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~lJe. Name Elements of Statistics (Math 106) - Exam 1 Spring 2009 - Brad Hartlaub Directions: Please answer all of the questions below and show your work. The point values for each problem are indicated in parentheses. You may use one sheet of formulas, our course web page, and Minitab during the exam. Don't spend too much time on anyone part of the exam. 1. Thirty percent of all automobiles undergoing an emissions inspection at a certain inspection station I f d~ fail the inspection. .4-':= I aufo,..-.ob"e'i I / .J rhcd /~ / {o., A -The U ~.,,- -V!fj)('I1=/~ ~ I'" {"'=. /1 1'.r1~e('fr'cVl. J. j'J\ a. Among 15 randomly selected cars, what is the probability that at most 5 fail the inspection? (5) '-ply { ~ ~ 5' ):= ~ I / t.tJdfu1o.ftVe. ~ 7c}/ b~/ ( ~t (! tI 7:l02 6retjJ(;> Ai .a~/~/\ '/ b. Among 15 randomly selected cars, what is the probability that exactly 5 fail the inspection? (5) ()(l:= §);; (§) Among 15 randomly selecte~f:.(~at is the probability that at least 5 fail the inspection? (5) 1(1.~~)=l- f(r tV ~ ~5):=:/- /(K~t!):::/- ..~/{/j91=6fjffdj /JJ. Among 25 randomly selected cars, what is the mean value of the number of cafs·that /'. tlJ>~ pass017!/4.7 inspection, and what is the standard deviation of the number that pass inspection? (10r- y /V o ~/){,jJO oJ !(2~.,7) L, :i1Ji.cJ pO-rs. ~ :: /) fJ 0-;;== ::: j)d/ ) f/rb/J 17/01 ;25"(. 7) = / 7~5 [d5(. lj(d7.= ;:J. ,)9/3 What is the probability that am~g 25 randomly selected cars, the number than pass is within 1 /7{- /J /. ~ f/(/~;Jdf7- standard deviation of the mean? (10) ';;,;J9/?= 15'.;2tJg7 /?,{ -f ;),J9/?= ;;; 1(lt /7· 79/j y.c-. ~y~/7)= //4..74/2 /,..1../ \ ~b/7 2. What type of graph or graphs would you plan to make in a study of each of the following issues? (18 - 3 each question) a. What makes of cars do students drive? b. c. d. e. /;~,- 9reft ~r (He (,Ia.,!) . I~ U How old are their cars? It (-~101r~M1/ cIalpt:~ >/e-7'J-Cl"cI-~a.t/J~I How many hours per week do stGdents study? /'dlor;ra-n dol;/l~~ ),4.-n-a.-u ,/ ~ How does the number of study hours change during th~ se;ester? -h:,."t! S"(!rrpf /101 ~~ Which radio stations are the most popular with students? het"--'j,~/4 (or ;:,"e (';fc..,-~) k 3. You are assigned to direct a study at Kenyon College to discover factors that are associated with strong academic performance. You decide to identify 20 students who have perfect GP As of 4.0, and then measure explanatory variables for them that you think may be important, such as high school GPA and average amount oftime spent studying per day. ~I 7't:> b. Describe a study design that would provide more useful information. (10) !even::J.! re~/o'?(e> a.re fJ0S"~,:lk. ;:::r eXd."'7/'.Ie/ f~"'/~ ./' (/.."..,< coale!ie rekclecl" ,i/"M f(tllA/ /74'-?1krr- 0) t b~dJe/ c,"7/ /J?c.(J ,t.V/t.~~4?/C~-'1 {~~0 rCr >I'~/If' C/ I / 1C7<4cTo.-w ~ct.6/!"c1>o.",,/k ""-<!C'i/,,,qr Cp«// I,. "/,(M"'/ ~ ~ c:t5~OCI"C'./'o4 j~ee4 ca~e Itt /l4 &.vZd stands lJ$erex/Ia;1~ry S 54<#1/until 6(!obtaining exa.~/~1'ec/ 4. An interviewer at the street comeiand i/e;{rYa'~ conducts interviews a quota in various groups representing the relative sizes of the groups in the population. For instance, the quota might be 50 factory workers, 100 housewives, 60 elderly people, 30 Hispanics, and so forth. This is called quota sampling. Is this a random sampling method? Explain, and discuss the potential advantages and disadvantages of this method. (15) d, -lis (f U' / f'ht! U ~t?d Ij~ 1/ v./t 14' j /1 JtJt>CCi I I(J()//e c/e, /, ~t1e! arrIve.. ('Ire$e~v(, '''77 ~eg/ --7t aL,,~~/ "o! a ra••L.." /S II. I v,/I;o I)' // / ~7f7J yl'(Z)Clc.'k /'.A/ _7_1... cJ~eoe./o __ (..0(0 -/ -1'0 ~~re}eJ ~ r / '// {freer / / jl 1f/1'1",/"'/'b/1 -I"h~ ('orr1e-r a r(' I~ IJrV1'fJtVP/ S'() / #~~t(M()4. r k /~n/i-e4/eclaf $c<r/dol / 5. Raw scores on behavioral tests are often transformed