STAT 3014/3914. Semester 2 Applied Statistics Solution to Tutorial 13

Transcription

1 STAT 304/394 Semester Appled Statstcs 05 Soluton to Tutoral 3. Note that s the total mleage for branch. a) -stage cluster sample Cluster branches N ; n 4) Element cars M 80; m 40) Populaton mean no. of cars per branch M M N Sample mean mleage per branch ȳ Sample mean mleage per car r n M The samplng method s a sngle-stage cluster samplng and the quantt to be estmated s R, the average mleage per car. The ordnar estmate of mean mleage per car s wth estmated varance var R c ) M R c ȳ M n ) s N n 67.5 The rato estmate of mean mleage per car s ) R c,r r M 6.404

2 wth estmated varance s r n rm ) n r M + r var R c,r ) M n ) s r N n 4 ) M ) Note: The ordnar estmate has a much larger varance 39.64) than the rato estmate 0.44). Ths s due to the great varablt n cluster szes M 60, 0, 0, 50 for the selected clusters). b) -stage cluster sample Cluster branches N ; n 4) Element cars M 80; m 40) Mean no. of car per branch M M N Estmated sample mean mleage per branch ˆȳ Estmated sample mean mleage per car ˆr n ŷ 6, 5.58, n 4 n ŷ 6, 5.58 n M Branch M M ˆr M ˆr) M ˆr) Sum Varance due to estmated ŷ s N nm varŷ ) N nm M m M ) s m [60 0 ) ) ] , , , 684.3) ,

3 We also have ) s ŷ nŷ n 3 [4, 6, , ) ], 76, s r ŷ ˆr n ŷ M + ˆr 3 4, 6, , , 600) 5, The ordnar estmate of the average mleage per car R s M ) wth estmated varance: var R c ) M R c ˆȳ M n ) s N n 80, ), 76, , 3, Thus var R c ) var R c ) + N nm varŷ ) 8, 3, , The rato estmate of the average mleage per car R s R c,r ˆr n ŷ n M 6, wth estmated varance var R c,r ) M n ) s r N n 80 4 ) 5, , Thus Note: var R c,r ) var R c,r ) + N nm varŷ ) 36, ,

4 . The ordnar estmate has a much larger varance 43.33) than the rato estmate 0.635). Ths s due to the great varablt n cluster szes N 60, 0, 0, 50 for the selected clusters).. Both estmates of ordnar and rato n -stage samplng have larger varance than the correspondng estmates n snge-stage samplng due to the ncrease n varablt n estmatng ŷ n the -stage samplng. However the ncrease ) s not great snce the subsample szes of 0 n each selected cluster are not too small.. -stage cluster sample ŷ , Plant M m ŷ M s M 5, ŷ 583, 98.67, ŷ M 6, , M 7978 Cluster plant N 90; n 0) Element machne M 4, 500; m 5) Mean no. of machne per plant M M 4, N 90 S Mean downtme per plant ŷ, n 0 S ŷ, Mean downtme per machne S M a) The ordnar estmate of the average downtme per machne Y s Y c ŷ M wth estmated varance for stage as varŷ c) M n ) s N n )

5 where s S ŷ ŷ) n S ŷ nŷ n Varance due to estmated ŷ s N varŷ nm ) N nm Hence S S M 3 [583, ) ] m M [ ) s m ) ) ] , ) varŷ c) varŷ c) + N varŷ nm ) S ) Error boundŷ c) ) b) The rato estmate of the average downtme per machne Y s S ŷ, Y c,r S M wth estmated varance for stage : varŷ c,r) where s r Hence M S ŷ M ŷ) n n ) s N n 50 0 ), S ŷ ŷ ŷm + ŷ M n 9 583, , , 978), varŷ c,r) varŷ c,r) + N varŷ nm ) S Error boundŷ c,r) )

