. The table displays the starting positions for the winners of 240 competitions. Test the claim that the probability of winning is the same regardless of starting position. Use α = 0.05. The results are based on 240 wins.

Question 10

A multinomial experiment with k = 4 cells and n = 400 produced the data shown in the following table.

Question 11

What is categorical data? 13.2 Testing Categorical Probabilities: One-Way Table 1 Use Chi-Square Distribution

Question 12

A new coffeehouse wishes to see whether customers have any preference among 5 different brands of coffee. A sample of 200 customers provided the data below. Test the claim that the probabilities show no preference. Use α = 0.01. Brand 1 2 3 4 5 Customers 32 65 55 30 18

Question 13

A multinomial experiment with k = 4 cells and n = 300 produced the data shown in the following table.

Question 14

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". 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 90% confidence interval for the multinomial probability associated with the "Disagree Somewhat" response was desired. Which of the following confidence intervals should be used?

A) (0.206, 0.327)
B) (0.216, 0.317)
C) (0.225, 0.309)
D) (0.201, 0.332)

Question 15

Use the appropriate table to find the following chi-square value: $\chi _ { .10 } ^ { 2 }$ for df = 6.

A) 10.645
B) 2.204
C) 9.236
D) 12.017

Question 16

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 α = 0.01.

Question 17

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table.

Question 18

What are characteristics of the trials in a multinomial experiment?

Question 19

A survey of entrepreneurs focused on their job characteristics, work habits, social activities, leisure time, etc. One question put to each entrepreneur was, "What make of car (U.S., Europe, or Japan) do you drive?" The responses (number in each category) for a sample of 100 entrepreneurs are summarized below. The goal of the analysis is to determine if the proportions of entrepreneurs who drive American, European, and Japanese cars differ. U.S. Europe Japan

Question 20

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

Question 21

Economists at USF are researching the problem of absenteeism at U.S. firms. A random sample of 100 U.S. organizations was selected to participate in a 1-year study. As part of the study, the economists had collected data on the following two variables for each company: shiftwork available (Yes or No), and union-management relationship (Good or Poor). As part of their analyses, the economists wanted to determine whether or not a company makes shiftwork available depends on the relationship between union and management. The collected data are shown below:

Question 22

Find the rejection region for a test of independence of two classifications where the contingency table contains r = 2 rows and c = 4 columns and α = .10.

Question 23

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 α = 0.01.

Question 24

A survey of entrepreneurs focused on their job characteristics, work habits, social activities, leisure time, etc. One question put to each entrepreneur was, "What make of car (U.S., Europe, or Japan) do you drive?" The responses (number in each category) for a sample of 100 entrepreneurs are summarized below. The goal of the analysis is to determine if the proportions of entrepreneurs who drive American, European, and Japanese cars differ. $$\begin{array} { c c c } \text { U.S. } & \text { Europe } & \text { Japan } \\\hline 40 & 35 & 25 \\\hline\end{array}$$ In order to determine whether the true proportions in each response category differ, a one-way chi-square analysis should be conducted. Suppose the p-value for the test was calculated to be p = 0.1738. What is the appropriate conclusion to make when testing at α = 0.10?

A) There is sufficient evidence to indicate the proportion of entrepreneurs driving the three makes of car differ.
B) There is insufficient evidence to indicate the proportion of entrepreneurs driving the three makes of car differ.
C) There is sufficient evidence to indicate the proportion of entrepreneurs driving Japanese cars is less than the proportion driving U.S. cars.
D) There is sufficient evidence to indicate the proportion of entrepreneurs driving the three makes of car are equal.

Question 25

A survey of entrepreneurs focused on their job characteristics, work habits, social activities, leisure time, etc. One question put to each entrepreneur was, "What make of car (U.S., Europe, or Japan) do you drive?" The responses (number in each category) for a sample of 100 entrepreneurs are summarized below. The goal of the analysis is to determine if the proportions of entrepreneurs who drive American, European, and Japanese cars differ.

Question 26

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. Assuming the row and column classifications are independent, find an estimate for the expected cell count of cell $E _ { 22 }$ . $$\begin{array} { l | l l l l } & \text { Football } & \text { Basketball } & \text { Soccer } & \text { Baseball } \\\hline \text { Home team wins } & 38 & 155 & 24 & 86 \\\text { Visiting team wins } & 30 & 96 & 20 & 77\end{array}$$

A) 106.4
B) 18.7
C) 144.6
D) 25.3

Question 27

Question 28

A survey of entrepreneurs focused on their job characteristics, work habits, social activities, leisure time, etc. One question put to each entrepreneur was, "What make of car (U.S., Europe, or Japan) do you drive?" The responses (number in each category) for a sample of 100 entrepreneurs are summarized below. The goal of the analysis is to determine if the proportions of entrepreneurs who drive American, European, and Japanese cars differ. $$\begin{array} { c c c } \text { U.S. } & \text { Europe } & \text { Japan } \\\hline 40 & 35 & 25 \\\hline\end{array}$$ In order to determine whether the true proportions 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 "Europe" response was desired. Which of the following confidence intervals should be used?