for easier comparison. A test of reading ability has mean 75 and standard deviation 10 when given to third graders. Sixth graders have mean score 82 and standard deviation lion the same test. To provide separate "norms" for each grade, we want scores in each grade to have mean 100 and standard deviation 20. a. What linear transformation will change third-grade scores x into new scores Xnew = a + bx that have the desired mean and standard deviation? (Use b > 0 to preserve the order of the scores.) (10) -XI1l!W :: a:../- b/51> tJw -=: 6 K 2tJ :::: b i0 4;=: '<l1ew ~~1! ~2J -q .::; 100 - 5. - bX 2 (7») a:::: - ~() b. The linear transformation that will change sixth-grade scores x mto new scores that have the desired mean and standard deviation is xnew = -49.0924 + 1.8182x. Nancy is a sixth-grade student who scores 78 on the test. What is her transformed score? David is a third-grade student who scores 78. Who scores higher within his or her grade? (10) ffJ I~i rfT7 Il--I l J}t1V( = j= -ff."7Zf~/'3lf2(7g) :;:;9;2.7,)7;2 -7'0 (-J(7f) ~ (66 J/ -I<JA;Vf// /ar-!£. hr'1iflf!- f'c()rC? W/(Qr'1 krj !,-aote. 6. The usual way to study the brain's response to sounds is to have subjects listen to "pure tones." The response to recognizable sounds may differ. To compare responses, researchers anesthetized macaque monkeys. They fed pure tones and also monkey calls directly to their brains by inserting electrodes. Response to the stimulus was measured by firing rate (electrical spikes per second) of neurons in various areas of the brain. The file p:\data\math\stats\monkey.mtw contains the responses for 37 neurons. a. One notable finding is that responses to monkey calls are generally stronger than responses to pure tones. Give a numerical measure that sypports this 4f~a'1 /J?ect{il~ t21 finding. b2..5(5) ~r-e J;~e -V /06--:~ 3'3 /"'-];5 b It(! call7jresponse2()~" b. FindIl/oAfaV the least Cdl! squares line/76.. for predicting morikey from5pure tone response. (5) -"1 £al/::: 93, 92 1--. c. Identify and interpjet the value of r2• 77~3~ (10) 1-~(/a~cI ::; ~ £(OJ' O~. ~. 1/0,7% 1)1' ~ vaC7a/'!jr d 51'''' 'I (:;w II / ~ ~~#-~ r~ u/ 4" 17 fC'Hr"o4 j,,,,1''',- R% •• <"~"" U /t:l7~) / / C"j{ U,/ d. ~f Ident1)' theIre! poi9tIt?(;~ with the largest C/?~) residual. (5) 4d~ 1"he-' t- Coal! ~ C••••• 6F I/a m<,/; e,y",,''''''; /q ,-p k.s. / /(21J:?, /. 5'tJ9"'~ "/') ~frd«41 e. One point is an outlier in 1he~x direction. Identify this point. How influential is this point on the correlation coefficient? (10) ~ /~ Ib,J va!ve ~p r= .. b3J 41> It'HJ- r; ()4 f. 13" '$' ,-tJP14ovPc! ~'5 jJ r /~t'~ $,J" vVl~ b.../kr r..•.~ b~ ;feN';; 0 ~(/t'O~5"f % /rr>~ ~ dl'C)~S"~ .&> C/77 w/4'7 ?as- 4 ""f/c,r /~~('/ Would you be willing to use your least squares regression line from part (b) to predict the monkey call response when the pure tone response is 550? Explain. (5) ~ ~ -It d~ (<'1)6 ,1ua/!o-J? AI" (17~fee) i..e$' Fa"7 ,4.." &;, wwl/jO c1da. e,k/odrf ~ 1"J"'5T'o# /1 ~ <0'(~ U/e~ j"co ",J/eJ /'? ~rv"/ ~ R<7 oj! >,t:«§ 40~ ~ Cd"'/' 7. Different varieties of the tropical flower Heliconia are fertilized by different species of hummingbirds. Over time, the lengths of the flowers and the form of the hummingbirds' beaks have evolved to match each other. The data on the lengths in millimeters of three varieties of these flowers on the island of Dominca are in the file p:\data\math\stats\heliconia.mtw. Use visual displays and descriptive statistics to compare the three distributions. Your comparison should address center, spread, and shape of the three distributions. What are the most important differences among the three varieties of flowers? (30) ;a ! JA>cr'flJP V~'e~ l'4aJ reel b"llt/ (,., .H"') ~~4 17. ~fJ8 16 ye/~w ,tr ;;23 57.. 711 15 J(;,iPo 70ii:;~S ~ : 1,213 / .• '79;; ( 71~.5'1 r -;) 50 Li2.~ I/o 71~ ?~ B, Q :775" Ji f ~~ .]7. 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