6 Note that s r > s because ˆρ < sx s x. 3. a) HH estmator: ) Sx-dgt random numbers are generated, gnorng and an numbers greater than If the lst are 0005, 85953, , sa, the selected hosptals are, 58, and such that hosptal appears twce n the sample. ) ) We have n 4, N, 58, 563, 60, 79, 470, 94, 84, p p and ) 9, p Ŷ HH 350, 00 + n p ) 40, Y HH n N p 350, , ).6757 $000) [ ) ] s /p n n p p 3 79, 470, 94, , ) 9, 463, 56 varŷ HH) s /p N n 9, 463, 56, 58 4 seŷ HH) p P HH nn s /p p 0 4, ) [ ) ] n n p p 3 9, ) 7, varŷ HH) s /p N n 7, 37.43, 58 4 seŷ HH)

7 The total estmates are respectvel, 58).6757) 40, $000) and, 58) ) 00. b) Wth IPPS and n 3: ) The HT estmate s: Y HT N π ) $000) 067 The total estmate of hosptal purchases and count for product Y s, 58)0.5849) 39, $000) It s hard to fnd ther s.e. unless we know π j. If we can assume the usual the draw-b-draw p for samplng wth replacement, the second order ncluson probabltes are π p ) p π p ) p π 58 p 58 ) p 58 ) / ) / ) / π, π + π [ p p ) 3 ] [ ) 3 ] }{{ 689}}{{},,,,, π,58 π + π 58 [ p p 58 ) 3 ] [ )3 ] π,58 π + π 58 [ p p 58 ) 3 ] [ )3 ] Note that the draw-b-draw samplng s a requred assumpton when calculatng the second order ncluson probabltes. We have π , π , π

8 varŷht,) π N π + ) πj π π j j S π <j π j π j [ )] seȳ HT, ) ) Wth sstematc IPPS samplng and n 3, but the se estmate s varŷ ss,pps) N S, seŷ ss,pps) The second order ncluson probabltes are Y ss,pps 0.69, same as ) ) n ) n π Ŷss π n [ ) ) , ) ) , ) ) ] , , j) π j sum, j) π j Sum,) JB,JS 4,9) AJ,DE 3,3) MB,JS 5,9) PJ,DE 3 6 4,6) AJ,JC 6,9) JC,DE ,6) PJ,JC 7,9) SC,DE ,7) AJ,SC 8,9) MC,DE ,7) PJ,SC ,8) AJ,MC 5,8) PJ,MC

9 Note that ths samplng scheme defnes a set of π and π j but t s not an ncluson probablt proportonal to sze IPPS) samplng as there s no X varable whch defnes π nx X even though π vares across ndvduals. Y HT N S varŷht,) π ) 6 5 π 9 / ) /3 + π ) π π π + π ) 5 6 ) + seŷht,) varŷ HT,) π π π, N π, π 3 ) π π π ) 60 ) ) 3 ) ) ) π π ) 5 /6 60 ) ve /3 Formula one can be appled to an samplng scheme whch defnes a set of π and π j but formula two can onl be appled to samplng schemes wthout replacement. Extra exercse. -stage cluster sample. We have N 30 and n 3. M m Sample data j Sample mean Sample var. s ŷ M 0 4, 3, 3, , 4, 0, ,, 0, Total M 3; M 35; ŷ 67.5; ŷ, ; ŷ M Addtonal varance due to those households of sze 4: N M m ) s N M m ) s n M m n M m 30 [0 4 ) ) ) ] ,

10 Rato estmator of total: Ŷ c,r M ŷ M s r n S S ŷ S M ŷ ŷ S ŷ M + ŷ M ), ) varŷc,r) N n N ) s r n 30 varŷc,r) varŷc,r) + Add. varance S 3 ) , , , , varŷc,r) 8, Nave estmator of total: Ŷ c N ŷ N n s n nŷ ), ) S varŷc) N n ) s N n + N M m ) s n M m 30 3 ) , , , , varŷc), The s.e. of ordnar estmate s slghtl larger but the two s.e. are qute close to each other because the cluster total ŷ s not hghl correlated to the cluster sze M and also the cluster szes M are all close to 0. We stll prefer rato estmator to ordnar estmator as t uses the nformaton of cluster sze M. The two estmators wll be the same f the cluster szes M are all equal.. We have M M N 46 4 a) -stage cluster samplng: and ȳ n

