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ch12 - Ch. 12 Inferences on Categorical Data 12.1...

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Unformatted text preview: Ch. 12 Inferences on Categorical Data 12.1 Goodness-of-Fit Test 1 Perform a Goodness-of-Fit Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Determine the expected counts for each outcome. n = 500 pi Expected Counts A) pi C) pi 0.40 0.15 0.30 0.15 73 151 72 Expected Counts 204 0.40 0.15 0.30 0.15 75 150 75 D) pi 0.40 0.15 0.30 0.15 Expected Counts 2000 750 1500 750 Expected Counts 200 B) pi Expected Counts 0.40 0.15 0.30 0.15 40 15 30 15 0.40 0.15 0.30 0.15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 2) A multinomial experiment with k = 4 cells and n = 300 produced the data shown in the following table. Cell 1 2 ni 69 65 3 80 4 86 Do these data provide sufficient evidence to contradict the null hypothesis that p1 = .20, p2 = .20, p3 = .30, and p4 = .30? Test using α = .05. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3) 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 data lists the number of wins for track runners in the different starting positions. Calculate the chi - square test statistic χ 2 to test the claim that the probabilities of winning are the same in the different positions. Use α = 0.05. The results are based on 240 wins. Starting Position 1 2 3 4 5 6 Number of Wins 45 32 50 33 44 36 A) 6.750 B) 9.326 C) 12.592 D) 15.541 Page 269 4) 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 data lists the number of wins for track runners in the different starting 2 positions. Find the critical value χ 0 to test the claim that the probabilities of winning are the same in the different positions. Use α = 0.05. The results are based on 240 wins. Starting Position 1 2 3 4 5 6 Number of Wins 50 32 45 44 33 36 A) 11.070 B) 9.236 C) 15.086 D) 12.833 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) 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 data lists the number of wins for track runners in the different starting positions. Test the claim that the probabilities of winning are the same in the different positions. Use α = 0.05. The results are based on 240 wins. Starting Position 1 2 3 4 5 6 Number of Wins 32 44 50 36 45 33 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 6) 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 χ 2 to test the claim that the probabilities show no preference. Use α = 0.01. Brand 1 2 3 4 5 Customers 65 32 18 55 30 A) 37.45 B) 45.91 C) 48.91 D) 55.63 7) A coffeehouse wishes to see if customers have any preference among 5 different brands of coffee. A sample of 2 200 customers provided the data below. Find the critical value χ 0 to test the claim that the probabilities show no preference. Use α = 0.01. Brand 1 2 3 4 5 Customers 55 18 30 32 65 A) 13.277 B) 9.488 C) 11.143 D) 14.860 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 8) 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 30 32 65 55 18 Page 270 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 9) A teacher figures 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 number of grades were recorded. Calculate the chi- square test statistic χ 2 to 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) 5.25 B) 6.87 C) 0.6375 D) 4.82 10) A teacher figures 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 number of grades were recorded. Find the 2 critical value χ 0 to 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 14 8 A) 13.277 B) 15.086 C) 9.488 D) 7.779 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 11) A teacher figures 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 number of 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 42 36 60 8 14 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 12) Each side of a standard six- sided die should appear approximately 1/6 of the time when the die is rolled. A player suspects that a certain die is loaded. The suspected die is rolled 90 times. The results are shown below. Calculate the chi - square test statistic χ 2 to test the playerʹs claim. Use α = 0.10. Number 1 2 3 4 5 6 Frequency 12 11 15 17 16 19 A) 3.067 B) 2.143 C) 5.013 D) 4.312 13) Each side of a standard six- sided die should appear approximately 1/6 of the time when the die is rolled. A player suspects that a certain die is loaded. The suspected die is rolled 90 times. The results are shown below. 