6,tr- 5 .?s-.-?> 6-a 6's*q s b 8 \-/ Class: Name: Date: Block: Normal Curves Introduction Definition Word Are aPprpp"iq1<- ftfr distrib,;tions u\hosa ShA"P unirn oda\ { syrnen enric Normal Models t'B{. \\shcr.fad r.l(M<nn,3b) Novfno.\ \\ode Cr) rv €" \ v.r ith nrl€a,o O c."rd SD o€ '* Standard Normal Model/Distribution r aql. :a;-Zs trg X T+9 Tr,Ls ! rls [\ la\t ? IS Sketch Normal Models usingthe 68'95-99'7 Rule L. Birth weights 2. \ of babies N(7.5Ib, 1'3 Ib) ACT Scores at a certain college, N(2L'2,4'4) t- qgY. -) tbQY't L, N(36, 4) 11 ,+ /. What percent of data are between3? and 40? ZL kZA + LIO frso- fit = rfi-S b87- What percent above the mean are between 36 and 40? 36 -+ q0 YfF) rnhs What percent of data are between 28 and 44? Lbe VL e Za+\orqqm-? * aYnfl'@ of What range contains n 50+ rr\t3S (WY"r3c{))= QO 36-acq) =ZLl rr3s a What range contains m-Ls . . data? 99.7o/o a 47.5o/o below the mean? a-- M s-WZu Where would the top !60/o of data be? fi*3 What percent of data is oltsid-e 24 and 48? L, reports that the diStribution of head circumferenc.e-Crnpglfale soldiers is approximately normal with a mean ofz.8 inchesbnd a standard deviation oflll1Eugll Helmet SiZeS: The army rvVhat percent of soldiers have a head circumference greater than 23.9in? Lz,{ - rn+ 13 -- A head circumference of 23.9 inches would be what percentile? loO-l(p = 8 tl+ \rVhat percent of soldiers have a head circumference between 20.6 and 23.9inches? 2olo ? Z\fr ?U.0 + rn-ZS \s m- m Lg q mtS What interval below the mean would contain !3.5o/o of soldiers? tz,5'/. = xi3s 5i-zs lz0 ,(o - 2l= S'peedS: Suppose it Akes you 20 minutes, on average, to drive to school with a standard deviation ofl minutes. Suppose a normal model is appropriate for the distribution of driving times' Dfiving How often will you arrive at school in less than??' minutes? arnts . l+ 7,.1 + lZ.S IOO-W :3tY. ,bb- How often will it take you more thanzrtminutes? z qr-l {ffi'Y\ntz3 think the distribution of your.driving times is unimodal and svmmetric in general? t.fOGrr Orccide'/.,f ,, l.p\iday , Do you U)u\dba{\rd^q, t,lL .. v{00\d Cc}.l-ls{-- {rrruStD' VorV. What does this say about the accuracy of your predictions? Explain. \\1 prcdtctiOnE ba*ad on qo(n^1) $nod€\ Nh\cvr OS4onfteq Un\rn0dql ' \nnmlvi c data' No+o.Ccr.)c^\e t Practice Problems 1. Environmental Protection Agency [EPA) fuel economy estimates for automobile models tested recently a mean of 24.8 mpg and a standard deviation of 6.2 mpg for highway driving. Assume that a Normal Model can,be applied. a. Draw the model for auto fuel economy. Clearly label it, showing what the 68- 95- 99.7 rule predicts about miles per gallon l' AEv /\ n.q B,b 21'fi 31 t-b. *\ljgryof t4 the central 680/o of autos to be found? of cars should get more than 31 mpg? ,s e. \-/ -> In what interval would About Ll 11,+ t8.G-3\ c. 21.7- should get between 31 and 37 mpg? Describe the gas mileage of the worst 2.5%o of all cars. Anl Crrr +ho$ hrs 0.$cro tnr\ec'rge o( Q"p1 D'q rnP3 bLb€\o(r, '\-/ 1 2. YJ Fifty-three men qualified for the men's alpine downhill race in Salt Lake City. The gold medal winner finished in l minute 39.13 seconds. All competitors times (in seconds) are found in the following list: Q? =-r&pt@@ 100.00 100.30 1,00.58 100.74 100.84 100.