Chapter 22 Sample Survey Learning Objectives Learn what a sample survey is and how it differs from an experiment as a method of collecting data Know about the methods of data collection for a survey Know the difference between sampling and nonsampling error Learn about four sample designs: (1) simple random sampling, (2) stratified simple random sampling, (3) cluster sampling, and (4) systematic sampling Lean how to estimate a population mean, a population total, and a population proportion using the above sample designs Understand the relationship between sample size and precision Learn how to choose the appropriate sample size using stratified and simple random sampling Learn how to allocate the total sample to the various strata using stratified simple random sampling 22 1 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 Solutions: a x = 215 is an estimate of the population mean b sx c 215 2(2.7386) or 209.5228 to 220.4772 a Estimate of population total = N x = 400(75) = 30,000 b Estimate of Standard Error = Ns x 20 50 800 50 2.7386 800 400 80 Nsx 400 320 400 80 c 30,000 2(320) or 29,360 to 30,640 a p = .30 is an estimate of the population proportion b 1000 100 (.3)(.7) sp .0437 1000 99 c .30 2(.0437) or .2126 to .3874 B = 15 n (70) 4900 72.9830 2 (15) (70) 67.1389 450 A sample size of 73 will provide an approximate 95% confidence interval of width 30 a x = 149,670 and s = 73,420 sx 771 50 73, 420 10, 040.83 771 50 approximate 95% confidence interval 149,670 2(10,040.83) or $129,588.34 to $169,751.66 b � X = N x = 771(149,670) = 115,395,570 22 2 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Sample Survey sx$ = N sx = 771(10,040.83) = 7,741,479.93 approximate 95% confidence interval 115,395,570 2(7,741,479.93) or $99,912,610.14 to $130,878,529.86 c 771 50 (.36)(.64) p = 18/50 = 0.36 and s p .0663 49 771 approximate 95% confidence interval 0.36 2(0.0663) or 0.2274 to 0.4926 This is a rather large interval; sample sizes must be rather large to obtain tight confidence intervals on a population proportion B = 5000/2 = 2500 Use the value of s for the previous year in the formula to determine the necessary sample size n (31.3) 979.69 336.0051 (2.5)2 (31.3) 2.9157 724 A sample size of 337 will provide an approximate 95% confidence interval of width no larger than $5000 a Stratum 1: = 138 Stratum 2: x2 = 103 Stratum 3: x3 = 210 b Stratum 1 x1 = 138 30 200 20 sx1 6.3640 200 20 138 2(6.3640) or 125.272 to 150.728 22 3 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 Stratum 2 x2 = 103 25 250 30 sx2 4.2817 250 30 103 2(4.2817) or 94.4366 to 111.5634 Stratum 3 x3 = 210 50 100 25 sx3 8.6603 100 25 210 2(8.6603) or 192.6794 to 227.3206 c 200 250 100 xst 138 550 103 550 210 550 = 50.1818 + 46.8182 + 38.1818 = 135.1818 sxst (550) (30) (25) (50) 200(180) 250(220) 100(75) 20 30 25 (550) 3,515,833.3 3.4092 approximate 95% confidence interval 135.1818 2(3.4092) or 128.3634 to 142.0002 a Stratum 1: N1 x1 = 200(138) = 27,600 22 4 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Sample Survey Stratum 2: N x2 = 250(103) = 25,750 Stratum 3: N x3 = 100(210) = 21,000 b N xst = 27,600 + 25,750 + 21,000 = 74,350 Note: the sum of the estimate for each stratum total equals N xst c sxst = 550(3.4092) = 1875.06 (see 7c) approximate 