### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Statistical Methods STT 200

MSU

GPA 3.72

### View Full Document

## 35

## 0

## Popular in Course

## Popular in Statistics and Probability

This 80 page Class Notes was uploaded by Aryanna Jerde on Saturday September 19, 2015. The Class Notes belongs to STT 200 at Michigan State University taught by Raoul Lepage in Fall. Since its upload, it has received 35 views. For similar materials see /class/207816/stt-200-michigan-state-university in Statistics and Probability at Michigan State University.

## Popular in Statistics and Probability

## Reviews for Statistical Methods

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/19/15

Week 31609 1 In outcomes are regarded as being equally likely Pevt ne nt M 7 an event is just a subset of the sample space S e j w f F H gt vi 7quot 1 JV l 2 a H Q out J P ent n S an event is just a subset of the sample space take the ratio of favorable outcomes to total outcomes a PEI MIME aIa H gt vi 7quot 1 JV l 2 a H Q out J P ent n S an event is just a subset of the sample space take the ratio of favorable outcomes to total outcomes D EWEW 1 1 egP1lll 336 1I12 39 ram 6 4 or 5 5 or 4 6 me an 1 an 2 M 1 PI aa 1 1I4 PIaAl2I41I2 fquot L mun14 FMJHJL I A prize is behind one of 3 doors Whatever door you guess the host will reveal another door behind which there is no prize and ask if you wish to switch to the remaining door not your original pick and not the opened one If your original pick is at random then your chance of winning if you always switch is 23 since you then only lose if you originally choose the prize door If you never switch your win rate is 13 1 1 5 Jack draws a bill rst Jill draws second from the two bills then remaining mmwne may 39 e by 535 91 1 1 5 a b 0 Jack rst Jill second draw BILLS not dollar amounts 50 rla l Jack draws a bill rst Jill draws second from the two bills then remaining unuusm azma l a snmueas aers WWIquot quotin l 5Jack rst Jill second from the two bills then remaining IISHHSI39N39H f IMMUNE 1 1 5 Jack rst Jill second from the two bills then remaining quot3985 IIHMIIIIIIE IIt NMIIHUSTMIF l2 lllll1l3 1 1 5Jack rst Jill second from the two bills then remaining IMMIMIMHMSJJ mmnmn mmlmmmmmu ammdmmmmm I wl ll l m 1 1 5Jack rst Jill second from the two bills then remaining H MHMMMII l draws ihout replacemen from R R R B B G I 1 m I MWWMMM M ml MI W B I from R R R B B Gnt 1mm 2 MMWEWWI umnanzmlamamlHm1931r m umunnmsa 3 wmumammh mmzmumgal Suppose that each birth is independently placed into one of 365 days The chance that all of a given number n of birthdays will differ is 364365 363365 366n365 2nd misses rst 3rd misses 1st and 2nd etc This is around equot n n1730 12 for n 23 That is around 50 of the time there would be no shared birthdays among 23 persons By complements there is around 12 probability of at least one instance of same birthdays among 23 persons and even greater in the real world Where some days have more births from R R R B B G H M 39r LZH mm 1 Wm WHVH w W w 2341 El LL 39 H T 3 fr M 2 x Mme Vr H of the deal does not matter eg no need to ght over where to sit at cards