STAT 3014/3914. Semester 2 Applied Statistics Solution to Tutorial 12

Transcription

1 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 , X X + X and X X /N 4500/ The data are C l x y x y y /x y r x C C Mean Var a) The total sales estmate usng the Hartley Ross estmator s Ŷ hr X r nȳ r x) + N ) n ) ) ) 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 C varr ) X ) n N ) s r ser ) n )

2 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 ) sey r ) X ser ) ) 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 s sr,a s y,a r A s xy,a + r As x,a s sr,a.86. ) 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 s sr,b s y,b r B s xy,b + r Bs x,b s sr,b 3.. ) Y st,sr W A RA X A + W B RB X B ) ) ) ) varŷ st,sr) WA n ) A s sr,a + WB n ) B s sr,b N A n A N B n B ) ) ) )

3 b) Combne rato estmate: X X A + X B 6, 300 +, 800 N A + N B, 000 +, X st W A x A + W B x B Y st W A y A + W B y B r c 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 s cr,a.39. ) 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 s cr,b ) Y st,cr R st,cr X varŷ st,cr) WA n ) A s cr,a + WB n ) B s cr,b N A n A N B n B ) ) ) ) > 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

4 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 Overall Mean Var The true varance of the mean estmator s the varance of these 9 sample means whch s k ) VarȲ ) ȳ k ȳ k 9 [ ) b) The sum of squares and mean sum of squares are SST o SSW SSB n or k k n yj Nȳ ) 33. j n y j ȳ ) j k n )s ) 3 k ȳ Nȳ ) 36.8 ). SST o SSW Sw SSW kn ) S SST o N < 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 The estmate for the average level of deldrn n ths 4

5 stretch of the rver and ts varance are Ȳ n s ȳ ).5 n s 3 s ȳ varȳ ) n s n N ns ) ȳ n s ȳ [ ) ) s ȳ n s ) 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 thousands [ varŷpst,) N n ) L Sl W l N n + L W n l )Sl [ 37 5 ) ) ) [ 37 5 ) ) ) ) 54, seŷpst,) 54, b) Estmate and ts s.e. for the total commsson receved usng post-stratfcaton 5

6 when the data s stratfed accordng to the length of stay n the company: Ŷ pst, l N l ȳ l thousands [ varŷpst,) N n ) L Sl W l N n + L W n l )Sl [ 37 5 ) ) ) [ 37 5 ) ) ) ) 8, seŷpst,) 8, 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 and the sample varance Hence Ŷ Nȳ , varŷ ) N n ) s y N n 37 5 ) seŷ ) 38, , 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

7 that n s neglgble as compared wth n. For a) For b) s L W lsl s L W ls l ) ) 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, , , , 63 ) [ N s n nw n L N s 70, 63 [ 3, n nw , 63 [, 0 0. n 3 nw , 63 [ n 4 nw , 63 ) Estmate of the total annual proft last year for all the tradng frms n that 7

8 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, n l ) ) seŷst) 69, 377, , , , 530 +, 070 5, % CI for Y st Ŷst z 0.05 seŷst), Ŷ + z 0.05 seŷst) ) ) , , , 70, , ) 53, , 86, ) 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, x 85 Ŷ r X R 7, , s r n y R x y + R x ) 99 8, , , 674) varŷr) N n ) s r 9, ) , 5, 984. N n 9, seŷr) 9, 5, , > seŷst) 8, 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 We gnore fpc snce N s not gven. 8

9 a) For sample mean usng SRS Y srs,4 ȳ ȳ st 9, y s n ) + n ȳ , y s n ) + n ȳ , yj,j y + y s y n yj nȳ ) , ),j varŷ srs,4) s y n 3, 67, , 67, seŷ srs,4), , 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 9, varŷ st) n w s ) + n w s ) + n [w ȳ Ŷ st,ds) + w ȳ Ŷ st,ds) ) ) [0. 3, 000 9, ) , 000 9, ), 000 +, , seŷ st,ds) 3, a) ) The total nventory X n thousand dollars last year for all dealers of the juce 9

10 drnk company X pst NW y + W y + W 3 y 3 ) 9, ) 9, , 564 [ vary pst ) N n ) L s l W l N n + L W n l )s l [ 9, ) 970 9, ) ) 970 9, ) 9, 985, 643 ) The populaton sze N for each stratum N N n 9, n 970 N N n 9, n 970 N 3 N n 3 9, 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, , , , ) N s n nw n L N s [ 5, ) , [ 0, n nw ) , [ 3, n 3 nw ) , b) The total nventory Y n thousand dollars ths year for all dealers of the juce 0

11 drnk company L ) Ŷ st,ds N w ȳ ) , , , 868 [ varŷst,ds) N w s + w n n ȳ Ŷ st,ds) { ) 9, ) ) [ ) ) + 88 } ) 970 9, ) 34, 737, 058.5