2 Find the critical value χ 0 to test the playerʹs claim. Use α = 0.10. Number 1 2 3 4 5 6 Frequency 11 17 15 16 12 19 A) 9.236 B) 1.610 C) 10.645 D) 11.071 Page 271 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 14) Each side of a standard six- sided die should appear approximately 1/6 of the time when the die is rolled. A player suspects that a certain die is loaded. The suspected die is rolled 90 times. The results are shown below. Test the playerʹs claim. Use α = 0.10. Number 1 2 3 4 5 6 Frequency 15 17 12 11 19 16 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 15) A random sample of 160 car crashes are selected and categorized by age. 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 χ 2 to test the claim that all ages have crash rates proportional to their driving rates. Use α = 0.05. Age Under 26 26 - 45 46 - 65 Over 65 Drivers 66 39 25 30 A) 75.101 B) 85.123 C) 101.324 D) 95.431 16) A random sample of 160 car crashes are selected and categorized by age. 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 2 the 45- 65 group, and 12% for the group over 65. Find the critical value χ 0 to test the claim that all ages have crash rates proportional to their driving rates. Use α = 0.05. Age Under 26 26 - 45 46 - 65 Over 65 Drivers 66 39 25 30 A) 7.815 B) 6.251 C) 11.143 D) 9.348 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 17) A random sample of 160 car crashes are selected and categorized by age. 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 driving rates. Use α = 0.05. Age Under 26 26 - 45 46 - 65 Over 65 Drivers 66 39 25 30 18) The results of a recent national survey reported that 70% of Americans recycle at least some of the time. As part of their final project in statistics class, Nayla and Roberto survey 5 random students on campus and ask them if they recycle at least some of the time. They then repeat this experiment 1000 times. The results of their research are shown below. X (number who recycle out of 5) 0 1 2 3 4 5 Frequency 2 25 137 306 359 171 Is there evidence to support the belief that the random variable X, the number of students out of 5 who recycle at least some of the time, is a binomial random variable with p = 0.7 at the α = 0.05 level? Page 272 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 19) As the number of the degrees of freedom increases, the χ 2 distribution A) Becomes more symmetric B) Becomes less symmetric C) Does not change shape as the degrees of freedom change D) Becomes exponential 20) A __________________ test is an inferential procedure used to determine whether a frequency distribution follows a defined distribution. A) Goodness- of- fit B) χ 2 C) F D) Normality 21) The degrees of freedom for a χ 2 goodness - of- fit test when there are 6 categories and a sample of size 1200 is A) 5 B) 6 C) 1199 D) 1205 12.2 Tests for Independence and the Homogeneity of Proportions 1 Perform a Test for Independence SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) Test thenull hypothesis of independence of the two classifications, A and B, of the 3 × 3 contingency table shown below. Test using α = .05. B B1 B2 A A1 A2 A3 19 55 31 40 23 42 B3 60 22 47 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) 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. Use α = 0.05. Opinion Approve Disapprove No Opinion Party 42 20 14 Republican 50 24 18 Democrat Independent 10 16 6 2 Find the critical value χ 0 , to test the claim of independence. A) 9.488 B) 7.779 C) 11.143 D) 13.277 Page 273 3) 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. Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Find the chi- square test statistic, χ 2 , to test the claim of independence. A) 8.030 B) 11.765 C) 7.662 D) 9.483 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 4) 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. Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Test the claim of independence. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 5) A researcher wants to determine if the number of minutes spent online per day is independent of gender. A 2 random sample of 315 adults was selected and the results are shown below. Find the critical value χ 0 to determine if there is enough evidence to conclude that the number of minutes spent online per day is related to gender. Use α = 0.05. Minutes spent online per day Gender 0 - 30 30 - 60 60 - 90 90 - over 45 Male 25 35 75 45 45 15 Female 30 A) 7.815 B) 9.348 C) 11.345 D) 6.251 6) A researcher wants to determine if the number of minutes spent online per day is independent of gender. A random sample of 315 adults was selected and the results are shown below. Calculate the chi - square statistic χ 2 to determine if there is enough evidence to conclude that the number of minutes spent online per day is related to gender. Use α = 0.05. Minutes spent online per day Gender 0 - 30 30 - 60 60 - 90 90 - over Male 25 35 75 45 Female 30 45 45 15 A) 18.146 B) 19.874 C) 20.912 D) 21.231 Page 274 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 7) A researcher wants to determine if the number of minutes spent online per day is independent of gender. A random sample of 315 adults was selected and the results are shown below. Is there enough evidence to conclude that the number of minutes spent online per day is related to gender? Use α = 0.05. Minutes spent online per day Gender 0 - 30 30 - 60 60 - 90 90 - over Male 25 35 75 45 Female 30 45 45 15 MULTIPLE CHOICE. Choose the one alternative thatbest completes the statement or answers the question. 8) A medical researcher is interested in determining if there is a relationship between adults over 50 who walk regularly and low, moderate, and high blood pressure. A random sample of 236 adults over 50 is selected and 2 the results are given below. Find the critical value χ 0 to test the claim that walking and low, moderate, and high blood pressure are independent. Use α = 0.01. Blood Pressure Low Moderate Walkers 35 62 Non - walkers 21 65 A) 9.210 High 25 28 C) 6.251 D) 5.991 B) 9.348 9) A medical researcher is interested in determining if there is a relationship between adults over 50 who walk regularly and low, moderate, and high blood pressure. A random sample of 236 adults over 50 is selected and the results are given below. Calculate the chi - square test statistic χ 2 to test the claim that walking and low, moderate, and high blood pressure are independent. Use α = 0.01. Blood Pressure Low Moderate Walkers 35 62 Non - walkers 21 65 A) 3.473 High 25 28 C) 18.112 D) 6.003 B) 16.183 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) A medical researcher is interested in determining if there is a relationship between adults over 50 who walk regularly and low, moderate, and high blood pressure. A random sample of 236 adults over 50 is selected and the results are given below. Test the claim that walking and low, moderate, and high blood pressure are independent. Use α = 0.01. Blood Pressure Low Moderate Walkers 35 62 Non - walkers 21 65 High 25 28 Page 275 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 11) 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 2 below. Find the critical value χ 0 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) 11.345 B) 12.838 C) 7.815 D) 9.348 12) 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 χ 2 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) 3.290 B) 2.919 C) 5.391 D) 4.192 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 13) 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 14) 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 15 - 21 21 21 - 30 68 30 - 40 65 40 - 50 60 Rock 45 55 47 39 Pop Classical 90 33 42 48 31 57 25 53 Page 276 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 15) In a χ 2 test of independence, the null hypothesis is that A) There is not an association B) There is an association C) Each element of each set has the same probability of occurrence D) The random variables are dependent 2 Perform a Test for Homogeneity of Proportions SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) A random sample of 400 men and 400 women was randomly selected and asked whether they planned to vote in the next election. The results are listed below. Perform a homogeneity of proportions test to test the claim that the proportion of men who plan to vote in the next election is the same as the proportion of women who plan to vote. Use α = 0.05. Men Women Plan to vote 230 255 Do not plan to vote 170 145 2) A random sample of 100 students from 5 different colleges was randomly selected, and the number who smoke was recorded. The results are listed below. Perform a homogeneity of proportions test to test the claim that the proportion of students who smoke is the same in all 5 colleges. Use α = 0.01. Colleges 1 2 3 4 5 Smoker 18 25 12 33 22 Nonsmoker 82 75 88 67 78 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3) In a chi- square test of homogeneity of proportions we test the claims that