Vis a Vis the cards dealt skill levels of players may matter fro R B P R1 R2 3625 PBl R2 2635 PG1 R2 1635 PR2 sum of above PR2 3625 2535 1635 630 630 330 1530 12 same as PRl R2 must come by way of exactly one of R1 B1 G1 23 f T i 91391quotquot Jig iir ltj f iif i 77 When the occurrence 0 one thing does not change the probabilities for the oth J I m n W w H E H M quotMH 1 way H l L j 3 1 H r H mm L 14 H m g 1 statistica i ind enden t MEIR mm IEZMM 31W Eli H H H g mm mm W R B B 612 Klar proposes the following If you are type aa for a hypothesized gene controlling handedness then you have 50 chance of being born non right handed and independently of this you have 50 chance of being non right Whorled hair If you are not type aa you are born right handed and right Whorled tune tavorallle ll 1 III 1 III 1 III 1 For IIGI SOIIS aa ill the gene of hamlellness mm mm 27 nunion nA nB nintersection 62 25 38 62 25 25 38 25 counted twice I F m Qt I39I39IIIII BM If there is 80 Chance of rain today and 55 Chance of rain tomorrow we cannot say what is the Chance of rain today or tomorrow H Hi If W ave also a 42 Chance of rain both days then P rain today or tomorrow 8 55 42 93 Emu an 17 H7 M1 If there is 80 Chance the left engine fails and 55 Chance the right engine fails and if these failures are INDEPENDENT then Pb0th fail 92 55 44 P at least one fails 8 55 44 91 Lecture outline 2 13 09 Part of the period will cover numerical examples as in 21109 The rest will be devoted to the points below 1 Important characterization of all points X y which lie on the line of regression y y 5y XX SX quot 2 2 S 0y y y Sloper rT SX OX X2 X2 2 TakingX 0in gives intercept 3quot i slope pg 198 3 For every X solving for y in y i I X i amp gives predicted y pt 0n regr line pred y 3quot X i slope 4 For an approximately ELLIPTICAL plot at a given X the distribution of y is approximately NORMAL with mean predicted y stddeV V 1 r2 Sy Notice that the mean depends upon X but the std deV does not 5 For an ELLIPTICAL PLOT the regression predictor 7 X i slope is optimal in the sense of least mean squared error of prediction 771700 zr09a We car buying public We at Kelley Blue Book asked 50039p0tential new ueetionsthatyield answers Eq that multi billion dollar question Their 4 f Asked quotHow likelywould you be to buy a Efrem General Motors if Iheywere to go bankruptquot42 l percent efnew vehicle buyers said they were eitherquotnot at all likelyquotor quotnetvery likelyquotte do 50 fawn2y my KW ox wa yarfw 7 W4C 467 Vt rc395iM quot 10m mm my7950wa M F 67m enm I wzam 1W W 70 may 7 2 W76 Wm farm 1 Fa la ii If 4 4 m A M f 267 M vpm 54 a gym 6490 mi MW ywf m 4 my Cf me eWm a my Wzagwoz xy 571712 7 oIFoKa 7277 mv f W a Do A 57 z a LDI I IYI IOHIEIWZ UNI cmm BEIGKDEI du 390 N5 W005 Ewan with a warrantyguarantaa frurn thug fadaral39 In anlh Cummunlnrins mnsumarg aria nutvary enthursiastic abnutmnsiv Such guarantee mnly rumwas the media vainnut t paints with 338 pareant saying th Ey39WEiFB Iaitl1gtrarquot39nutat all likelyquot Ijr quotnut vary Iiicrai39yquot In whicl in such a gituatinn 4 J 4 N07 00 6ow MmeWV 54 Ufaa f y gjf W174 I II mAt j 5mg 2 C7 75 00 