Exam 13: Categorical Data Analysis

A multinomial experiment with k = 3 cells and n =100 has been conducted and the results are shown in the table. $\quad$ $\quad$ $\quad$$\text { Cell }$ 1 2 3 46 32 22 Construct a 99% confidence interval for the multinomial probability associated with cell 2. 3 Perform One-Way Chi Square Test

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

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}$ 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. Use the chi-square distribution to determine the rejection region for this test when testing at $\alpha = 0.025$ . A) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 7.37776$ B) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 9.34840$ C) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 5.99147$ D) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 7.81473$

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. Calculate the value of the test statistic for the desired analysis. A) $\chi ^ { 2 } = 0.15$ B) $x ^ { 2 } = 0.70$ C) $x ^ { 2 } = 11.88$ D) $\chi ^ { 2 } = 0.07$

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. Use the chi-square distribution to determine the rejection region when testing at $\alpha = .05$ . A) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 0.351846$ B) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 9.48773$ C) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 7.81473$ D) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 0.710721$

A multinomial experiment with k = 4 cells and n = 300 produced the data shown in the following table. $\quad$ $\quad$ $\quad$$\text { Cell }$ 1 2 3 4 65 69 80 86 Do these data provide sufficient evidence to contradict the null hypothesis that $p _ { 1 } = .20 , p _ { 2 } = .20 , p _ { 3 } = .30$ , and $p _ { 4 } = .30$ ? Test using $\alpha = .05$ .

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. Which of the following statements is not necessary for the analysis to be valid?

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. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\mathrm { B }$ 19 40 60 55 23 22 31 42 47

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. When calculating the test statistic, what values for the expected counts should be used in the calculation? A) $\mathrm { E } _ { 1 } = 46 , \mathrm { E } _ { 2 } = 44 , \mathrm { E } _ { 3 } = 9$ B) $\mathrm { E } _ { 1 } = 33.33 , \mathrm { E } _ { 2 } = 33.33 , \mathrm { E } _ { 3 } = 33.33$ C) $\mathrm { E } _ { 1 } = 100 , \mathrm { E } _ { 2 } = 100 , \mathrm { E } _ { 3 } = 100$ D) $\mathrm { E } _ { 1 } = 0.45 , \mathrm { E } _ { 2 } = 0.46 , \mathrm { E } _ { 3 } = 0.09$

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 Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Test the claim of independence. Use α = .05.

Consider the accompanying contingency table. Column Row 1 2 3 1 13 15 16 2 19 26 11 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

A multinomial experiment with k = 4 cells and n = 400 produced the data shown in the following table. $\quad$ $\quad$ $\quad$$\text { Cell }$ 1 2 3 4 47 206 104 43 Previous studies in this area have shown that $p _ { 1 } = p _ { 2 } = p _ { 3 } = p _ { 4 } = .25$ . Construct a $95 \%$ confidence interval for the multinomial probability associated with cell 2. A) $( 0.466,0.564 )$ B) $( 0.474,0.556 )$ C) $( 0.086,0.149 )$ D) $( 0.473,0.557 )$

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. WRONG DOSAGE USE NUMBER OF ADEs (1) Lack of knowledge of drug 29 (2) Rule violation 17 (3) Faulty dose checking 13 (4) Slips 9 (5) Other 27 TOTAL 95 In order to determine whether the true percentages of ADEs in the five "cause" categories differ, a chi-square analysis was conducted. Use the chi-square distribution to determine the rejection region when testing at $\alpha =$ $0.025$ . A) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 11.1433$ B) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 12.8325$ C) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 9.48773$ D) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 14.4494$ 2 Construct Confidence Interval

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. M, CE M, RP F, RP M, CE M, CE F, CE F, RP, M, CE M, RP F, RP F, CE M, RP F, RP F, CE M, RP F, RP M, RP F, CE M, CE M, CE M, RP F, CE F, RP M, CE M, CE F, CE M, RP F, RP M, CE M, CE F, CE F, RP, M, CE M, RP F, RP M, CE M, RP F, RP F, CE M, RP F, RP M, RP M, CE M, CE M, CE M, RP F, CE F, RP M, CE F, CE a. Create a contingency table for the data. b. Perform a $\chi ^ { 2 }$ -test to determine if there is any evidence that gender and choice of research project or comprehensive exams are not independent. Use $\alpha = 0.05$ .

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. A) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 9.34840$ B) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 9.48773$ C) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 5.99147$ D) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 7.81473$

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. Starting Position 1 2 3 4 5 6 Number of Wins 45 36 33 44 50 32 A) $\chi ^ { 2 } > 11.070$ В) $x ^ { 2 } > 9.236$ C) $\chi ^ { 2 } > 15.086$ D) $\chi ^ { 2 } > 12.833$

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. Grade A B C D F Number 36 42 60 8 14

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 }$ 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. 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?

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. Age Country Rock Pop Classical 15-21 21 45 90 33 21-30 68 55 42 48 30-40 65 47 31 57 40-50 60 39 25 53

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