A) Different populations have the same proportions of individuals with the same characteristics. B) Across a single sample the proportion of individuals with the same characteristic is the same as the population. C) The proportion of individuals with a given characteristic doesnʹt change over time. D) The proportion of a population having a given characteristic is based on the homogeneity of the population. Page 277 12.3 Testing the Significance of the Least-Squares Regression Model 1 Understand the Requirements of the Least -Squares Regression Model MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) One of the requirements for conducting inference on the least - squares regression model is that the A) Mean of the response variable changes at a constant rate while the standard deviation remains constant B) Mean of the explanatory variable changes at a constant rate while the standard deviation remains constant C) Mean of the response variable remains constant while the standard deviation changes at a constant rate D) Mean of the explanatory variable remains constant while the standard deviation changes at a constant 2) The least- squares regression model for one explanatory variable is given by the equation A) y i = β 0 + β 1 x i + εi B) y = mx + b C) y - y i = m(x - x i) D) y i = Ai x + Ci Bi i 2 Compute the Standard Error of the Estimate SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in the 15 minute period following the addition of food. The data showing the number of grunts and the age of the warthog (in days) are listed below: Number of Grunts, y 95 73 44 49 68 45 67 22 27 Age (days), x 123 139 153 158 165 172 181 187 193 Compute the standard error, the point estimate for σ. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) Find the standard error of estimate, se, for the data below, given that y = 2x + 1. x 1 2 3 4 y3 5 7 9 A) 0 B) 1 C) 2 D) 3 ^ Page 278 3) Find the standard error of estimate, se, for the data below, given that y = - 2.5x. x - 1 - 2 - 3 - 4 y2 6 7 10 A) 0.866 B) 0.675 C) 0.532 ^ ^ D) 0.349 4) Find the standard error of estimate, se, for the data below, given that y = 2.097x - 0.552. x y 0 2 3 -4 -5 -3 4 1 -1 -2 - 10 - 8 9 1 - 2 - 6 - 1 3 6 - 8 A) 0.976 B) 0.990 C) - 0.990 ^ D) 0.980 5) Find the standard error of estimate, se, for the data below, given that y = - 1.885x + 0.758. x y 1 -1 -2 0 2 3 -4 -5 -3 4 11 6 - 6 - 1 3 4 1 - 4 - 5 8 A) 0.613 B) 0.981 C) 0.312 ^ D) 0.011 6) Find the standard error of estimate, se, for the data below, given that y = - 0.206x + 2.097. x y 1 2 3 -4 -5 -3 4 -1 -2 0 11 - 6 8 - 3 - 2 1 5 - 5 6 7 A) 6.306 B) 3.203 C) 5.918 D) 8.214 7) The data below are the final exam scores of 10 randomly selected statistics students and the number of hours ^ they studied for the exam. Find the standard error of estimate, se, given that y = 5.044x + 56.11. Hours, x Scores, y A) 6.305 3 5 2 8 2 4 4 5 6 3 65 80 60 88 66 78 85 90 90 71 B) 7.913 C) 8.912 D) 9.875 8) The data below are the temperatures on randomly chosen days during a summer class and the number of ^ absences on those days. Find the standard error of estimate, se, given that y = 0.449x - 30.27. Temperature, x Number of absences, y A) 0.934 72 85 91 90 88 98 75 100 80 3 7 10 10 8 15 4 15 5 B) 1.162 C) 1.007 D) 0.815 Page 279 9) The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly ^ selected adults. Find the standard error of estimate, se, given that y = 1.488x + 60.46. 38 41 45 48 51 53 57 61 65 Age, x Pressure, y 116 120 123 131 142 145 148 150 152 A) 4.199 B) 6.981 C) 5.572 D) 3.099 10) The data below are the number of absences and the final grades of 9 randomly selected students from a ^ statistics class. Find the standard error of estimate, se, given that y = - 2.75x + 96.14. Number of absences, x Final grade, y A) 1.798 0 3 6 4 9 2 15 8 5 98 86 80 82 71 92 55 76 82 B) 4.531 C) 3.876 D) 2.160 11) A manager wishes to determine the relationship between the number of miles (in hundreds of miles), the managerʹs sales representatives travel per month, and the amount of sales (in thousands of dollars) per month. ^ Find the standard error of estimate, se, given that y = 3.53x + 37.92. Miles traveled, x Sales, y A) 22.062 2 3 10 7 8 15 3 1 11 31 33 78 62 65 61 48 55 120 B) 15.951 C) 10.569 D) 5.122 12) In order for applicants to work for the foreign - service department, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that applicants have studied a particular language and the grades