37537 W78 oog J Z 72 Mg pew ap wzra d 023 aag 39 Cr Mue avga x s c4 7 ma MC M07 8555 rwm WF m 700 nQf a p40 a6 Jami5 s 02 4 0 M Maquot 5 va 7W 476 7 m amp 7 ameJof Mquot P a39m zyF Asked quotHow likelywould you be to buy a carfrom General Motors iftheywere allowr had to renegotiate their contracts and agreements with suppliers and laborersquot the Given thatscenario only 315 percent ofnew vehicle buyers said theywere either quotr very likelyquot to purchase a GM vehicle and theywere out numbered bythe 415 perc either verylikelyquotorquotextremelylikelyquot to do so gig7quot 51 MI Purl5339 WUbli wwmmmmm aV IerIt ctr KrWKWWMn 49 ve a m ngA Up 4 9d8 g ag572 39v 004 Jmf7f 10 7 4923 71wa r nL 150 WW fwdsz mf30 FarMD XsL 5145277 3 vr wmmm ocgo yewfags 57270 W C idemquot WanM6 2 417225 m 77 W7 939 f may 7 ESFW zszcj N if fm77J5y la g 04f 33 m w 74 W quot 55 OEr Wad 7 77217lt CI 62137 Faea 39 d o 4 rarraw 7o Mil1639ij CZquot F05 4 l f 27 TfTai my 05 5 41 74160 my 50101 a aimy amp 4739 39 quot 39 f y 07 ngMWMt jj ooJ 5 747 agoM9719 632w 7 4237 4 Inf Dig tpAF 42 Waer m jz if 51quot 076 9 FLVK WW 0055 jam1w 3m 96 a 3770200 MM m car buying public We at Kelley Blue Book asked 500 potential new uestionsthatyield answers to that multi billipn dollar question Their rm MWV Asked quotHow likelywould you be to buy a Efrem General Motors if theywere to go bankruptquot42 l percent pfnew vehiole buyers said they were eitherquotnpt at all likelyquotor quotnotvery likelyquot to do so I I I d 16 M MO M aamp e far07 592 art aw ArKkDFm39afsm u gammy 107 laa gym3 FM 6am i 2amp7V y Wt 57 z 31 307 4744 INKV n MFWl ayg I Maxirquot 7 T 159K 0 5721 ar p WW451 I W zf figqt fowlWIWI m W i l n mm W glam6 x4 rimVal 9f BurWm 5347 a ONO WSW gammamarl UNI mm sacnnu ol39b39 39 39 9lt g lm In WC W0 Ewan with a warrantyguarantaa frurn the fald aral39 cylnnssummrsz are mat vary anthuaiastic abnut mnsim Such guarantee mnly mmas thug naedla yamgut t paints with 333 pareant saying th lay39wcam Eith rarquot39nntat all likelyquot Dr quotnut 39u39rary Iikal39yquot In whlcl a in such a Eituatiun zzni 20 lamae F m M m y v 60W 466 444mm 715V Ngf FWMJFW 2 5 To 5 76 masM 375 2023 C4113 5 LS No Zea51 mfagg jquot WV Mf 2 746 f a wa mgagw lama awn V50 My 07 I023 oFfoo W fallacyst ipf nsl WP OF QO 70 War21 15 mfg o P 9 f y 7 In anlh Cummunlnrins 1393 I VVBO f TZ f fatNU X lL l7UHVFU QJNUAIX M A AC 1 v A 0 m 1370 15W 70 f9 NOquot I 1 K mfv W00 2 EVKtquot 0amp3200NJ 5 70 f 4 u f N V A quotquot 2 w o 55b 7405 0 by 5 07 m y 7Jt7v x7 r x i5727277 757 29 e g fS39 7 97 fWLz W N Cm 9 4 m 3 PC F WWW c7 291 22 WWW F F yw j 0666 mew00 av rcm w No 5X X mfg we 55725 Wyea23 at EtaO Swad u EMUfag m f yZ m w 2 am gt rOA W 1077 j WW7 J Wz Mw WWW oo STT 200 Spring 2009 Lecture Outline 1 23 09 Estimated Margin of Error for E when sampling from a nor mal population This method applies exactly for every n gt 1 provided the population from which we are sampling has an Xscore distribution which is normal Consult pg 606 pp 586 596 Refer to 1 14 09 For every n gt 1 all that is needed is to replace 196 by an entry from the t Table A 97 Appendix D for quotdegrees of freedomquot df n 