Exam 13: Categorical Data Analysis

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

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. RESPONSE NUMBER Agree Strongly 60 Agree Somewhat 110 Disagree Somewhat 80 Disagree Strongly 50 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 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 40 35 25 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 multinomial experiment with k = 3 cells and n =30 has been conducted and the results are shown in the table. $\quad$ $\quad$ $\quad$$\text {Cell}$ 1 2 3 16 12 2 Explain why the sample size is not large enough to test whether $p _ { 1 } = .55 , p _ { 2 } = .40$ , and $p _ { 3 } = .05$ .

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. Age Under 26 26-45 46-65 Over 65 Drivers 66 39 25 30

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

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: $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text {Relation}$ Shiftwork Good Bad Total No 11 22 33 Yes 25 42 67 Total 36 64 100 In order to determine whether the shiftwork responses depend on the relationship responses, a two-way chi-square analysis should be conducted. When calculating the test statistic, what values for the expected counts should be used in the calculation? A) $\mathrm { E } _ { 11 } = 11 , \mathrm { E } _ { 12 } = 22 , \mathrm { E } _ { 21 } = 25 , \mathrm { E } _ { 22 } = 42$ B) $\mathrm { E } _ { 11 } = 36 , \mathrm { E } _ { 12 } = 33 , \mathrm { E } _ { 21 } = 64 , \mathrm { E } _ { 22 } = 67$ C) $\mathrm { E } _ { 11 } = 11.88 , \mathrm { E } _ { 12 } = 21.12 , \mathrm { E } _ { 21 } = 24.12 , \mathrm { E } _ { 22 } = 42.88$ D) $\mathrm { E } _ { 11 } = 0.065 , \mathrm { E } _ { 12 } = 0.036 , \mathrm { E } _ { 21 } = 0.032 , \mathrm { E } _ { 22 } = 0.018$

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. Calculate the chi-square test statistic $\chi ^ { 2 }$ used to test the claim that the probabilities show no preference. Brand 1 2 3 4 5 Customers 55 32 18 30 65 A) $37.45$ B) $45.91$ C) $48.91$ D) $55.63$

What are characteristics of the trials in a multinomial experiment?

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. Age Under 26 26-45 46-65 Over 65 Drivers 66 39 25 30 A) $\chi ^ { 2 } > 7.815$ B) $\chi ^ { 2 } > 6.251$ C) $\chi ^ { 2 } > 11.143$ D) $x ^ { 2 } > 9.348$

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 Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Find the chi-square test statistic χ2 used to test the claim of independence.

A teacher finds that final grades in the statistics department are distributed as: $\mathrm { A } , 25 \% ; \mathrm { B } , 25 \% ; \mathrm { C } , 40 \% ; \mathrm { D } , 5 \%$ ; F, 5%. At the end of a randomly selected semester, the following grades were recorded. Calculate the chi-square test statistic $\chi ^ { 2 }$ used to determine if the grade distribution for the department is different than expected. Use $\alpha = 0.01$ Grade A B C D F Number 36 42 60 14 8 A) $5.25$ B) $6.87$ C) $3.41$ D) $4.82$

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. RESPONSE NUMBER Agree Strongly 60 Agree Somewhat 110 Disagree Somewhat 80 Disagree Strongly 50 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. A) $\chi ^ { 2 } = 22.54$ B) $x ^ { 2 } = 0.25$ C) $x ^ { 2 } = 75$ D) $x ^ { 2 } = 28.0$

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 $E _ { 22 }$ . Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6

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 40 35 25 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 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 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$$\text {Diagnosis}$ Treatment Better Worse Total Control 4 33 37 DM 12 21 33 Honey 24 11 35 Total 40 65 105 In order to determine whether the treatment group is independent of the coughing diagnosis, a two-way chi-square test was conducted. Calculate the value of the test statistic for the desired analysis. A) $\chi ^ { 2 } = 28.54$ B) $\chi ^ { 2 } = 9.72$ C) $\chi ^ { 2 } = 15.79$ D) $\chi ^ { 2 } = 25.51$

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

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: $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text {Relation}$ Shiftwork Good Bad Total No 11 22 33 Yes 25 42 67 Total 36 64 100 Use the chi-square distribution to determine the rejection region for this test when testing at $\alpha = 0.05$ . A) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 3.84146$ B) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 5.99147$ C) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 5.02389$ D) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 7.81473$

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 Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Find the rejection region used to test the claim of independence. A) $x ^ { 2 } > 9.488$ B) $x ^ { 2 } > 7.779$ C) $x ^ { 2 } > 11.143$ D) $\chi ^ { 2 } > 13.277$