STAT 304/394 Semester Appled Statstcs 05 Soluton to Tutoral. Note that a sngle sample of n 5 customers s drawn from a populaton of N 300. We have n n + n 6 + 9 5, X X + X 4500 + 00 45700 and X X /N 4500/300 8.6667. The data are C l x y x y y /x y r x C 04 0 04 0.094-4.045 43 60 43 60.89 9.96 3 8 75 8 75 0.946 -.0363 4 56 80 56 80.0938.3990 5 75 300 75 300.0909.4637 6 98 90 98 90 0.9596-7.746 C 7 37 50 0 0.0949 0.0000 8 89 00 0 0.058 0.0000 9 9 5 0 0.0504 0.0000 0 63 60 0 0 0.954 0.0000 03 0 0 0.0680 0.0000 07 00 0 0 0.9346 0.0000 3 59 80 0 0.3 0.0000 4 63 75 0 0.905 0.0000 5 87 90 0 0.0345 0.0000 Mean 45.6667 53.6667 77. 8.048 0.0000 Var. 4454.95 5398.095 4.7 957.86 0.006 58.68 a) The total sales estmate usng the Hartley Ross estmator s Ŷ hr X r nȳ r x) + N ) n 45700.048) + 300 ) 486.35 553.6667.04845.6667) 5 b) The estmate for the rate of ncrease of sales of Brand I product and ts s.e. are y C r 8 x 77..0493 C varr ) X ) n N ) s r ser ) 0.00055 0.0349 n 8.6667 5 ) 58.68 0.00055 300 5
The rate of ncrease s 4.9%. c) The estmate for the total sales of Brand I product and ts s.e. are Ŷ r X r 4500.0493) 5705.96 sey r ) X ser ) 45000.0349) 575.5568. a) Separate rato estmate: n A n B 0; N A, 000 and N B, 500; X A 6, 300 and X B, 800; W A 000 and W 500 B 500; 500 For A: s y,a 0.36; s x,a 9.99; s xy,a 0.8; y 87; x 7.8; r A y A 8.7 x A 7.8.05. s sr,a s y,a r A s xy,a + r As x,a 0.36 8.7 7.8 0.8 + 3.477 s sr,a.86. ) 8.7 9.99 7.8 For B: s y,b 3.4; s x,b 5.45; s xy,b 0.356; y 78; x 46; r B y B 4.6 x B 7.8 0.59. s sr,b s y,b r B s xy,b + r Bs x,b 3.4 4.6 7.8 0.356 + 9.77 s sr,b 3.. ) 4.6 5.45 7.8 Y st,sr W A RA X A + W B RB X B ) ) ) ) 000 8.7 500 4.6 6.3 + 8.53 9.87. 500 7.8 500 7.8 varŷ st,sr) WA n ) A s sr,a + WB n ) B s sr,b N A n A N B n B ) 000 0 ) ).86 500 500 000 0 + 0 ) 3. 500 500 0 0.40.
b) Combne rato estmate: X X A + X B 6, 300 +, 800 N A + N B, 000 +, 500.64 X st W A x A + W B x B 0.4 7.8 + 0.6 7.8.80. Y st W A y A + W B y B 0.4 8.7 + 0.6 4.6 0.4. r c 0.4.80 0.8677966. For A: s y,a 0.36; s x,a 9.99; s xy,a 0.8; y 87; x 7.8; s cr,a s y,a r C s xy,a + r Cs x,a 0.36 0.4.80 0.8 + 5.79 s cr,a.39. ) 0.4 9.99.80 For B: s y,b 3.4; s x,b 5.45; s xy,b 0.356; y 78; x 46; s cr,b s y,b r C s xy,b + r Cs x,b 3.4 0.4.80 0.356 + 6.08 s cr,b 4.00. ) 0.4 5.45.80 Y st,cr R st,cr X 0.8677966.64 0.0. varŷ st,cr) WA n ) A s cr,a + WB n ) B s cr,b N A n A N B n B ) 000 0 ) ).39 500 500 000 0 + 0 ) 4.00 500 500 0 0.66 > 0.40 varŷ st,sr). Note that varŷ st,sr) < varŷ st,cr). Hence separate rato estmator s preferred snce. n A n B 0 s not too small and. RA.05, RB 0.59 s not smlar. c) Dscusson If the stratum sample szes n h are large enough say, 0) so that the separate rato estmator Ŷ st,sr does not have large bases and that the varance approxmaton works adequately, use the separate rato estmator. 3
If the stratum sample szes n h are very small and the stratum rato R h Y h X h constant over strata, the combned rato estmator Ŷ st,cr may perform better. s 3. We have n s 3, n 4, k 9, N 36 and total sample sze n 34). a) The sample means and varances are Samples 3 4 5 6 7 8 9 Overall 0 0 0 0 5 6 6 5 6 5 5 5 3 4 3 4 4 0 0 Mean.00.5.75.50.5.5.50.00.00.8 Var..667 4.97 7.583 5.667 3.583 6.97 3.000 4.667 4.667 3.806 The true varance of the mean estmator s the varance of these 9 sample means whch s k ) VarȲ ) ȳ k ȳ k 9 [47.475 9.8 ) 0.0673. b) The sum of squares and mean sum of squares are SST o SSW SSB n or k k n yj Nȳ 30 36.8 ) 33. j n y j ȳ ) j k n )s 3.667 + + 4.667) 3 k ȳ Nȳ 4 +.5 + + ) 36.8 ). SST o SSW 33. 3. Sw SSW kn ) 3 7 4.85 S SST o N 33. 36 3.806 < 4.85 Snce S w > S, systematc samplng s more effcent. c) Wth random starts of, 4 and 8, the selected sample means are ȳ.5,.5,. We have n s ȳ 5.35. The estmate for the average level of deldrn n ths 4
