Exam 10: Categorical Data Analysis

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?

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 $\alpha = .10$ .

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: 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 $\alpha = 0.05$ ?

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

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 $E _ { 22 }$ Football Basketball Soccer Baseball Home team wins 38 154 27 84 Visiting team wins 31 98 22 72

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. 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.. Starting Position 1 2 3 4 5 6 Number of Wins 45 50 36 33 32 44

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 $k = 5$ and $\alpha = .025$ .

The sampling distribution for ?² works well when expected counts are very small.

A teacher finds that final grades in the statistics department are distributed as: A, 25\%; B, 25\%; C, $40 \% ; \mathrm { D } , 5 \% ; \mathrm { F } , 5 \%$ . 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 42 36 60 8 14

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 30 18 32 55 65

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 $\alpha = 0.025$ .

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: 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 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.

What are characteristics of the trials in a multinomial experiment?

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

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?

What is categorical data?

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 $\alpha =$ $0.05$ . 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.

The null hypothesis for a test of data resulting from a multinomial experiment is given as $p _ { 1 } = p _ { 2 } = p _ { 3 } = p _ { 4 } = .25$ What is the alternative hypothesis for the test?

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. Use the chi-square distribution to determine the rejection region when testing at $\alpha = 0.025$ .