95% confidence interval 74,350 2(1875.06) or 70,599.88 to 78,100.12 a Stratum 1 p1 = .50 200 20 (.50)(.50) s p1 .1088 19 200 50 2(.1088) or 2824 to .7176 Stratum 2 p2 = .78 250 30 (.78)(.22) s p2 .0722 29 250 78 2(.0722) or 6356 to .9244 Stratum 3 p3 = .21 100 25 (.21)(.79) s p3 .0720 24 100 21 2(.0720) 22 5 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 or 066 to .354 200 250 100 (.50) (.78) (.21) .5745 550 550 550 b pst c s pst (550) (.5)(.5) (.78)(.22) (.21)(.79) 250(220) 100(75) 200(180) 19 29 24 (550) (473.6842 325.4483 51.8438) .0530 d approximate 95% confidence interval 5745 2(.0530) or 4685 to .6805 10 a n 300(150) 600(75) 500(100) (20) 2 (1400) 300(150) 600(75) 500(100) (140, 000) 92.8359 196, 000, 000 15,125, 000 Rounding up we choose a total sample of 93 300(150) n1 93 30 140, 000 600(75) n2 93 30 140, 000 500(100) n3 93 33 140, 000 b With B = 10, the first term in the denominator in the formula for n changes n (140, 000) (140, 000) 305.6530 49, 000, 000 15,125, 000 (10) (1400) 15,125, 000 Rounding up, we see that a sample size of 306 is needed to provide this level of precision 300(150) n1 306 98 140, 000 600(75) n2 306 98 140, 000 22 6 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Sample Survey 500(100) n3 306 109 140,000 Due to rounding, the total of the allocations to each strata only add to 305. Note that even though the sample size is larger, the proportion allocated to each stratum has not changed c n (140, 000) (140, 000) 274.6060 (15, 000) 56, 250, 000 15,125, 000 15,125, 000 Rounding up, we see that a sample size of 275 will provide the desired level of precision The allocations to the strata are in the same proportion as for parts a and b 300(150) n1 275 98 140, 000 600(75) n2 275 88 140, 000 500(100) n3 275 98 140, 000 Again, due to rounding, the stratum allocations do not add to the total sample size. Another item could be sampled from, say, stratum 3 if desired 11 a b x1 = 29.5333 x2 = 64.775 x3 = 45.2125 x4 = 53.0300 Indianapolis 13.3603 38 29.533 2 38 29.533 10.9086(.9177) or 19.5222 to 39.5438 Louisville 25.0666 45 64.775 2 45 64.775 17.7248(.9068) or 48.7022 to 80.8478 St. Louis 22 7 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 19.4084 80 45.2125 2 80 45.2125 (13.7238) (.9487) or 32.1927 to 58.2323 Memphis 29.6810 70 10 53.0300 2 70 10 53.0300 18.7719(.9258) or 35.6510 to 70.4090 c 38 45 80 70 pst .4269 233 233 233 233 10 d p (1 p1 ) N1 ( N1 n1 ) 38(32) 33.7778 n1 p (1 p2 ) N ( N n2 ) 45(37) 55.7478 n2 p (1 p3 ) N ( N n3 ) 80(72) 192.8571 n3 10 10 p (1 p4 ) N ( N n4 ) 70(60) 116.6667 n4 s pst 33.7778 55.7478 192.8571 116.6667 (399.0494) .0857 (233) (233) approximate 95% confidence interval 4269 2(.0857) or 2555 to .5983 12 a St. Louis total = N1 x1 = 80 (45.2125) = 3617 In dollars: $3,617,000 22 8 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Sample Survey b Indianapolis total = N1 x1 = 38 (29.5333) = 1122.2654 In dollars: $1,122,265 c 38 45 80 70 xst 29.5333 64.775 45.2125 53.0300 48.7821 233 233 233 233 N1 ( N1 n1 ) s12 (13.3603) 38(32) 36,175.517 n1 N ( N n2 ) s22 (25.0666) 45(37) 130, 772.1 n2 N ( N n3 ) s32 (19.4084) 80(72) 271, 213.91 n3 N ( N n4 ) s42 (29.6810) 70(60) 370, 003.94 n4 10 sxst 36,175.517 130, 772.1 271, 213.91 370, 003.94 (233) (808,165.47) 