stretch of the rver and ts varance are Ȳ n s ȳ.5 +.5 + ).5 n s 3 s ȳ varȳ ) n s n N ns ) ȳ n s ȳ [5.35 3.5 ) 0.065 ) s ȳ n s ) 0.065 0.039 36 3 Extra exercse. a) Estmate and ts s.e. for the total commsson receved usng post-stratfcaton when the data s stratfed accordng to branches: Ŷ pst, l N l ȳ l 6.34 + 0 56.30 + 5 55.90 37.58 thousands [ varŷpst,) N n ) L Sl W l N n + L W n l )Sl [ 37 5 ) 89.38 + 0 833. 37 37 5 37 5 + 5 ) 799.045 37 5 + ) 5 7 89.38 + 833. + 5 37 37 37 799.045 [ 37 5 ) 7.75 + 5.06 +.5958)+ 37 553.6 + 608.008 + 475.078) 5 37 3.300589 + 7.7430505) 54, 77.93409 seŷpst,) 54, 77.93409 3.76539 b) Estmate and ts s.e. for the total commsson receved usng post-stratfcaton 5
when the data s stratfed accordng to the length of stay n the company: Ŷ pst, l N l ȳ l 7 3.8 + 64.74 + 8 89.08 0.8 thousands [ varŷpst,) N n ) L Sl W l N n + L W n l )Sl [ 37 5 ) 7 8.94 + 74.63 + 8 ) 60.989 37 37 5 37 5 37 5 + ) 0 5 9 8.94 + 74.63 + 5 37 37 37 60.989 [ 37 5 ) 3.9496 + 3.7754 + 0.879)+ 37 69.6984 + 7.988 + 47.80) 5 37 5.595835 +.04658808) 8, 436.56053 seŷpst,) 8, 436.56053 9.8505637 c) Estmator n b) s better. The auxlary varable of the length of stay n the company leads to more effcent estmator because the resultng strata s more nternally homogeneous w.r.t. the commsson receved. d) For SRS, the sample mean 57.84 6 and the sample varance707.05538. Hence Ŷ Nȳ 37 57.84 6, 40.3 6 varŷ ) N n ) s y N n 37 5 ) 707.05538 37 5 seŷ ) 38, 367.37576 95.875993. 38, 367.37576 Note that seŷpst,) < seŷ ) < seŷpst,). Only the stratfcaton usng the length of servce n the company mproves the effcency of the estmator. e) Varance reducton n poststratfcaton varŷsrs) varŷpst) N [ n N ) s y n n N N [ n N ) n s N [ n s ) L L W l Sl ) n L W l Sl ) W l S l n n L L W l )Sl W l )S l Assumng n s suffcently small so that n ) and n s suffcently large so N N 6
that n s neglgble as compared wth n. For a) For b) s L W lsl 0 707.055 89.38 + 37 37 707.055 84.854354 07.8389 s L W ls l 707.055 833. + 5 ) 37 799.045 7 8.94 + 37 37 74.63 + 8 ) 37 60.989 707.055 9.067 577.9538 Hence s n b) s much larger and leads to larger reducton of varance for the estmator. For a), s L W lsl s negatve whch mples an ncrease of varance for the estmator usng post-stratfcaton. Ths s due to the fact that post-stratfcaton does not help n reducng the sample varance n each stratum but the sample sze n each stratum s much smaller.. a) Method A: If the stratfcaton leads to more nternally homogeneous strata, the stratfed SRS wll be preferred. Snce S l wll be small f the resultng strata are nternally homogeneous and hence varŷ st) ) L W l n l s l N l n l wll be small as well. The mean estmator usng the stratfed SRS and proportonal allocaton method s the same as the mean estmator usng SRS because Ŷ st l n n ȳ ȳ. N l N ȳl Method B: If the auxlary varable X s postvely and hghly correlated to Y, the varable of nterest and the populaton mean X or total X s known for the auxlary varable, ths method s preferred. Note that s r n 00 y R 00 x y + R 00 x wll be small f X and Y are hghly correlated. b) ) For Neyman allocaton, L N l s l, 940 3. + 3, 530 6.3 +, 0 0. + 070 6.5 70, 63 ) [ N s 940 3. n nw n L N 00 3.3 3 s 70, 63 [ 3, 530 6.3 n nw 00 3.49 3 70, 63 [, 0 0. n 3 nw 3 00 30.8 30 70, 63 [ 070 6.5 n 4 nw 4 00 5.00 5 70, 63 ) Estmate of the total annual proft last year for all the tradng frms n that 7
