/A/ mttrt?c,&l6m5 INFERENCE FOR DISTRIBUTIONS OF CATEGORICAL DATA Exercses, nuts! A cmpany clams that each batch f ttse&n ta-ns 5 2%-cas-hews, 27% almnds, 13% macadama nuts, and 8% brazl nuts. T test ths clam, a qualty cntrl nspectr takes a randm sample f 1 5 nuts frm the latest batch. The ne-way table belw dsplays the sample data. Nut: Cashew Almnd Macadama Brazl Cunt: 83 29 2 18 (a) State apprprate hyptheses fr perfrmng a test f the cmpany's clam. (b) Calculate the expected cunts fr each type f nut. Shw yur wrk. jaw, nuts! Calculate the ch-square statstc fr the data n Exercse 1. Shw yur wrk. /OOA> f W AJTS l»-l S
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Ch-Square Gdness-f-Ft Tests f 7) Brds n the trees Researchers studed the behav r f brds that were searchng fr seeds and nsects n an Oregn frest. In ths frest, 54% f the trees were Duglas frs, 4% were pndersa pnes, and 6% were ther types f trees. At a randmly selected tme durng the day, the researchers bserved 156 red-breasted nuthatchesr7 were seen n Duglas frs, 79 n pndersa pnes, and 7 n ther types f trees.2 D these data suggest that nuthatches prefer partcular types f trees when they're searchng fr seeds and nsects? Carry ut a ch-square gdness-f-ft test t help answer ths questn. Ct m &<?. VA Benfrd's law Faked numbers n tax returns, nvces, r expense accunt clams ften dsplay patterns that aren't present n legtmate recrds. Sme patterns are bvus and easly avded by a clever crk. Others are mre subtle. It s a strkng fact that the frst dgts f numbers n legtmate recrds ften fllw a mdttrrtwtt as &enfrdvtaw-.3-cau the-frst dgt f a randmly chsen recrd X fr shrt. Benfrd's law gves ths prbablty mdel fr X (nte that a frst dgt can't be ): I Frst dgt (X): 1 2 3 4 5 6 7 8 ' Prbablty:.31.176.125.97.79.67 58 51 46 (?) N ch-square A schl's prncpal wants t knw, f students spend abut the same amunt f tme ; n hmewrk each nght f the week. She asks a randm sample f 5 students t keep track f ther hmewrk tme fr a week. The fllwng table dsplays thaaverage arnunt_pf tme (n mnutes) students reprted per nght: Nght: Sunday Mnday Tuesday Wednesday Thursday Frday Saturday Average 13 18 115 14 99 37 62 tme: Explan carefully why t wuld nt be apprprate t perfrm a ch-square gdness-f-ft test usng these data. 1 A frensc accuntant wh s famlar wth Benfrd's law nspects a randm sample f25 nvces frm a cmpany that saccused Fcmmttng fraud. The table belw dsplays the sample data. Frst dgt: Cunt: I 1 61 2 5 3 43 4 34 5 25 6 16 7 [} 9 1 5 6 - (a) Are these data ncnsstent wth Benfrd's law? Carry ut an apprprate test at the a =.5 level t supprt yur answer. If yu fnd a sgnfcant result, perfrm a fllw-up analyss. (b) Descrbe a Type 1 errr and a Type II errr n ths settng, and gve a pssble cnsequence f each. Whch d yu thnk s mre serus? Jc4 a 4. ere. 11) pg688 Mendel and the peas Gregr Mendel (1822-1884), an Austran mnk, s cnsdered the father f genetcs. Mendel studed the nhertance f varus trats n pea plants. One such trat s whether the pea s smth r wrnkled. Mendel predcted a rat f 3 smth peas fr every 1 wrnkled pea. In ne experment, he bserved 423 smth and 133 wrnkled peas. The data were prduced n such a way~"thatlke~randrn and Independent cndtns are met. Carry ut a ch-square gdness-f-ft test based n Mendel's predctn. What d yu cnclude?
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