11 ) Ordnar estmator for mean per element: Y c M ˆȳ s varŷc) n S M n ) s N nŷ ) 3 0, ) n seŷ c) ) ) Ordnar estmator s preferred because the cluster sze dffers onl slghtl and t s easer to compute. The rato and ordnar estmates should be close as the cluster szes are smlar. b) -stage cluster samplng: ) Calculaton: j m Mark j ŷ M M j m M s 5 7, 6, 4, 8, / , 9, 0, 9, 7, 7 50/ , 6, 4, 8, / , 8, 9, 7, 0 33/ , 7, 4, 5, 4 9 6/ ,, 5, 4,, 0 7/ , 6, 4, 3, 9, 8/ , 7, 5, 7, / We have M 80; M 806; ŷ ; ) Ordnar estmator for mean per element: ˆȳ ŷ n Y c M ˆȳ s n ŷ 3, ; ŷ nˆȳ ) 7 3, ) ŷ M 4,

12 Add. var. nnm M m ) s M m nnm ) s M m ) varŷ c) varŷ c) + Add. var. due to ŷ M n ) s N n + nnm M m ) s M m 8 ) seŷ c) ) The frst sample of 4 classes consst manl classes of hgh marks. As a result, Y c wll overestmate the true mark. Also the s.e. seŷ c) based on manl classes of large total marks wll underestmate the true s.e.. Ths s the result of samplng error. v) Snce the addtonal 4 classes are manl classes of low marks, ths shows great varablt of marks across classes. The two-stage cluster samplng s preferred as more classes can be selected from the classes wth more varablt n class totals. 3. a) ) When the varablt n cluster sze M s large, -stage cluster samplng s preferred as we can subsample from those large cluster. As a result, the total sample sze s easer to control. ) When the varablt of Y, the varable of nterest wthn the cluster s relatvel less than that between clusters, the -stage cluster samplng s preferred as t enables the selecton of more clusters for a gven total sample sze of elements. The selecton of more cluster s necessar as the varablt between clusters s hgh. b) We have n 6, N 48, M 5, 00 and M M 5,00 N ) M ȳ /M Total

13 M 748; M 99, 88; ŷ, 869; We have ȳ 5.705, ȳ and ŷ 60, 35; ŷ M 43, 798 ȳ n 6 Y s ȳ n ȳ n ) ) varŷ ) n ) s ȳ N n 6 ) seŷ ) ) We have ȳ Ordnar estmate for mean per element: n 6 Y M ȳ s nȳ ) 5 60, ) 5, varŷ ) n M n N ) s n 6 ) 5, seŷ ) ) Rato estmate of mean per element: Y 3 r, s r n M r M + r M ) , , , 88) varŷ 3) n N M ) s r n seŷ 3) ) v). Y 3 s preferred when M s unknown, the varaton of M s hgh and M s hghl and postvel correlated to because M s not requred n the estmaton of Y. Moreover the strong and postve relatonshp between M and s accounted for n the rato estmate. Moreover Y s a based estmator smlar to the rato estmator Y 3 but Y s an unbased estmator. 3

14 . Y s preferred when M s known and the varaton of M s low because M s requred but M s not requred n the estmaton of Y. 4. One-stage cluster samplng s preferred to -stage cluster samplng when. The cluster sze s small so that sub-samplng s unnecessar and result n too small the sample sze.. The varablt of elements wthn cluster s hgh so that we would lke to nclude all unts wthn cluster nto the sample. The usual rule s to have a hgher samplng fracton from cluster of hgher varablt between elements n the cluster. 3. Easer to mplement. Two-stage cluster samplng s preferred to -stage cluster samplng when. The cluster sze s large so that t s nfeasble to nclude all unts wthn cluster nto the sample.. If the cluster sze vares a lot across clusters, t would be dffcult to control the total sample sze for a -stage cluster samplng. 3. If the varablt of elements wthn cluster s low, t s unnecessar to nclude all elements wthn a cluster nto the sample. 4. If the varablt across clusters s hgh, a -stage cluster sample enables a sample of more clusters than a -sate cluster sample gven the same sample sze. 4