they received on the proficiency exam. Find the ^ standard of estimate, se, given that y = 6.91x + 46.26. Number of years, x Grades on test, y A) 4.578 3 4 4 5 3 6 2 7 3 61 68 75 82 73 90 58 93 72 B) 3.412 C) 5.192 D) 6.713 13) In an area of the Midwest, records were kept on the relationship between the rainfall (in inches) and the yield ^ of wheat (bushels per acre). Find the standard error of estimate, se, given that y = 4.379x + 4.267. Rain fall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 A) 3.529 B) 4.759 C) 2.813 D) 1.332 14) In linear regression, what is the unbiased estimator of σ called? A) Standard error of the estimate C) Variance B) Standard deviation D) Sample standard deviation Page 280 3 Verify That the Residuals Are Normally Distributed MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) For the least- squares regression model, y i = β 0 + β 1 x i + εi, the predictor variable must be normally distributed. To show that this is true, the ________________ must also be normal. A) Residuals C) Variability 4 Conduct Inference on the Slope SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) Test the claim, at the α = 0.05 level of significance, that a linear relation exists between the two variables, for the ^ data below, given that y = - 2.5x. x - 1 - 2 - 3 - 4 6 7 10 y2 B) Explanatory variable D) Standard error of the estimate 2) Test the claim, at the α = 0.01 level of significance, that a linear relation exists between the two variables, for the ^ data below, given that y = 2.097x - 0.552. x y 0 2 3 -4 -5 -3 4 1 -1 -2 - 10 - 8 9 1 - 2 - 6 - 1 3 6 - 8 3) Test the claim, at the α = 0.10 level of significance, that a linear relation exists between the two variables, for the ^ data below, given that y = - 1.885x + 0.758. x y 1 -1 -2 0 2 3 -4 -5 -3 4 11 6 - 6 - 1 3 4 1 - 4 - 5 8 4) The data below are the temperatures on randomly chosen days during a summer class and the number of absences on those days. Test the claim, at the α = 0.05 level of significance, that a linear relation exists between ^ the two variables, given that y = 0.449x - 30.27. Temperature, x Number of absences, y 72 85 91 90 88 98 75 100 80 3 7 10 10 8 15 4 15 5 5) The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults. Test the claim, at the α = 0.01 level of significance, that a linear relation exists between the two ^ variables, given that y = 1.488x + 60.46. 38 41 45 48 51 53 57 61 65 Age, x Pressure, y 116 120 123 131 142 145 148 150 152 Page 281 6) A breeder of thoroughbred horses wishes to model the relationship between the gestation period and the length of life of a horse. The breeder believes that the two variables may follow a linear trend. The information in the table was supplied to the breeder from various thoroughbred stables across the state. Horse Gestation period x (days) 416 279 298 307 Life Length y (years) 24 25.5 20 21.5 Horse Gestation period x (days) 356 403 265 Life Length y (years) 22 23.5 21 1 2 3 4 5 6 7 Test the claim, at the α =0.05 level of significance, that a linear relation exists between the gestation period and the length of life of a horse. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) If a hypothesis test of the linear relation between the explanatory and the response variable is of the type where H0 : β 1 = 0 , then we are testing the claim that H1 : β 1 > 0 A) The slope of the least square regression model is positive B) The slope of the least squares regression model is negative C) A relationship exist without regard to the sign of the slope D) No linear relationship exists 5 Construct a Confidence Interval about the Slope of the Least -squares Regression Model MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Construct a 95% confidence interval about the slope of the true least - squares regression line, for the data ^ below, given that y = - 2.5x. x - 1 - 2 - 3 - 4 6 7 10 y2 A) (- 4.165, - 0.835) B) (- 6.226, 1.226) C) (- 3.630, - 1.370) D) (- 3.731, - 1.269) 2) Construct a 99% confidence interval about the slope of the true least - squares regression line, for the data ^ below, given that y = 2.097x - 0.552. x y 0 2 3 -4 -5 -3 4 1 -1 -2 - 10 - 8 9 1 - 2 - 6 - 1 3 6 - 8 A) (1.738, 2.456) B) ( - 1.177, 5.371) C) (1.787, 2.407) D) (1.749, 2.445) Page 282 3) Construct a 90% confidence interval about the slope of the true least - squares regression line, for the data ^ below, for the data below, given that y = - 1.885x + 0.758. x y 1 -1 -2 0 2 3 -4 -5 -3 