Exam 13: Categorical Data Analysis

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

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. Football Basketball Soccer Baseball Home team wins 39 156 25 83 Visiting team wins 31 98 19 75 A) $\chi ^ { 2 } > 11.345$ B) $\chi ^ { 2 } > 12.838$ C) $\chi ^ { 2 } > 7.815$ D) $x ^ { 2 } > 9.348$

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. A) $\chi ^ { 2 } > 6.251$ B) $\chi ^ { 2 } > 13.362$ C) $\chi ^ { 2 } > 22.307$ D) $\chi ^ { 2 } > 15.507$

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 $\alpha = , 025$ . A) $X ^ { 2 } > 11.143$ B) $\chi ^ { 2 } > 12.832$ C) $x ^ { 2 } > 0.484$ D) $X ^ { 2 } > 14.449$

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. When calculating the test statistic, what values for the expected counts should be used in the calculation? A) $\mathrm { E } _ { 1 } = 60 , \mathrm { E } _ { 2 } = 110 , \mathrm { E } _ { 3 } = 80 , \mathrm { E } _ { 4 } = 50$ B) $\mathrm { E } _ { 1 } = 75 , \mathrm { E } _ { 2 } = 75 , \mathrm { E } _ { 3 } = 75 , \mathrm { E } _ { 4 } = 75$ C) $\mathrm { E } _ { 1 } = 300 , \mathrm { E } _ { 2 } = 300 , \mathrm { E } _ { 3 } = 300 , \mathrm { E } _ { 4 } = 300$ D) $\mathrm { E } _ { 1 } = 0.25 , \mathrm { E } _ { 2 } = 0.25 , \mathrm { E } _ { 3 } = 0.25 , \mathrm { E } _ { 4 } = 0.25$

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

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 38 155 24 86 Visiting team wins 30 96 20 77

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. Grade A B C D F Number 42 36 60 14 8 A) $\chi ^ { 2 } > 13.277$ B) $\chi ^ { 2 } > 11.143$ C) $\chi ^ { 2 } > 9.488$ D) $x ^ { 2 } > 7.779$

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.

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

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 $\alpha = 0.01$ . Football Basketball Soccer Baseball Home team wins 39 156 25 83 Visiting team wins 31 98 19 75

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. Football Basketball Soccer Baseball Home team wins 39 156 25 83 Visiting team wins 31 98 19 75

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. 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) $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 }$ , where $\mu _ { 1 }$ represents the average number of Internet users in one of the found response categories D) $\mathrm { H } _ { 0 } : \mathrm { p } 1 = 60 , \mathrm { p } _ { 2 } = 110 , \mathrm { p } 3 = 80 , \mathrm { p } 4 = 50$ , where $\mathrm { p } _ { \mathrm { i } }$ represents the proportion of Internet users in one of the four response categories

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. Brand 1 2 3 4 5 Customers 65 18 30 55 32 A) $\chi ^ { 2 } > 13.277$ B) $\chi ^ { 2 } > 9.488$ C) $\chi ^ { 2 } > 11.143$ D) $\chi ^ { 2 } > 14.860$

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

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

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 95% confidence interval for the multinomial probability associated with the "Agree Strongly" response was desired. Which of the following confidence intervals should be used?

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? 13.3 Testing Categorical Probabilities: Two-Way (Contingency) Table 1 Use Chi-Square Distribution

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 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 $\alpha = 0.025$ . A) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 11.1433$ B) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 9.34840$ C) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 9.21034$ D) Reject $\mathrm { H } _ { 0 }$ if $\chi ^ { 2 } > 7.37776$

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