STAT 3507 Exercise Questions Solutions

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STAT 3507 Exercise Questions Solutions Click Link Below To Buy: http://hwcampus.com/shop/stat-3507-exercise-questions-solutions/ Bernoulli Sampling Questions Cluster Sampling Questions Ratio, Regression and Difference Estimation Stratified Random Sampling Systematic Sampling Two-stage Cluster Sampling Questions 1-A university professor who is correcting 600 written examinations decides to get a preliminary idea of the passing rate on the test. He decides to use a simple randomized scheme to single out a smaller number of exam copies for first-hand correction. In passing through the pile of exams, he tosses an ordinary six-sided die, once for each exam copy. If the die shows a 6, he corrects the corresponding exam, otherwise not. Suppose the sample selected in this way consists of 90 students and that 60 out of these are found to have passed. a. Identify the sampling design implemented. b. Estimate the total number of students who passed the exams c. Compute a 95% confidence interval, based on the normal approximation, for the number of passing students (among the 600). d. Repeat (b) and (c) using the alternative (improved) estimator. 2- Table: Number of Employees and Range Division Number of employees Cumulative range 1 1200 1— 1200 2 450 1201 —1650 3 2100 1651— 3750 4 860 3751— 4610 5 2840 4611 — 7450 6 1910 7451— 9350 7 390 9361 — 9750 8 3200 9751 - 12950 12950 i. Suppose 2011, 7972 and 10281 are the random numbers generated between 1 and 12950. Using this information what are the clusters selected? ii. Suppose the total number of sick days used by the three sampled divisions during the past quarter are respectively, y, = 4320 y2 = 4160 y3 = 5790 . Estimate the average number of sick clays used per person for the entire firm and place a bound on the error of estimation. Ratio and Regression Estimators (Examples) under Simple Random Sampling Without Replacement Set 1: Estimation of Ratio Let us consider a community having eight community areas. Suppose that we wish to estimate the ratio R of total pharmaceutical expenses Y to total medical expenses X among all persons in the community. To do this, a simple random sample of two community areas is to be taken and every household in each sample community area is to be interviewed. The data for the community areas am as given in Table 1 below: Table 1: Pharmaceutical Expenses and Total Medical Expenses Among All Residents of Eight Community Areas Community Area Total Pharmaceutical Expenses, Y ($) Total Medical Expenses, X ($) 1 100,000 300.000 2 50,000 200.000 3 75,000 300.000 4 200,000 600.000 5 150,000 450,000 6 175,000 520.000 7 170,000 680.000 8 150,000 450.000 Total 1,070,000 3,500,000 Suppose that community areas 2 and 5 were selected in the sample. a. State the target population for the study b. What are the elements in this sampling design? c. What am the sampling units? d. Estimate the ratio of total pharmaceutical expenses to total medical expenses. e. Estimate the variance of the estimate in (d). f. Based on the sample data, how many community areas would have to be sampled if it is desired to estimate the population ratio with 95% certainty to within 5% of its true value? Set 2: Estimation of Total Suppose that a road having a length of 24 miles traverses areas that can be classified as urban and rural and that the road is divided into eight segments, each having a length equal to 3 miles. A sample of three segments is taken, and on each segment sampled, special equipment is installed for purposes of counting the number of total motor vehicle miles traveled by cars and trucks on the segment during a particular year. In addition, a record of all accidents occurring on each sample segment is kept. The number of truck miles and the number of accidents in which a truck was involved during a certain period are given in Table 2 for each of the eight segments in the population. Suppose that we take a simple random sample of three segments for purposes of estimating the total number of truck miles traveled on the road. Table 2: Truck Miles and Number of Accidents Involving Trucks by Type of Road Segment Segment Type Truck Miles x 1000 Number of Accidents Urban 6327 oo in Cs CT in ,-1 cr% ct Rural 2555 Urban 8691 Urban 7834 V, Rural 1586 Rural 2034 Rural 2015 Rural 3012 Suppose the segments 1, 3 and 4 were selected in the sample. a. Estimate the total number of truck miles traveled on the mad using the customary and ratio estimators. b. Estimate the 95% confidence interval for the total number of truck miles using the customary and the ratio estimators. c. How do these estimators compare? d. Based on the sample data, how many road segments would have to be sampled if it is desired to estimate the total number of truck miles with 95% certainty to within 10% of its true value? Use the customary estimator and the ratio estimator. Example I: Cavities and Post -stratification Two