&5t 74t.27 101.56 {01.8$ 10?:15 103.04 103.19 L03.73 ro3.d 1os.4rd 106.6 108.84 a) L09.75 T0-0.Er L0a.74 101.05 L0L.66 9+#8 a99G 100.37 100.39 r00.76 1"00.81 t}L.24 LDr.25 104.3u L01.69 102.52 103.33 105.25 tor.6{ 107.65\ t02.3\ 103.m fiL.70 t02.54 103.63 105.35 108-'37 LL4.42 The mean ti*, Jfconds, with a standard deviation ofb=il seconds. If the Normal Model is appropriate, what percent of times will be less than 99.7 seconds? -rrlfrIZt q13 = mqe -ls h'sn lb'l o€ -+h< dq| b) o.. What is the actual percent of times less than 99.7 seconds? F7 5 53{ cJ Why do you think the two percentages don't agree? flnedala is n,tt Slmrnetric o%rtnri cnoda\ So ttrc \\o r rrrq\ tnode \ an"f+ be e?Vli<6 \7 d) Create a histogram of these times. What do you see? $r .\) <a5 c{_ a- +' I qb 3. fr m u n tsi;tf€;trti ,4 14 lob 16 (to ll2- lFl A friend t"rr, yoP"oubo.,$)"nilri.?$ffiJ,",th the number ofyears reaching experience among current college professors. He remembers the mean but can't recall whether the standard deviation was 6 months, 6 years or l6years. Tell him which one it most likely was, and why. qq.-+'1" aw a1 of dala Shu:V\b-c lczrVltgrn 3Standard d<rr\qhons 6ronn +lnx rr*'.n: PrOr{-fts.orE uso a\\y hau<- b,hrte4'6 O Cnd 5O \r5.o.C <)/-?4{\enc€- b r-non+\ns i9 to: 3 vrn c'r\\ Hc 1*C..;od \t{\-th $+- \r-s toou\d b4' T* 1:: '3 ltol's \s lD" \arge b&"^tse n" o*e\&sncn€on' is\oqe a1 furn ffitAn 6Pans bt{ gcc'tcs urhtcln 3 su *han \^la' d ev7w+X to -\1ZI%tco.[ Choice iS b grs" o,.w t 4. Some IQ tests are standardized to a Normal Model, with a mean of L00 and a standard deviation of 16. a. Draw the model for these IQ scores. Clearly label it, showing what the 6895-99,7 Rule predicts about the scores qq,?). V b. In what interval would you expect the central95o/o of IQ scores to be found? 6b -tTL c. Abotrtwhat percent of people should have IQ scorei aboVe 116? tGY d. Aboutwhat percent of people should have IQ scores between 68 and 84? 13,5y, v e. About what percent of people should have IQ scores above L32? (^r ),6"/- \ \ 5. fohn Beale of Stanford, CA, recorded the speeds of cars driving past his house, where the speed limit read 20 mph. The mean of 100 readings was 23.84 mph, with standard deviation 3.56 a. How many standard deviations from the mean would a car going at the speed limit be? Qt= e9:ry'8\= T*'{' \.oB b. $ nPlour -t.OB *he mzan \Mhich would be more unusual, a car traveling 34 mph or a car going 10 mph? Why? Vt1=3,r\'22,91 3.* ? ro = = ?.?g A cctc gotns \\o .nvt^' lD4zft\= *2,%8 3, sLo ), c. To see how many cars actually speed, fohn subtr".ttQO-*ifilftom all the speeds. What is the mean speed now? What is the new standard deviation? trran : a3.5Lt -aO =\39n1 tD: d d. fohn's friend in Berlin wants to study the speeds so fohn converts all the original miles per hour readings into kilometers per hour by multiplying all speeds by 1.609 km/hr. What is the mean now? What is the new standard deviation? bn<qn ; a 3,Sq ( l, troQ): 3g SD , j .f,Co 0 .(rrs9)= -3b 5.13 V'" lr' E* ln \)