3.8583 (233)2 approximate 95% confidence interval xst 2sxst 48.7821 2(3.8583) or 41.0655 to 56.4987 In dollars: $41,066 to $56,499 d approximate 95% confidence interval Nxst 2 Nsxst 233(48.7821) 2(233)(3.8583) 11,366.229 1797.9678 or 9,568.2612 to 13,164.197 In dollars: $9,568,261 to $13,164,197 22 9 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 n 13 50(80) 38(150) 35(45) (30)2 2 (123) 50(80) 38(150) 35(45) (11, 275) 27.3394 3, 404, 025 1, 245,875 Rounding up we see that a sample size of 28 is necessary to obtain the desired precision 50(80) n1 28 10 11, 275 38(150) n2 28 14 11, 275 35(45) n3 28 4 11, 275 b n 50(100) 38(100) 35(100) (30) 2 (123) 50(100) 38(100) 35(100) 123(100) 3, 404, 025 123(100) 33 50(100) n1 33 13 12,300 38(100) n2 33 10 12,300 35(100) n3 33 9 12,300 This is the same as proportional allocation . Note that for each stratum N nh n h N 14 a xc xi 750 15 Mi 50 � X M xc = 300(15) = 4500 pc b 15 .30 M i 50 ( xi xc M i ) = [ 95 15 (7) ]2 + [ 325 15 (18) ]2 + [ 190 15 (15) ]2 + [ 140 15 (10)]2 = (10)2 + (55)2 + (35)2 + (10)2 = 4450 22 10 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Sample Survey 25 sxc (25)(4)(12) 4450 1.4708 s X� Ms xc = 300(1.4708) = 441.24 (ai pc M i ) = [ 1 .3 (7) ]2 + [ 6 .3 (18) ]2 + [ 6 .3 (15) ]2 + [2 .3 (10) ]2 = (1.1)2 + (.6)2 + (1.5)2 + (1)2 = 4.82 25 4.82 s pc .0484 (25)(4)(12) c approximate 95% confidence Interval for Population Mean: 15 2(1.4708) or 12.0584 to 17.9416 d approximate 95% confidence Interval for Population Total: 4500 2(441.24) or 3617.52 to 5382.48 e approximate 95% confidence Interval for Population Proportion: 30 2(.0484) or 2032 to .3968 15 a 10,400 xc 80 130 � X M xc = 600(80) = 48,000 13 pc .10 130 b ( xi xc M i ) = [ 3500 80 (35) ]2 + [ 965 80 (15) ]2 + [ 960 80 (12) ]2 + [ 2070 80 (23) ]2 + [ 1100 80 (20) ]2 + [ 1805 80 (25) ]2 = (700)2 + (235)2 + (0)2 + (230)2 + (500)2 + (195)2 = 886,150 22 11 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 30 sxc (30)(6)(20) 886,150 7.6861 approximate 95% confidence Interval for Population Mean: c 80 2(7.6861) or 64.6278 to 95.3722 s �X = 600(7.6861) = 4611.66 approximate 95% confidence Interval for Population Total: 48,000 2(4611.66) or 38,776.68 to 57,223.32 d (ai pc M i ) = [ 3 .1 (35) ]2 + [ 0 .1 (15) ]2 + [ 1 .1 (12) ]2 + [4 .1 (23) ]2 + [ 3 .1 (20) ]2 + [ 2 .1 (25) ]2 = (.5)2 + (1.5)2 + (.2)2 + (1.7)2 + (1)2 + (.5)2 = 6.68 6.68 30 s pc .0211 (30)(6)(20) approximate 95% confidence Interval for Population Proportion: 10 2(.0211) or 0578 to .1422 16 a xc 2000 40 50 Estimate of mean age of mechanical engineers: 40 years b pc 35 .70 50 Estimate of proportion attending local university: .70 c ( xi xc M i ) = [ 520 40 (12) ]2 + ∙ ∙ ∙ + [ 462 40 (13) ]2 = (40)2 + (7)2 + (10)2 + (11)2 + (30)2 + (9)2 + (22)2 + (8)2 + (23)2 + (58)2 = 7292 22 12 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Sample Survey 120 10 sxc (120)(10)(50 /12) 7292 2.0683 approximate 95% confidence Interval for Mean age: 40 2(2.0683) or 35.8634 to 44.1366 d (ai pc M i ) = [ 8 .7 (12) ]2 + ∙ ∙ ∙ + [ 12 .7 (13) ]2 = (.4)2 + (.7)2 + (.4)2 + (.3)2 + (1.2)2 + (.1)2 + (1.4)2 + (.3)2 + (.7)2 + (2.9)2 = 13.3 13.3 120 10 s pc .0883 (120)(10)(50 /12) approximate 95% confidence Interval for Proportion Attending Local University: 70 