Question

Please help with these questions.
1. (10 points) According to an airline, flights on a certain route are on time 82% of the time. Suppose 12 flights are randomly selected and the number of on time flights is recorded. (a) Find the mean and standard deviation for the number of on time flights. (b) What is the probability that exactly 9 of the 12 flights will be on time? (c) What is the probability that at most 10 of the flights will be on time? 2. (8 points) Suppose a geyser has a mean time between cruptions of 60 minutes and that the interval of time between eruptions is normally distributed with a standard deviation of 18 minutes. (a) What is the probability that a randomly selected time interval between cruptions is longer than 69 minutes? (b) What is the probability that a random sample of eight time intervals between eruptions has a mean longer than 69 minutes?

Answer 1

cal n X=0 $ Ans (1) Let & be the no. of flights on time out of n=12 randomly selected Hights. Then xv Binomial (n=12, p=0.82) with pmf P(x=x)= @ PX C1-; 2050,1,2,ug h The mean of number of on time flights in E(X)= [X PIX=X) = npe 1.280.02 = 9.84 The Jandard deviation of no. of on time flights is r = Vvaniy) = Vnpei-p) = 1280,02x0.18 6. 1.33 (6) The prob that exactly g of the 12 flights will be on time in P(X=9) = (3) 0.82,90.101? = 0.215) The prob that atmost 10 of the Hights will be on time is - Plx>10) = 1 PUX=11)-PCX 512) = 1- () 0.02)"0,10)' (2010.02)2(0,10)" = 1-0.2434 -0.09 24 = 0.6642 PLX 510)

18 n=8 (2) let y be the time interval b/w eruptions. Then given that is N(N=60, 051@)So Z= xy NCO,!) Cal The prob that a randomly selected time interval 6/W eruption in longer than 69 minutes is 1(X76973 PLZ) 69060) = P(23 035) El-p1250.5)= 1-616:5)= 1-0,691550.3005 (6) The prob that a randomly selected sumple ay eight (128) time blw eruptions has a mean longer than 69 minutes is PC tn >69)= P ( Xn-4, 69 -- 44 ; since In GN (M, 82) 0/55 0/5h = P(Z > GOO) = P127 1,4142 ) 1-$11.4142) = 1-0,9213= 0.0787, 14, let X1, X2, X3 be a random sample from a population with mean o let X-X2 + 2X3 be an estimator oso. Then Elô): E ( X1-X2+273 ) Since Eli)=0 Hi=1,2,3 3 [ ELY) E (X2)+2 ELM37] = [ 0-0 +20] =0 Since Elô) + 0,80 the estimator ô is not an unbiased estimator of o. (6) The variance of estimator o is Van lô )= Van (Yv-X2 +243) = von (X; - X2 42X) Isince xi's are independent (var(x1) + Var (Y27 + 4 Var(x37] since Var(X) Vcl X n ) = VaslY3) 2 varlx) 3 (al 3 varly 3 3