4 11 6 - 6 - 1 3 4 1 - 4 - 5 8 A) (- 2.010, - 1.760) B) (- 3.025, - 0.745) C) (- 1.979, - 1.791) D) (- 2.008, - 1.762) 4) The data below are the temperatures on randomly chosen days during a summer class and the number of absences on those days. Construct a 95% confidence interval about the slope of the true least - squares ^ regression line, for the data below, given that y = 0.449x - 30.27. Temperature, x Number of absences, y A) (0.367, 0.530) 72 85 91 90 88 98 75 100 80 3 7 10 10 8 15 4 15 5 B) (- 1.760, 2.658) C) (0.385, 0.513) D) (0.371, 0.527) 5) The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults.Construct a 95% confidence interval about the slope of the true least - squares regression line, for ^ the data below, given that y = 1.488x + 60.46. 38 41 45 48 51 53 57 61 65 Age, x Pressure, y 116 120 123 131 142 145 148 150 152 A) (1.098, 1.877) B) (- 8.443, 11.419) C) (1.175, 1.801) D) (1.108, 1.868) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) A breeder of Thoroughbred horses wishes to model the relationship between the gestation period and the length of life of a horse. The breeder believes that the two variables may follow a linear trend. The information in the table was supplied to the breeder from various thoroughbred stables across the state. Horse Gestation period x (days) 416 279 298 307 Life Length y (years) 24 25.5 20 21.5 Horse Gestation period x (days) 356 403 265 Life Length y (years) 22 23.5 21 1 2 3 4 5 6 7 Construct a 90% confidence interval about the slope of the true least - squares regression line. Page 283 12.4 Confidence and Prediction Intervals 1 Construct Confidence Intervals for a Mean Response MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Construct a 95% confidence interval about the mean value of y, given x = - 3.5, y = 2.097x - 0.552 and se = 0.976. x y 2 3 -4 -5 -3 4 1 -1 -2 0 - 10 - 8 9 1 - 2 - 6 - 1 3 6 - 8 A) (- 8.921, - 6.862) B) (- 10.367, - 5.417) C) (- 4.598 ,- 1.986) D) (- 12.142 , - 6.475) ^ 2) The data below are the scores of 10 randomly selected students from a statistics class and the number of hours they studied for the exam. Construct a 95% confidence interval about the mean value of y, the score on the final ^ exam, given x = 7 hours, y = 5.044x + 56.11 and se = 6.305. Hours, x Scores, y 3 5 2 8 2 4 4 5 6 3 65 80 60 88 66 78 85 90 90 71 B) (74.54, 108.30) C) (77.21, 110.45) D) (79.16, 112.34) A) (82.840, 99.996) 3) The data below are the temperatures on randomly chosen days during a summer class and the number of absences on those days. Construct a 95% confidence interval about the mean value of y, the number of days ^ absent, given x = 95 degrees, y = 0.449x - 30.27 and se = 0.934. Temperature, x 72 85 91 90 88 98 75 100 80 Number of absences, y 3 7 10 10 8 15 4 15 5 A) (11.378, 13.392) B) (9.957, 14.813) C) (4.321, 6.913) D) (6.345, 8.912) 4) In order for applicants to work for the foreign - service department, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that applicants have studied a particular language and the grades they received on the proficiency exam. Construct ^ a 95% confidence interval about the mean value of y, given x = 2.5, y = 6.91x + 46.26, and se = 4.578. Number of years, x Grades on test, y A) (58.28, 68.79) 3 4 4 5 3 6 2 7 3 61 68 75 82 73 90 58 93 72 B) (51.50, 75.57) C) (55.12, 87.34) D) (47.32, 72.13) 5) In an area of the Midwest, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Construct a 95% confidence interval about the mean value of y, the yield, given ^ x = 11 inches, y = 4.379x + 4.267 and se = 3.529. 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Rainfall (in inches), x Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 A) (49.41, 55.47) B) (43.56, 61.32) Page 284 C) (40.54 , 64.15) D) (39.86, 65.98) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) A company keeps extensive records on its new salespeople on the premise that sales should increase with experience. A random sample of seven new salespeople produced the data on experience and sales shown in the table. Months on Job 2 4 8 12 1 5 9 Monthly Sales y ($ thousands) 2.4 7.0 11.3 15.0 0.8 3.7 12.0 Construct a 90% confidence interval about the mean value of y when x = 5 months. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) How does a confidence interval differ from a prediction interval? A) Confidence intervals are used to measure the accuracy of the mean response of all the individuals in the population, while a prediction interval is used to measure the accuracy of a single individualʹs predicted value. B) Confidence intervals are used to measure the accuracy of a single individualʹs predicted value, while a prediction interval is used to measure the accuracy of the mean response of all the individuals in the population. C) Confidence intervals are constructed about the predicted values of y while prediction intervals a constructed about a particular value of x D) Confidence intervals are constructed about the predicted values of x while prediction intervals a constructed about a particular value of y 8) When constructing a confidence interval about the mean response of y in a linear regression, the t - distribution is used with __________________ degrees of freedom. A) n - 2 B) n - 1 C) n + k - 2 D) n1 + n2 - 2 2 Construct Prediction Intervals for an Individual Response MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Construct a 95% prediction interval for y given x = - 3.5, y = 2.097x - 0.552 and se = 0.976. x y 2 3 -4 -5 -3 4 1 -1 -2 0 - 10 - 8 9 1 - 2 - 6 - 1 3 6 - 8 A) (- 10.367, - 5.417) B) (- 3.187, - 2.154) C) (- 4.598, - 1.986) D) (- 8.921, - 6.862) ^ Page 285 2) The data below are the scores of 10 randomly selected students from a statistics class and the number of hours they studied for the exam. Construct a 95% prediction interval for y, the score on the final exam, given x = 7 ^ hours, y = 5.044x + 56.11 and se = 6.305. Hours, x Scores, y 3 5 2 8 2 4 4 5 6 3 65 80 60 88 66 78 85 90 90 71 B) (55.43, 78.19) C) (77.21, 110.45) D) (82.840, 99.996) A) (74.54, 108.30) 3) The data below are the temperatures on randomly chosen days during a summer class and the number of absences on those days. Construct a 95% prediction interval for y, the number of days absent, given x = 95 ^ degrees, y = 0.449x - 30.27 and se = 0.934. 72 85 91 90 88 98 75 100 80 Temperature, x 7 10 10 8 15 4 15 5 Number of absences, y 3 A) (9.957, 14.813) B) (3.176, 5.341) C) (4.321, 6.913) D) (11.378, 13.392) 4) In order for applicants to work for the foreign - service department, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that applicants have studied a particular language and the grades they received on the proficiency exam. Construct ^ a 95% prediction interval for y given x = 2.5, y = 6.91x + 46.26, and se = 4.578. Number of years, x Grades on test, y A) (51.50, 75.57) 3 4 4 5 3 6 2 7 3 61 68 75 82 73 90 58 93 72 B) (60.23, 91.42) C) (55.12, 87.34) D) (58.28, 68.79) 5) In an area of the Midwest, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Construct a 95% prediction interval for y, the yield, given x = 11 inches, ^ y = 4.379x + 4.267 and se = 3.529. Rainfall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 A) (43.56, 61.32) B) (41.68, 63.21) C) (40.54, 64.15) D) (49.41, 55.47) Page 286 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) A breeder of thoroughbred horses wishes to model the relationship between the gestation period and the length of life of a horse. The breeder believes that the two variables may follow a linear trend. The information in the table was supplied to the breeder from various thoroughbred stables across the state. Horse Gestation period x (days) 416 279 298 307 Life Length y (years) 24 25.5 20 21.5 Horse Gestation period x (days) 356 403 265 Life Length y (years) 22 23.5 21 1 2 3 4 5 6 7 Construct a 95% prediction interval about the value of y when x = 300 days. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) Is a confidence interval or a prediction interval the wider interval at the same level of significance? A) Prediction Interval B) Confidence interval Page 287 Ch. 12 Inferences on Categorical Data Answer Key 12.1 Goodness-of-Fit Test 1 Perform a Goodness-of-Fit Test 1) A 2 2) Since χ 2 = 3.056 < χ .05 = 7.815 (df = 3), we do not reject the null hypothesis. There is not sufficient evidence that the cell proportions differ from those given in the null hypothesis. 3) A 4) A 2 5) critical value χ 0 = 11.070; chi - square test statistic χ 2 = 6.750; fail to reject H0 ; There is not sufficient evidence to reject the claim. It seems that the probability of winning in different lanes is the same. 6) A 7) A 2 8) critical value χ 0 = 13.277; chi - square test statistic χ 2 = 37.45; reject H0 ; There is sufficient evidence to reject the claim that customers show no preference for the brands. 