1 Question It is thought that diameters x of a production run of steel pins are random and follow a normal distribution having unknown mean and standard deviation p 039 A random sample of 5 such pins is selected from production and their sample mean diameter T is calculated What is the margin of error for T and what is the 95 confidence interval for p The X measurements are 10030 10034 10031 100292 10031 1a As with 11409 we require the sample mean and sam ple standard deviation s T 10031 s 0000181879 2 l lecture12309 nb Care must be taken When calculating s Rounding and other errors can easily affect the result If you use a calculator39s built in sample standard deViation routine be sure that it is using the n 1 diVisor It is especially important for very small 11 since it does make a difference 1b The estimated margin of error for T is calculated 2776 L 2776 W 0000225797 n 5 Value t 2776 is from tTable A97 Appendix D You find 2776 by locating con dence 95 at the bottom of the table and running up the column to row 4 degrees of freedom df 51 4 The ttable uses df to denote quotdegrees of freedom quot In our application of the table df n1 because in assessing the mar gin of error of Y we have to estimate the parameter 039 by s Statistically this has the effect of reducing the sample size by 1 thus df n1 1c Claim made for estimated margin of error Around 95 of random samples of n 5 from a normal population have i estimated margin of error cover the population mean u lecture12309 nb 13 Our random sample of n 5 parts has produced the interval Y i estimated margin of error 10031 i 0000225797 100287 100333 This interval is called a quot95 confidence interval for pquot 1d If the population is normal and all calculations are made with perfect accurac the EXACTLY 95 of samples of 5 are quotgood samplesquot whose 95 confidence interval Y i estimated margin of error covers the true value of 1 Is our sample of 5 a quotgood onequot We don39t know Nonetheless the method of reporting the sampling results does give a pretty good idea of the reliability of the findings 2 Other confidence levels Look at the next to bottom row of the t table it has an 00 symbol at the left of that row denoting large sample size Notice the entry 2576 of this row and that directly below this is confidence level 99 We will however use the entry for df 5 1 4 To obtain a 99 confidence interval p in the n 5 case you would substitute 4604 for 2576 So the 99 confidence interval for p takes the form 7 i 4604 not 2576 STT 200 Spring 2009 Lecture Outline 1 21 09 Estimated Margin of Error for difference of sample means El 2 This follow on to lecture 1 14 09 extends the margin of error concept given there for a single sample average 7 We require samples sizes n1 Nl nl n2 N2 n2 be all quotlarge enoughquot so that a bell curve approximation is justified I will not introduce Student s t at this point so the readings of pp 616 619 are relevant except the bottom of 619 A Question What is the difference ll 12 Where ul average number of graphics per page in pages 1 through 383 u2 average number of graphics per page in pages 384 through 767 of our textboook