2(.0883) or 5234 to .8766 17 a 17(37) 35(32) 57(44) 11, 240 xc 36.9737 17 35 57 304 Estimate of mean age: 36.9737 years b Proportion of College Graduates: 128 / 304 = .4211 Proportion of Males: 112 / 304 = .3684 c ( xi xc M i ) = [ 17 (37) (36.9737) (17) ]2 + ∙ ∙ ∙ + [ 57 (44) (36.9737) (44) ]2 = (.4471)2 + (174.0795)2 + (25.3162)2 + (460.2642)2 + (173.1309)2 + (180.3156)2 + (94.7376)2 + (400.4991)2 = 474,650.68 150 sxc (150)(8)(40) 474, 650.68 2.2394 approximate 95% confidence Interval for Mean Age of Agents: 22 13 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 36.9737 2(2.2394) or 32.4949 to 41.4525 d (ai pc M i ) = [ 3 .4211 (17) ]2 + ∙ ∙ ∙ + [ 25 .4211 (57) ]2 = (4.1587)2 + (.7385)2 + (2.9486)2 + (10.2074)2 + (.1073)2 + (3.0532)2 + (.2128)2 + (.9973)2 = 141.0989 150 s pc (150)(8)(40) 141.0989 .0386 approximate 95% confidence Interval for Proportion of Agents that are College Graduates: 4211 2(.0386) or 3439 to .4983 e (ai pc M i ) = [ 4 .3684 (17) ]2 + ∙ ∙ ∙ + [ 26 .3684 (57) ]2 = (2.2628)2 + (.8940)2 + (2.5784)2 + (3.6856)2 + (3.8412)2 + (1.5792)2 + (.6832)2 + (5.0012)2 = 68.8787 150 s pc (150)(8)(40) 68.8787 .0270 approximate 95% confidence Interval for Proportion of Agents that are Male: 3684 2(.0270) or 3144 to .4224 18 a p = 0.19 sp (0.19)(0.81) 0.0206 363 Approximate 95% Confidence Interval: 0.19 2(0.0206) or 0.1488 to 0.2312 b p = 0.31 22 14 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Sample Survey sp (0.31)(0.69) 0.0243 363 Approximate 95% Confidence Interval: 0.31 2(0.0243) or 0.2615 to 0.3585 c p = 0.17 sp (0.17)(0.83) 0.0197 373 Approximate 95% Confidence Interval: 0.17 2(0.0197) or 0.1306 to 0.2094 d The largest standard error is when p = .50 At p = .50, we get sp (0.5)(0.5) 0.0262 363 Multiplying by 2, we get a bound of B = 2(.0262) = 0.0525 For a sample of 363, then, they know that in the worst case ( p = 0.50), the bound will be approximately 5% e If the poll was conducted by calling people at home during the day the sample results would only be representative of adults not working outside the home. It is likely that the Louis Harris organization took precautions against this and other possible sources of bias 19 a Assume (N n) / N 1 p = .55 sp b (0.55)(0.45) 0.0222 504 p = .31 sp (0.31)(0.69) 0.0206 504 22 15 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 c The estimate of the standard error in part (a) is larger because p is closer to .50 d Approximate 95% Confidence interval: 55 2(.0222) or 5056 to .5944 e Approximate 95% Confidence interval: 31 2(.0206) 2688 to .3512 20 a sx 3000 200 3000 204.9390 3000 200 Approximate 95% Confidence Interval for Mean Annual Salary: 23,200 2(204.9390) or $22,790 to $23,610 b N x = 3000 (23,200) = 69,600,000 sx$ = 3000 (204.9390) = 614,817 Approximate 95% Confidence Interval for Population Total Salary: 69,600,000 2(614,817) or $68,370,366 to $70,829,634 c p = .73 3000 200 (.73)(.27) sp .0304 3000 199 Approximate 95% Confidence Interval for Proportion that are Generally Satisfied: 73 2(.0304) or 6692 to .7908 d If management administered the questionnaire and anonymity was not guaranteed we would expect a definite upward bias in the percent reporting they were “generally satisfied” with their job. A procedure for guaranteeing anonymity should