9) A 10) A 2 11) critical value χ 0 = 13.277; chi - square test statistic χ 2 = 5.25; fail to reject H0 ; There is not sufficient evidence to support the claim that the grades are different than expected. 12) A 13) A 2 14) critical value χ 0 = 9.236; chi- square test statistic χ 2 = 3.067; fail to reject H0 ; There is not sufficient evidence to support the claim of a loaded die. 15) A 16) A 2 17) critical value χ 0 = 7.815; chi- square test statistic χ 2 = 75.101; reject H0 ; There is sufficient evidence to reject the claim that all ages have the same crash rate. 2 18) Since χ 2 = 0.717 < χ 0.05 = 11.070 (with 5 degrees of freedom), we do not reject the null hypothesis. There is evidence at the α = 0.05 level that X is a binomial random variable with n = 5 and p = 0.7. 19) A 20) A 21) A 12.2 Tests for Independence and the Homogeneity of Proportions 1 Perform a Test for Independence 2 1) Since χ 2 = 42.857 > χ .05 = 9.488, we reject the null hypothesis. There is sufficient evidence to reject the claim that A and B are independent. 2) A 3) A 2 4) critical value χ 0 = 9.488; chi- square test statistic χ 2 = 8.030; fail to reject H0 ; There is not sufficient evidence to reject the claim of independence. Page 288 5) A 6) A 2 7) critical value χ 0 = 7.815; chi- square test statistic χ 2 = 18.146; reject H0 ; There is sufficient evidence to reject the claim of independence. 8) A 9) A 2 10) critical value χ 0 = 9.210; chi- square test statistic χ 2 = 3.473; fail to reject H0 ; There is not sufficient evidence to reject the claim of independence. 11) A 12) A 2 13) critical value χ 0 = 11.345; chi - square test statistic χ 2 = 3.290; fail to reject H0 ; There is not sufficient evidence to reject the claim of independence. 2 14) critical value χ 0 = 16.919; chi - square test statistic χ 2 = 91.097; reject H0 ; There is sufficient evidence to reject the claim of independence. 15) A 2 Perform a Test for Homogeneity of Proportions 2 1) critical value χ 0 = 3.841; chi- square test statistic χ 2 = 3.273; fail to reject H0 ; There is not sufficient evidence to reject the claim. 2 2) critical value χ 0 = 13.277; chi - square test statistic χ 2 = 14.336; reject H0 ; There is sufficient evidence to reject the claim. 3) A 12.3 Testing the Significance of the Least-Squares Regression Model 1 Understand the Requirements of the Least -Squares Regression Model 1) A 2) A 2 Compute the Standard Error of the Estimate 1) s = 15.35 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 3 Verify That the Residuals Are Normally Distributed 1) A Page 289 4 Conduct Inference on the Slope 1) Since t = - 6.455 < - t0.025 = - 4.303, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 2) Since t = 19.510 > t 0.005 =3.355, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 3) Since t = - 27.929 < - t0.05 = 1.860, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 4) Since t = 13.031 > t 0.025 = 2.365, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 5) Since t = 9.034 > t 0.005 = 3.499, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 6) Since t = 0.814 < t 0.025 = 2.571, we do not reject the null hypothesis. There is not sufficient evidence to support the claim of a linear relationship between the two variables between the gestation period and the length of life of a horse. 7) A 5 Construct a Confidence Interval about the Slope of the Least -squares Regression Model 1) A 2) A 3) A 4) A 5) A 6) b1 = 0.01087 s b = 0.0134 1 t0.05 = 2.015 90% confidence interval = ( - 0.0161, 0.0379) 12.4 Confidence and Prediction Intervals 1 Construct Confidence Intervals for a Mean Response 1) A 2) A 3) A 4) A 5) A ^ 6) For x = 5, y = - 0.25 + 1.315(5) = 6.325 The confidence interval is of the form: ^ 1 (x - x)2 + y ± tα/2s n SSxx Confidence coefficient 0.90 = 1 - α ⇒ α = 1 - 0.90 = 0.10. α/2 = 0.10/2 = 0.05. From a Studentʹs t table, t0.05 = 2.015 with n - 2 = 7 - 2 = 5 df. The confidence interval is: 1 (5 - 5.8571) 2 + ⇒ 6.325 ± 1.233 ⇒ (5.092, 7.558) 6.325 ± 2.015(1.577) 7 94.8571 7) A 8) A 2 Construct Prediction Intervals for an Individual Response 1) A 2) A 3) A 4) A 5) A Page 290 6) The prediction interval is of the form: ^ 1 (x - x)2 y ± tα/2s 1 + + n SSxx y = 18.89 + 0.01087(300) = 22.151 Confidence coefficient 0.95 = 1 - α ⇒ α = 1 - 0.95 = 0.05. α/2 = 0.05/2 = 0.025. From a Studentʹs t table, t0.025 = 2.571 with n - 2 = 7 - 2 = 5 df. The 95% prediction interval is: 1 (300 - 332)2 ⇒ 22.151 ± 5.528 ⇒ (16.623, 27.679) 22.151 ± 2.571(1.971) 1 + + 7 21,752 7) A ^ Page 291 ...
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