A solution Randomly sample 17 pages from 001 through 383 Independently of this sample 19 pages from 383 through 767 384 pages Estimate u1 u2 by the difference of sample means fl nghere 71 average mean of graphics amp picture counts for 17 sample pages in 001 383 Y2 average mean of graphics amp picture counts for 19 sample pages in 384 767 The estimated margin of error in this setup is 2 2 S N n 32 Nz nz n1 N1 1 n Nz l 2 1 lecture12109 nb The samples I ve randomly sampled 36 pages by perusing the table of random digits A 94 skipping over those outside the range 001 to 767 consult the table Those in bold have graph ics and I ve indicated the number of graphics in parentheses graphic sub components of a graphic display are not counted individually It is a serious point needing clarification for seri ous work but let s just suppose that we recounting graphics drawing attention to markedly differ entpansofthepage 716 32 1 463 473 200 1 731 727 1 39 759 quot skip 944 quot 43 quot skip 890 and 877 quot 764 3 3641 132 1 512 678 98 3 181 3 27 1 133 622 2 quotskip 922quot 666 1 3101 quotskip 844quot 7201 quot skip 945quot 639 112 5 285 1 quot skip duplicate 112 quot 429 471 1 quot skip duplicate 112quot 647 1 quot skip 770 quot 183 1 71 359 412 585 1 428 42 2 Here are the 36 sample pages sorted 271 321 39 422 43 71 983 1125 1321 133 1813 1831 2001 2851 3101 359 3641 412 428 429 463 4711 473 512 5851 6222 639 6471 6661 678716 7201 7271 731759 7643 Those sample pages less than 384 number 17 and constitute a random sample of 17 from 383 The others number 19 and constitute a random sample of 19 from the 384 pages 384 767 lecture12109 nb l3 1a The sample average number of graphics per page is for pages lt 384 11020035103111101 n1 17 123529 The sample standard deviation 51 see page 64 is s1 114564 The sample average number of graphics per page is for pages gt 383 Y2 0000100120110011003 n2 0656347 The sample standard deviation s see page 64 is s2 0781736 The difference of sample means is J71 372 123529 0656347 0656347 1b Estimated margin of error of the estimator 71 72 is calculated as follows 2 2 196 S1 N1n1 S2 Nznz n1 N1 1 n2 N2 1 13125 383 17 061111 384 19 17 383 1 19 384 1 0633975 S WQOO 3 75 f s z 6mm 3 495 m lt29 ZDe IOlC T 74 of W A a Lif 6 v 2 r DalEWlt 2amp5quot ifl 4W5 37 0 14 W fag COKZFCTEO C 63 72 5 W t Wt AorzW7o 444 G 5mm g3 f egzr 54 jam 16w 7 5 og5 5 7402 pews Hm P SEWgd K 6L 6 Q 7 H jaw0 7 I I 7 p g quot av 5xaEW z V Uoot UK 1me 3 Pawy 54 a 6 YA53 L mootf P6540 J a A ma M 55 19 WNW ex b 4 m 5 PCch 3 770770 L I A a quotWCZOWJ lt9 0392 160 5 AJ Wdl 39 59465 F Om MA 616 3 7 5 WC 39 Emmav5 67 4846 W 72m WMA PEc zvm 7 3quot y 5 3 a Kl 62 f E 7 75 AR i 57 KL F m 777 W F we PL 4660 74427 a x62 FaeEm m 3 3 gt 570 798 EL 85 2908 PRzK7P53 Ez or if PCK 70El W53 5V garL W 5 F 0 7a 4O f 74mm L 19074 lt7 2 2er My pa ctr7 7 397 7 51 Z 77 Clt P szljm k 2 r I 25 C4 f 6 1 Z J g STT 200 Spring 2009 Lecture Outline 1 14 09 Estimated Margin of Error for T Consult pp 62 65 This follow on to lecture 1 12 09 extends the margin of error concept to the sample average 7 I am again consid ering the case in Which n and N n