Solutions to Ma381, Exam 2 (purple version), F04

Solutions to Ma381, Exam 2 (purple version), F04 - (16 pts...

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Unformatted text preview: (16 pts.) Multiple Choice. For each problem, circle the single best response. 1. If X and Y are independent random variables and a,b,c are constants, then Var(aX + bY + c) = a2 Var(X) + b2 Var(Y) b. Sometimes c. Never 2. If X and Y are random variables and a,b,c are constants, then E(aX +bY +c) = aE(X) +bE(Y) +c b. ometimes c. Never 3. The value of <D(—5.0) is Ve near b. Very nearly 1 0; Cannot be accurately determined 4. If X and Y are independent random variables with common standard deviation 0, then the standard deviation of %X —-;-Y is var-(av-éw l c —a 2 =(%)1V2W-LZ) 4- (“st-)2 V4.43) .. v6 1 1- : l7 ‘l. .- {(2.5 6 + '2’; 0" 156' a” 5’ -r ~ 1 5 $8“- 5‘3 m 5. (18 pts.) The (total) ankle diameter of a sample of 507 adults is displayed in the histogram below. As you can see, it is reasonable to assume that ankle diameter is normally distributed. Suppose that a good—fitting normal curve to this data is one with a mean, u, of 13.90 cm, and a standard deviation, 0', of 1.25 cm. X =- MKi—é- WM 1o 11 12 13 14 15 16 17 18 Ankle Diameter (cm) Assume that the sample of 507 adults is representative of the U.S. adult population as a whole. /’ a. Using the stated normal approximation, estimate the proportion of U.S. adults with ankle diameters 15 cm or more. W‘XfizlS): I—- got—13.90) (“‘5 >1- £072) -.- /-,s/o(, [email protected] b. Using the stated normal approximation, estimate the proportion of U.S. adults with ankle diameters between 12 and 16 cm. VC/zszsm) -.- g<f44240 _ Eek/3.90 52; #15 = $0.62) —- £04.52) 1'. .752; ’— ,oe+3 [email protected] (X: ;. annI/t’s fame—Z. 6. (30 pts.) Suppose, in a single game of chance, the winnings have a mean of -$2 and a standard deviation of $5. Assume the winnings in different games are i.i.d.. a. For a sequence of 49 such games estimate the chance of winning losing no more than $10. That is, compute the chance of winning -$10 or more. P<xz+~+ zfisé as») ,= m if: 2 ”V47? 1*,(f’x7 -..D/‘H—-("‘-2> Wm “ 5N4? [37’ =PC222.5I) = P- $0.51) b. For a sequence of 49 games, compute the (total) expected winnings. 5(Zl+“'* qu) = fiery-w 60:”) W 4 c. For a sequence of 49 games, compute 316 standard deviation of the (total) winnings. V24" (Z«*‘~-+X+4) ‘d. , '-> vamp-HM Vat—(2,.) =— §¢+ + 4" W 4‘74“"; = 47/15) = 025 ail-”*2” : Wt“- ’ 4 7. (21 pts.) A density model for the life-span of a randomly chosen person from a certain ulation e. . coun is . P0P ( E W) 2: > L‘ R’s a f(x)= 137353000“er 05165100 0 otherwise ' Note that no person lives beyond 100 years of age with this model. a. Write, but don ’t evaluate an integral for the mean lifetime of an individual from this population. . 1” ,g M‘ $0 'K 75,. 7c? (loo-K)? 4:: b. Write, but dgn’t evaluat , an expression for the standard deviation of lifetime for an individual from this population. c. Write, b . nt evaluate an integral for the chance that an individual from this population lives to be at most 65 years of age. FZ $365.): 35 ,4, 76,000 —-7t)2-L70 8. (15 pts.) When making a “Four-Number Bet” of $1 in roulette, you net $8 if the roulette ball falls into one of 4 slots among 38 equally likely slots. You net -$l if the roulette ball falls into one of the other 34 slots. 3. Compute the mean net winnings in one $1 four-number bet. b. Compute the Standard deviation in winnings in one $1 four-number bet. 512E”1(%)* (5’).”(338 * Hg)" ? 7.62??? ,._\ “may...
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