A) (0.271, 0.428)
B) (0.265, 0.440)
C) (0.257, 0.443)
D) (0.227, 0.473)

Question 29

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 $E _ { 22 }$ . $$\begin{array} { l | c c c } &{ \text { Opinion } } \\\hline \text { Party } & \text { Approve } & \text { Disapprove } & \text { No Opinion } \\\text { Republican } & 42 & 20 & 14 \\\text { Democrat } & 50 & 24 & 18 \\\text { Independent } & 10 & 16 & 6\end{array}$$

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

Question 30

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table.

Question 31

A drug company developed a honey-based liquid medicine designed to calm a child's cough at night. To test the drug, 105 children who were ill with an upper respiratory tract infection were randomly selected to participate in a clinical trial. The children were randomly divided into three groups - one group was given a dosage of the honey drug, the second was given a dosage of liquid DM (an over-the-counter cough medicine), and the third (control group) received a liquid placebo (no dosage at all). After administering the medicine to their coughing child, parents rated their children's cough diagnosis as either better or worse. The results are shown in the table below:

Question 32

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table.

Question 33

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 45-65 group, and 12% for the group over 65. Find the rejection region used to test the claim that all ages have crash rates proportional to their number of drivers. Use α = 0.05.

Question 34

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. Determine if the grade distribution for the department is different than expected. Use α = 0.01.

Question 35

Test the null hypothesis of independence of the two classifications, A and B, of the 3 × 3 contingency table shown below. Test using α = .10.

Question 36

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table.

Question 37

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 α = 0.05.

Question 38

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 45-65 group, and 12\% for the group over 65 . Calculate the chi-square test statistic $\chi ^ { 2 }$ used to test the claim that all ages have crash rates proportional to their driving rates.

$\begin{array}{l|cccc}\text { Age } & \text { Under 26 } & 26-45 & 46-65 & \text { Over 65 } \\\hline \text { Drivers } & 66 & 39 & 25 & 30\end{array}$

A) 75.101
B) 85.123
C) 101.324
D) 95.431

Question 39

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

Question 40

The null hypothesis for a test of data resulting from a multinomial experiment is given as . What is the alternative hypothesis for the test? 13.3 Testing Categorical Probabilities: Two-Way (Contingency) Table 1 Use Chi-Square Distribution

Question 41

Economists at USF are researching the problem of absenteeism at U.S. firms. A random sample of 100 U.S. organizations was selected to participate in a 1-year study. As part of the study, the economists had collected data on the following two variables for each company: shiftwork available (Yes or No), and union-management relationship (Good or Poor). As part of their analyses, the economists wanted to determine whether or not a company makes shiftwork available depends on the relationship between union and management. The collected data are shown below:

Question 42

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 α = 0.01.

Question 43

Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". 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. Which of the following statements is not necessary for the analysis to be valid?

A) The 300 internet users sampled are independent from one another.
B) The probabilities for the four response outcomes remain the same from one internet user to the next.
C) The expected cell counts all must be 30 or more.

Question 44

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. Party Approve Disapprove No Opinion $$\begin{array} { l | c c c } & { \text { Opinion } } \\\hline \text { Party } & \text { Approve } & \text { Disapprove } & \text { No Opinion } \\\text { Republican } & 42 & 20 & 14 \\\text { Democrat } & 50 & 24 & 18 \\\text { Independent } & 10 & 16 & 6\end{array}$$ Find the chi-square test statistic χ2 used to test the claim of independence.

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

Question 45

The χ2-test for independence is a useful tool for establishing a causal relationship between two factors.

Question 46

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. Test the claim that the number of home team and visiting team wins is independent of the sport. Use α = 0.01.

Question 47

In a test of independence, it is safe to conclude that the events are independent when the value of χ2 is very small.

Question 48

The sampling distribution for χ2 works well when expected counts are very small.

Question 49

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. Party Approve Disapprove No Opinion Test the claim of independence. Use α = .05.

Question 50

A business professor conducted a campus survey to estimate demand among all students for a protein supplement for smoothies and other nutritional drinks. Each of 113 students, randomly selected from all students on campus, provided the following information: (1) How health conscious are you? (Very, Moderately, Slightly, Not very) (2) Do you prefer protein supplements in your smoothies? (Yes, No) As part of his analysis, the professor claims that whether or not the student prefers a protein supplement in smoothies is independent of health consciousness level (Very, Moderate, Slightly, or Not very). Use the chi-square distribution to determine the rejection region for this test when testing at α = 0.05.