are both quotlarge enoughquot so that a bell curve approximation is justified I will not introduce Student s t at this point so the readings of pp 586 606 are not particularly relevant A Question What is the average number p of pictures or graphics per page from pp 1 to 767 of our textbook This quotpopulation averagequot u Greek pronounced quotmuquot is ordinarily estimated by the sample average 7 What is the estimated margin of error in this setup I ve randomly sampled 36 pages by perusing the table of random digits A 94 skipping over those outside the range 001 to 767 consult the table Those in bold have graphics and I ve indicated the number of graphics in parentheses graphic sub components of a graphic display are not counted individually It is a serious point needing clarification for serious work but let s just suppose that we recounting graphics drawing attention to markedly different parts of the page 716 0321 463 473 2001 731 7271 039 759 skip 944 043 skip 890 and 877 7643 3641 1321 512 678 0983 1813 0271 133 6222 skip 922 6661 3101 skip 844 7201 skip 945 639 1125 2851 skip duplicate 112 429 4711 skip duplicate 112 6471 skip 770 1831 071 359 412 5851 428 0422 Here are resulting scores x number of graphics on each page of 36 sample pages 0 1 0 0 1 0 1 0 0 0 3 1 1 0 0 3 3 1 0 2 1 1 1 0 5 1 0 1 1 1 0 0 0 1 0 2 2 l lecture11409 nb 1a The sample average number of graphics per page is 2 11131133121115111112 n 36 16 014 12 23 31 5 36 3 0888889 The sample standard deviation s see page 64 is computed 16 0 08888892 14 1 08888892 2 2 08888892 3 3 088888921 5 08888892 36 1 114087 1b Point estimate of p Y 0888889 and sample standard deviation s 114087 Estimated margin of error of the estimator 7 is calculated as follows 5 N n 114087 767 36 196 E i 196 W 7671 036407 1c Claim made for estimated margin of error Around 95 of random samples of n 36 pages from the book of N 767 pages Will produce an interval Y estimated margin of error that Will cover the actual value of u average number of pictures graphics per page of pages 1 through 767 of the textbook Our random sample of n 36 pages from the N 767 pages of the book has produced the interval Y estimated margin of error 0888889 i 036407 05248196 125296 This interval is called a quot95 confidence interval for uquot 5779200 204 0 W KO 0 ya f 7 flt 5 ouf IF Pg o 7K 70 0 4f 7IK4 0F 2gt y j m xgg aA 617677 7057 a 3 0 1 engJWTa v ig 2150 SF 74g Iyo 199576 5295 5 74 amp FII4Wzr4 40 ffmavcf 76 77 0w Cd m y gv 6mm M14 47 L WOV gig 43856 WWIx7 V24 ggygV a 1 79346 WL XQ dWC SagMix fcx AWEUV 4440 3530 Sham g m 00 2300 05 m m z WJQD Y A 7700 79 A7 Z39567 la ale 7754 06 FWMy80m A 4274523 W 357 of 723 4 lb 7 fie23 0F 39 r 23 c 57 570 P l 00 alt 57 VMQESA 0F 57 39 A x 7 12377 RFC 510 VVo nm L o 0 6 W CZpZ LOKWMW flag 3 an 196 ng 2 1 684 807 745993 CT 690 A 5 F9 Elba UPmay F 42 EMAZr m o 39 9 ML OUK NAMO COW 372 07 U a jA 0000 gt 3 8 LjL fo il F Waspm a K 39 PK 6 N d N N FM my 160 Camry0N 0 170067 quot 39 Raw777546 M 61 m sfo 2 Jrow faves6 7 5 r TuTOA Moamp7252 5 6 37747941 a I 522 6lt 0lt5Z7 4 J VaMC JLm 557 7ch 7 y mm a C go frp aFQ S aff r ffz jfo ltgt Ff7p 71WFA 05 we 1 