dentists conduct a survey on the condition of teeth of 200 children in a village. The first dentist selects using a simple random sampling 20 children among 200, and counts the data in the sample according to the number of teeth with cavities. The results are presented in the Table I. The second dentist examines the 200 children but with the sole goal of determining who has no cavities. He notices that 50 children are in this category. Table I: Teeth with cavities Number of teeth with cavities 0 1 2 3 4 5 6 7 8 Number of children 8 4 2 2 1 2 0 0 1 a) Estimate the mean number of teeth with cavities per child in the village using only the results of the first dentist. What is the accuracy of the unbiased estimator obtained? Estimate this accuracy and the associated confidence interval. b) Propose another estimator for the mean number of teeth with cavities per child using the results of the two dentists. Calculate the new estimate, and estimate the gain in efficiency obtained. Example II: Foot Size The director of a business that makes shoes wants to estimate the average length of right feet of adult men in a city. Let _v be the characteristic length of right foot (in centimetres) and x be the height of the individual (in centimetres). The director knows moreover from the results of a census that the average height of adult men in this city is 168 cm. To estimate the average foot length, the director draws a simple random sample without replacement of 100 adult men. The results are the following: A A Y=169, 7:24, sn,=l5. st=lO, s‘_=2. Knowing that 400,000 adult men live in this city, a) Calculate the I-lorvitz-Thompson (customary) estimator, the ratio estimator, the difference estimator and the regression estimator. b) Estimate the variances of these four estimators. c) Which estimator would you recommend to the director? d) Express the literal difference between the estimated variance of the ratio estimator and the estimated variance of the regression estimator, as a function A A i j of X , Y and the slope bof the regression of _\~' on x in the sample. Comment on this. The lll'lIl'i - ~ < t gt ment of a particular com ' ' - panv is interested in cst' t‘ - - * - .tment ol , '_ _- _ _ . _ _ _ from wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error.

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STAT 3507 Exercise Questions Solutions Click Link Below To Buy: http://hwcampus.com/shop/stat-3507-exercise-questions-solutions/ Bernoulli Sampling Questions Cluster Sampling Questions Ratio, Regression and Difference Estimation Stratified Random Sampling Systematic Sampling Two-stage Cluster Sampling Questions 1-A university professor who is correcting 600 written examinations decides to get a preliminary idea of the passing rate on the test. He decides to use a simple randomized scheme to single out a smaller number of exam copies for first-hand correction. In passing through the pile of exams, he tosses an ordinary six-sided die, once for each exam copy. If the die shows a 6, he corrects the corresponding exam, otherwise not. Suppose the sample selected in this way consists of 90 students and that 60 out of these are found to have passed. a. Identify the sampling design implemented. b. Estimate the total number of students who passed the exams c. Compute a 95% confidence interval, based on the normal approximation, for the number of passing students (among the 600). d. Repeat (b) and (c) using the alternative (improved) estimator. 2- Table: Number of Employees and Range Division Number of employees Cumulative range 1 1200 1— 1200 2 450 1201 —1650 3 2100 1651— 3750 4 860 3751— 4610 5 2840 4611 — 7450 6 1910 7451— 9350 7 390 9361 — 9750 8 3200 9751 - 12950 12950 i. Suppose 2011, 7972 and 10281 are the random numbers generated between 1 and 12950. Using this information what are the clusters selected? ii. Suppose the total number of sick days used by the three sampled divisions during the past quarter are respectively, y, = 4320 y2 = 4160 y3 = 5790 . Estimate the average number of sick clays used per person for the entire firm and place a bound on the error of estimation. Ratio and Regression Estimators (Examples) under Simple Random Sampling Without Replacement Set 1: Estimation of Ratio Let us consider a community having eight community areas. Suppose that we wish to estimate the ratio R of total pharmaceutical expenses Y to total medical expenses X among all persons in the community. To do this, a simple random sample of two community areas is to be taken and every household in each sample community area is to be interviewed. The data for the community areas am as given in Table 1 below: Table 1: Pharmaceutical Expenses and Total Medical Expenses Among All Residents of Eight Community Areas Community Area Total Pharmaceutical Expenses, Y ($) Total Medical Expenses, X ($) 1 100,000 300.000 2 50,000 200.000 3 75,000 300.000 4 200,000 600.000 5 150,000 450,000 6 175,000 520.000 7 170,000 680.000 8 150,000 450.000 Total 