ndustry and ts CI estmate: Ŷ st l N l ȳ l, 940)8.7) + 3, 530)5.) +, 0)0.4) +, 070)44.8) 70, 4 varŷst) L N l n l N l ) s l, 940 3, 940, 0 30, 0 69, 377, 544.89 n l ) 6.604 3 ) 5.536 30 seŷst) 69, 377, 544.89 8, 39.38393 + 3, 530 3 3, 530 +, 070 5, 070 95% CI for Y st Ŷst z 0.05 seŷst), Ŷ + z 0.05 seŷst) ) 67.6996 + 3 ) 3.576 70, 4.96 8, 39.38393, 70, 4 +.96 8, 39.38393) 53, 888.5359, 86, 539.464) c) Estmate of the total annual proft last year for all the tradng frms n that ndustry usng rato estmaton and ts s.e. estmate: R ȳ, 96.5.335 5 x 85 Ŷ r X R 7, 730.335 5 69, 835.5697 s r n y R x y + R x ) 99 8, 67.335 5 5, 707 +.335 5 8, 674) 00.036097 varŷr) N n ) s r 9, 650 00 ) 00.036097 9, 5, 984. N n 9, 650 00 seŷr) 9, 5, 984. 9, 604.8470 > seŷst) 8, 39.38393. d) Would prefer the total estmator usng stratfed SRS rather than the total estmator usng SRS and rato estmate. It s possble that the correlaton between X and Y may not be strong enough as large frm sze does not necessary lead to hgher annual proft. However the relatonshp between the frm sze and annual proft should generally be postve so that stratfcaton usng the frm sze should lead to more nternally homogeneous strata w.r.t. the annual proft. 5 3. We gnore fpc snce N s not gven. 8
a) For sample mean usng SRS Y srs,4 ȳ ȳ st 9, 333. 3 y s n ) + n ȳ 400 99 + 00, 000.44584 0 0 y s n ) + n ȳ 00 99 + 00 8, 000.8099 0 0 yj,j y + y.44584 0 0 +.8099 0 0.7783 0 0 s y n yj nȳ ) 99.7783 00 300 9, 333. 3 ),j varŷ srs,4) s y n 3, 67, 079.59 3, 67, 079.59 300 seŷ srs,4), 090.6386 09.96., 090.6386 Snce there s large varaton between the strata, the s.e. of SRS estmator s large. Post-stratfcaton estmator s not feasble as the populaton szes are unknown. b) For double samplng for stratfcaton Y st,ds L w ȳ 00 00, 000 + 8, 000 9, 333. 3 300 300 varŷ st) n w s ) + n w s ) + n [w ȳ Ŷ st,ds) + w ȳ Ŷ st,ds) 0 0. 3 400) + 0 0. 6 00) + 300 [0. 3, 000 9, 333. 3) + 0. 68, 000 9, 333. 3), 000 +, 85.8585 3, 85.859 seŷ st,ds) 3, 85.859 7.69389 4. a) ) The total nventory X n thousand dollars last year for all dealers of the juce 9
drnk company X pst NW y + W y + W 3 y 3 ) 9, 4000.3 8.5 + 0.55 7. + 0.5 7) 9, 400 6.06 3, 564 [ vary pst ) N n ) L s l W l N n + L W n l )s l [ 9, 400 970 ) 970 9, 400 0.3 6. + 0.55 5. + 0.5 55.5) + 0.7 6. + 0.45 5. + 0.85 55.5) 970 9, 400 0.06458 + 0.000074) 9, 985, 643 ) The populaton sze N for each stratum N N n 9, 400 7 n 970 N N n 9, 400 50 n 970 N 3 N n 3 9, 400 88 n 970 5, 440 0, 00 3, 760 ) Neyman allocaton for a sample of sze n 97 nto the three strata L N l s l N s + N s + N 3 s 3 5, 440 6. + 0, 00 5. + 3, 760 55.5 0, 04.857 ) N s n nw n L N s [ 5, 440 6. 97 97.6) 0.96 0, 04.857 [ 0, 00 5. n nw 97 97.507) 49.6 49 0, 04.857 [ 3, 760 55.5 n 3 nw 3 97 97.77) 6.89 7 0, 04.857 b) The total nventory Y n thousand dollars ths year for all dealers of the juce 0
drnk company L ) Ŷ st,ds N w ȳ ) 7 50 88 9, 400 9.7 + 8.3 + 970 970 970 3.5 9, 400 8.45 357, 868 [ varŷst,ds) N w s + w n n ȳ Ŷ st,ds) { ) 9, 400 7 5.3 + ) 50 8.5+ 970 49 970 ) 88 60. + [ 7 9.7 8.45) 7 970 97 970 + 50 970 8.3 8.45) + 88 } 3.5 8.45) 970 9, 400 0.308 + 0.056) 34, 737, 058.5