7 g 39 gJ C jg ZRa 397 if mg Wis 9717 862 6 in M K F 6 0 f x 9J ogy 7yFcnu AoT 5672 lA ygzamp rjo v35 39 39 WW MFA7 439 00 prewa 5400 54WD 5 5 7455 Ice J rrao rs39 a I WEVWSP WKKW W15 quot05 07500775 quot kvrrm fqa OF 477501 U 7070NS 0 i E quot12gtquot tWm OE39 4 Dow W39 4 0W m 7r przrmg S 4Z f y ymw 9quot gigA ANOO 7 quot quot PFC IV m v 4 39 Tux OI J 427539 6 V 39t v 7 tWV DFL f 43577 Z 1 K6 1quot 0JZ7 W074 7 v 5 1905 BF a 64 1L 0 L0 dVVW 170 b Amie 77m zf CZ COW 11 5 W Vf M f bW gr Mfg f azgy O W0 07 7 W3 7 W576 7 at Z 07 FCWQE 0077071621 garmm lFzW 70 40V 643 7mm 439 20 99153on jam7 memmn 2amp4 Ame fWm 556 W MWJ I me Ibo f A1739 WP 41 AM 62 raw 06 lo 0747 5537294053 39 Ao c 4 VA Y 7 V 0amp7wa f5 m v Za39c 0Q7ANEZgo IT 7544 M F W 465 M V6 4 23 P 7 0 3 7 3 az TOAI L Raf 47734 A Pam7 579F 15 mg q z t 6 57 570 E za I K II 2 12 g 00 Ni VC fu L 577 V I cw gl p rd 95756 a 4V 4 ngWm 95 W g 1 m sz39glr 17457 quot W M23 7 lt meox WORM 39 V00 50 7 Kt39KbMi r 52 5mm 2 quot M 4 7de jg oquot I 39 W29 7mg g7 M 471 one 39Z3 7 W5 71007 4amp7 lij 0 coyg39es 2 9S 4739 Wt DON TWwF9Z 0 7345 40550 ColE 11 mm N m m 38ltltn ShhDENT L157 H g 0000 v 338 m K237 OW 96 76 K 3729q 2351 7465 7AKE39 Skro W 7 6 4 645 F606 mo 477mb JZZQMMV m0 57 OA 9d C1 7 V0 4 N0 W M 9 pix7f 7 W xativeV o W mom d 39 4 quot W T W Q KOZX Z a I 1 Zeb mSa U 639 ya54 23 lapw 53957 yrA If 982 a L g v 130 To M7 37 ng d AWN you way UK 2 0 570570 mega 615k 78 a 70 7 pm7iov Qv v39W f ew4 4 Sn EQQOK 597 5 5 7A 130 774iquot 576F zafrKKFX m a 57 F 7569i 5 A717 5 6 J39o E c 7 0 570777W67A0Ff EOFXO 4g lt7L07 j lt V0 Z6 WW57VE7J CIVZDM a i nggw 675 7 10sz 8 1 96 0012 6 5 2f ox F xxx27 9a CZ 7162 WW Wage W5 aver Loz39jb1NWAW afQ OMMJ W mav 407 WZGZ39 Cr 7 7L oJ 2 a 9 fWM znwwfl J O e r cf 39 j gco 3475 yvza fact56031 4599 47777 3107 5 76575 Ff m WAY 7 2 WM WMMCMA M we 17C 2 T 23555 7 4700 1570 55 W 4 g 579704 L 3 x 0 4126426 4 QFJ c39f 1 F f 700 677211 7537 5707 r 7 65 57w757c 700 0M6 gtgt 0 FM fleam 95765 b 575627 0 37 412511 407 6 ltDDAA WS c look 0 quot5 3 gm7 57WJZ 70 45 As7 nD quot 2lt Jam0R5 rm b F160 9 77 6 PM WF 72757 WKIOUS39 560715 52 Z 39z t 74545 Urc EFLOmega 9F fme new 5 7o We7FJD Mw5 foe cm7 007 l ov SOA W 50 Wo g J 5357 7ypcquot 6390004FSS 0F 2 77quot veal gammam 7000 27 AWLr 0Fme 5 407 ST ggygg gg 055 02d 4 m 58 6 2r 4 fwmw yji mo 73 gt572 L quot 7 quot i 1 g ML 2 g M 571T 15707 967 737 fx o L 4 2 23La em20me 2m 2 1 LF Zur 7 a 72W Ir ZNr Wmf f g I 7674117 gt 2 ZF zue 34 L 61797 15 D 34 ZIZ M Q n z Lu 9 2495quot 0 JJ 4 c camE0 DFcnlt wry50 L I f l J J W Jr Z Lfrn J 279187 217 IIIMDQ XL 9747 55 Ada 25 7 W 4922 cum es 7w 546 748x 3 Skaw 52rng Fog mm of 757 2 Wilmer Ufa f 7 i7m I757 L 7Forn3 37 EXOEC39Z O So yo 20 077ltJL w 20 agme a 2quot 90 3f mam We 00 390570xmfzssl 2 L L L 257677571 f 601627 3 Z MVEXPZ 30 f0 20 530 E Av 2r 1 30 W lgZZf 7167 j 5 3 WW 0 41 CW 20 I gDF 3 1 2 yo PgtO Nb EEW fiev10 7gtp 9 2 KM1 0 75 7 af manf y 0er 5m j 539 407 KW60M J MfrV X dblAWE WV A 52C 1 F 5FA quot QE P 03557 Faemum It CotN1 FOE mm UMD FK W 3 f 0F VOFp 570760 7 5 97a N076 71 IVA6 62 397 s 5 cm dmw W449 0 0g 2 L mamV54 Elm7 3 VXamp0 J I 0 33756537r9 3395 654 0gt0 