1,070,000 3,500,000 Suppose that community areas 2 and 5 were selected in the sample. a. State the target population for the study b. What are the elements in this sampling design? c. What am the sampling units? d. Estimate the ratio of total pharmaceutical expenses to total medical expenses. e. Estimate the variance of the estimate in (d). f. Based on the sample data, how many community areas would have to be sampled if it is desired to estimate the population ratio with 95% certainty to within 5% of its true value? Set 2: Estimation of Total Suppose that a road having a length of 24 miles traverses areas that can be classified as urban and rural and that the road is divided into eight segments, each having a length equal to 3 miles. A sample of three segments is taken, and on each segment sampled, special equipment is installed for purposes of counting the number of total motor vehicle miles traveled by cars and trucks on the segment during a particular year. In addition, a record of all accidents occurring on each sample segment is kept. The number of truck miles and the number of accidents in which a truck was involved during a certain period are given in Table 2 for each of the eight segments in the population. Suppose that we take a simple random sample of three segments for purposes of estimating the total number of truck miles traveled on the road. Table 2: Truck Miles and Number of Accidents Involving Trucks by Type of Road Segment Segment Type Truck Miles x 1000 Number of Accidents Urban 6327 oo in Cs CT in ,-1 cr% ct Rural 2555 Urban 8691 Urban 7834 V, Rural 1586 Rural 2034 Rural 2015 Rural 3012 Suppose the segments 1, 3 and 4 were selected in the sample. a. Estimate the total number of truck miles traveled on the mad using the customary and ratio estimators. b. Estimate the 95% confidence interval for the total number of truck miles using the customary and the ratio estimators. c. How do these estimators compare? d. Based on the sample data, how many road segments would have to be sampled if it is desired to estimate the total number of truck miles with 95% certainty to within 10% of its true value? Use the customary estimator and the ratio estimator. Example I: Cavities and Post -stratification Two dentists conduct a survey on the condition of teeth of 200 children in a village. The first dentist selects using a simple random sampling 20 children among 200, and counts the data in the sample according to the number of teeth with cavities. The results are presented in the Table I. The second dentist examines the 200 children but with the sole goal of determining who has no cavities. He notices that 50 children are in this category. Table I: Teeth with cavities Number of teeth with cavities 0 1 2 3 4 5 6 7 8 Number of children 8 4 2 2 1 2 0 0 1 a) Estimate the mean number of teeth with cavities per child in the village using only the results of the first dentist. What is the accuracy of the unbiased estimator obtained? Estimate this accuracy and the associated confidence interval. b) Propose another estimator for the mean number of teeth with cavities per child using the results of the two dentists. Calculate the new estimate, and estimate the gain in efficiency obtained. Example II: Foot Size The director of a business that makes shoes wants to estimate the average length of right feet of adult men in a city. Let _v be the characteristic length of right foot (in centimetres) and x be the height of the individual (in centimetres). The director knows moreover from the results of a census that the average height of adult men in this city is 168 cm. To estimate the average foot length, the director draws a simple random sample without replacement of 100 adult men. The results are the following: A A Y=169, 7:24, sn,=l5. st=lO, s‘_=2. Knowing that 400,000 adult men live in this city, a) Calculate the I-lorvitz-Thompson (customary) estimator, the ratio estimator, the difference estimator and the regression estimator. b) Estimate the variances of these four estimators. c) Which estimator would you recommend to the director? d) Express the literal difference between the estimated variance of the ratio estimator and the estimated variance of the regression estimator, as a function A A i j of X , Y and the slope bof the regression of _\~' on x in the sample. Comment on this. The lll'lIl'i - ~ < t gt ment of a particular com ' ' - panv is interested in cst' t‘ - - * - .tment ol , '_ _- _ _ . _ _ _ from wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error. A simple random sample of 10 customers was taken at a local grocery store and the following data were obtained: Customer Total Annual Income x Annual Amount Spent on (in thousands) Food y (in thousands) 25 3.7 32 4.1 40 4.5 35 3.9 29 3.8 42 4.6 50 4.8 38 4.0 41 4.5 44 4.4 ‘5\ooo\1o\u14>wr\>-- Assuming that N is effectively infinite, estimate R , the population ratio (R = Y/X ) , and obtain its standard error.

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