reduce the bias 22 16 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Sample Survey 21 a p = 1/3 380 30 (1/ 3)(2 / 3) sp .0840 29 380 Approximate 95% Confidence Interval: 3333 2(.0840) or 1653 to .5013 b � X = 760 (19 / 45) = 320.8889 c p = 19 / 45 = .4222 760 45 (19 / 45)(26 / 45) sp .0722 44 760 Approximate 95% Confidence Interval: 4222 2(.0722) or 2778 to .5666 d 380 10 760 19 260 pst .3717 1400 30 1400 45 1400 25 p (1 ph ) (1/ 3)(2 / 3) N h ( N h nh ) h 380(350) 29 nh 760(715) (19 / 45)(26 / 45) (7 / 25)(18 / 25) 260(235) 44 24 = 1019.1571 + 3012.7901 + 513.2400 = 4545.1892 s pst 4545.1892 .0482 (1400) Approximate 95% Confidence Interval: 3717 2(.0482) or 2753 to .4681 22 17 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 22 a � X = 380 (9 / 30) + 760 (12 / 45) + 260 (11 / 25) = 431.0667 Estimate approximately 431 deaths due to beating b 380 760 12 260 11 pst .3079 1400 30 1400 45 1400 25 N h ( N h nh ) ph (1 ph ) nh = (380) (380 30) (9 / 30) (21 / 30) / 29 + (760) (760 45) (12 / 45) (33 / 45) / 44 + (260) (260 25)(11 / 25) (14 / 25) / 24 = 4005.5079 s pst 4005.5079 .0452 (1400) Approximate 95% Confidence Interval: 3079 2(.0452) or 2175 to .3983 c 380 21 760 34 260 15 pst .7116 1400 30 1400 45 1400 25 N h ( N h nh ) ph (1 ph ) nh = (380) (380 30) (21 / 30) (9 / 30) / 29 + (760) (760 45) (34 / 45) (11 / 45) / 44 + (260) (260 25) (15 / 25) (10 / 25) / 24 = 3855.0417 s pst 3855.0417 .0443 (1400) Approximate 95% Confidence Interval: 7116 2(.0443) or 6230 to .8002 d � X = 1400 (.7116) = 996.24 Estimate of total number of victims is 996 22 18 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Sample Survey 23 a n 3000(80) 600(150) 250(220) 100(700) 50(3000) (20) 2 2 (4000) 3000(80) 600(150) 250(220) 100(700) 50(3000) 366, 025, 000, 000 170.7365 1, 600,000, 000 543,800, 000 Rounding up, we need a sample size of 171 for the desired precision b 3000(80) n1 171 68 605, 000 600(150) n2 171 25 605, 000 250(220) n3 171 16 605,000 100(700) n4 171 20 605, 000 50(3000) n5 171 42 605, 000 24 a 14(61) 7(74) 96(78) 23(69) 71(73) 29(84) 18,066 xc 75.275 14 96 23 71 29 240 Estimate of mean age is approximately 75 years old b 12 30 10 22 84 pc .35 14 96 23 71 29 240 (ai pc M i ) = [12 .35 (14) ]2 + [ 2 .35 (7) ]2 + [30 .35 (96) ] 2 + [ 8 .35 (23) ]2 + [ 10 .35 (71) ]2 + [ 22 .35 (29) ]2 = (7.1)2 + (.45)2 + (3.6)2 + (.05)2 + (14.85)2 + (11.85)2 = 424.52 100 424.52 s pc .0760 (100)(6)(48) Approximate 95% Confidence Interval: 35 2(.0760) or 198 to .502 22 19 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 � X = 4800 (.35) = 1680 Estimate of total number of Disabled Persons is 1680 22 20 © 2010 Cengage Learning. All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part ... = 760 (19 / 45) = 320.8889 c p = 19 / 45 = . 4222 760 45 (19 / 45)(26 / 45) sp .0 722 44 760 Approximate 95% Confidence Interval: 4222 2(.0 722) or 2778 to .5666 d 380 ... (.78)( .22) s p2 .0 722 29 250 78 2(.0 722) or 6356 to .9244 Stratum 3 p3 = .21 100 25 (.21)(.79) s p3 .0720 24 100 21 2(.0720) 22 5 © 2010 Cengage Learning. All Rights Reserved... May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 22 or 066 to .354 200 250 100 (.50) (.78) (.21) .5745 550 550 550 b pst c s pst (550) (.5)(.5) (.78)( .22) (.21)(.79) 250 (220 ) 100(75) 200(180)