6 amt140W a NABurm fL 7 966 230 00 our Iivt Ref 9K 0 Mia an About Variation 4 73 A Billion Dollar Misunderstanding In the late 19905 the Bill and Melinda Gates Foundation began funding an effort to encourage the breakup of large schools into smaller schools Why It had been noticed that smaller schools were more common among the best performing schools than one would expect in time the Annenberg Foundation the Carnegie Corporation the Center for Collaborative Edu cation the Center for School Change Harvard39s Change Leadership Group the Open Society Institute Pew Charitable Trusts and the US Department of Educa tion39s Smaller Learning Communities Program all supported the effort Well over a billion dollars was spent to make schools smaller But was it all based on a misunderstanding of sampling distributions Statisti cians Howard Wainer and Harris Zwerling13 looked at the mean test scores of schools in Pennsylvania They found that indeed 12 of the topscoring 50 schools were from the smallest 3 of Pennsylvania schoolssubstantially more than the 3 we39d naively expect But then they looked at the bottom 50 There they found that 18 were small schools The explanation Mean test scores are well means We are looking at a rough realeworld simulation in which each school is a trial Even if all Pennsylvania schools were equivalent we d expect 9 their mean scores to vary How much The CLT tells us that means of test scores vary according to Smaller schools have by definition smaller n s so the sampling distributions of their mean scores naturally have larger standard deviations It39s natural then that small schools have both higher and lower39mean scores On October 26 2005 The Seattle Times reported Whe Gates Foundation announced last week it is moving away from its emphasis on converting large high schools into smaller ones and instead giving grants to specially selected school districts with a track record of academic improvement and effective leadership Education leaders at the Foundation said they concluded that improving classroom instruction and mobilizing the resources of an entire district were more important first steps to improving high schools than r breaking down the size 01 or 4 187 jga 27y 4W5 130 9744 57WV75 hwvm V5ta 39 bo M 47jav 7 mama 01 1 W63 5696 firf 75 4 WNW V VSD Z 4447 5 Mc 057 7795 WS39 hW A f 72757 J20 V oo 57 Z 39 72 4 78 5 we fee ax WINZia VD NS wrw m zroo 4 5755775 L 1 2 W z r r 66675 m to y g Fear 539 M41160 2 78 130 72 m i QUZFS 39 7993 5 12 Ala445 fezc39 t chVer 7 6 eetWP MEGAs TM F ou 7amp4 I quote X 7 L 99 057 47 View fay W Maw 4197 65 rm 22 sz g 776 IMP VJLafyz X7L 517 67 WW My W WZ W Z y 5 V44 X WWW 7 4 0 Mr L 4M y Vm i m T Fggyy 393 JV 5 ffZYy7 yy L my ahwh 55477LZZ 6 X7 5 LL XVampLJ 7 KA7X 297 quot 4 4w 46 Kg WV X m rumw my QWKM urn7 74f39O Fn lL 1 m 6 Tag6 70 579 N 76447 quot 03 me if iquot 4 39 if 7t 6 4 5 47 EMFV7 7quot 77 quot W A 430 4 s 3 J L WE 5N7 Mama 3 m V5 LCM7773 x g 4315 J l 0 3375 6557 a J395 654 39 0gt0 6730 00 N 36506

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made $280 on my first study guide!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.