Question 51

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. Calculate the chi-square test statistic $\chi ^ { 2 }$ used to test the claim that the number of home team and visiting team wins is independent of the sport. Use $\alpha = 0.01$ .
$\begin{array}{l|cccc} & \text { Football } & \text { Basketball } & \text { Soccer } & \text { Baseball } \\\hline \text { Home team wins } & 39 & 156 & 25 & 83 \\\text { Visiting team wins } & 31 & 98 & 19 & 75\end{array}$

A) 3.290
B) 2.919
C) 5.391
D) 4.192

Question 52

A drug company developed a honey-based liquid medicine designed to calm a child's cough at night. To test the drug, 105 children who were ill with an upper respiratory tract infection were randomly selected to participate in a clinical trial. The children were randomly divided into three groups - one group was given a dosage of the honey drug, the second was given a dosage of liquid DM (an over-the-counter cough medicine), and the third (control group) received a liquid placebo (no dosage at all). After administering the medicine to their coughing child, parents rated their children's cough diagnosis as either better or worse. The results are shown in the table below: $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Diagnosis }$
$\begin{array}{lccc}\text { Treatment } & \text { Better } & \text { Worse } & \text { Total } \\\text { Control } & 4 & 33 & 37 \\\text { DM } & 12 & 21 & 33 \\\text { Honey } & 24 & 11 & 35 \\\text { Total } & 40 & 65 & 105\end{array}$
In order to determine whether the treatment group is independent of the coughing diagnosis, a two -way chi-square test was conducted. Suppose the p-value for the test was calculated to be p = 0.0016. What is the appropriate conclusion to make when testing at α = 0.05?

A) There is sufficient evidence to indicate the treatment group is independent of the coughing diagnosis.
B) There is sufficient evidence to indicate the treatment group is dependent on the coughing diagnosis.
C) There is insufficient evidence to indicate the treatment group is dependent on the coughing diagnosis.
D) There is insufficient evidence to indicate the treatment group is independent of the coughing diagnosis.

Question 53

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. Use α = 0.05. Party Approve Disapprove No Opinion

Question 54

Economists at USF are researching the problem of absenteeism at U.S. firms. A random sample of 100 U.S. organizations was selected to participate in a 1-year study. As part of the study, the economists had collected data on the following two variables for each company: shiftwork available (Yes or No), and union-management relationship (Good or Poor). As part of their analyses, the economists wanted to determine whether or not a company makes shiftwork available depends on the relationship between union and management. The collected data are shown below:

Question 55

A professor chose a random sample of 50 recent graduates of an MBA program and recorded the gender of each graduate (M or F) and whether the graduate chose to complete his or her degree requirements by completing a research project (RP) or by taking comprehensive exams (CE). The results are shown below.

Question 56

Consider the accompanying contingency table. a. Convert the values in row 1 to percentages by calculating the percentage of each column total falling in row 1. b. Create a bar graph with row 1 percentage on the vertical axis and column number on the horizontal axis. c. What pattern do you expect to see if the rows and columns are not independent? Is this pattern present in your graph? 13.4 A Word of Caution about Chi-Square Tests 1 Understand Assumptions of Chi-Square Tests

Question 57

A business professor conducted a campus survey to estimate demand among all students for a protein supplement for smoothies and other nutritional drinks. Each of 113 students, randomly selected from all students on campus, provided the following information: (1) How health conscious are you? (Very, Moderately, Slightly, Not very) (2) Do you prefer protein supplements in your smoothies? (Yes, No) As part of his analysis, the professor claims that whether or not the student prefers a protein supplement in smoothies is independent of health consciousness level (Very, Moderate, Slightly, or Not very). Identify the appropriate alternative hypothesis that the professor should use in the test of hypothesis he desires.

A) HA: Preference and Health Consciousness level are independent variables.
B) HA: Preference and Health Consciousness level are dependent variables.
C) HA: There is interaction between the Preference and Health Consciousness variables.
D) HA: Preference and Health Consciousness level are mutually exclusive variables.

Question 58

A drug company developed a honey-based liquid medicine designed to calm a child's cough at night. To test the drug, 105 children who were ill with an upper respiratory tract infection were randomly selected to participate in a clinical trial. The children were randomly divided into three groups - one group was given a dosage of the honey drug, the second was given a dosage of liquid DM (an over-the-counter cough medicine), and the third (control group) received a liquid placebo (no dosage at all). After administering the medicine to their coughing child, parents rated their children's cough diagnosis as either better or worse. The results are shown in the table below:

Question 59

When using any procedure to perform a hypothesis test, the user should always be certain that the experiment satisfies the assumptions given with the procedure.

Question 60

The data below show the age and favorite type of music of 779 randomly selected people. Test the claim that age and preferred music type are independent. Use α = 0.05.