Question

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Assignment-07: Problem 1 Previous Problem Problem List Next Problem (8 points) The wait time (after a scheduled arrival time) in minutes for a train to arrive is Uniformly distributed over the interval [0, 15]. You observe the wait time for the next 95 trains to arrive. Assume wait times are independent. Part a) What is the approximate probability (to 2 decimal places) that the sum of the 95 wait times you observed is between 670 and 796? Part b) What is the approximate probability (to 2 decimal places) that the average of the 95 wait times exceeds 7 minutes? Part c) Find the probability (to 2 decimal places) that 92 or more of the 95 wait times exceed 1 minute. Please carry answers to at least 6 decimal places in intermediate steps. Part d) Use the Normal approximation to the Binomial distribution (with continuity correction) to find the probability (to 2 decimal places) that 56 or more of the 95 wait times recorded exceed 5 minutes.

Answer 1

Here x ~ Uco, 15) , n=95. we want p(670 L EX <796), Now we know that X= {X/n. 1. divide it by n we get, P (670 <EXi <796) =P ( 670 Lexi 2 T 95 n 796 95 ) = P(7.05 < Ý < 8.38) If x r u ca,b) then its mean, & variance is given by & u= atb = 0+15 = 7.5 XNUCO, 15) oz (0-2)= (15)2 - 18.75 (b -a) 18.75 12 12

E = √18.75 = 4.33 p(7.05 < x < 8.33) = P17.05-7.5 T 4.33 95 L X-1 28.33-7.01 75 4331895 = P(-1.01 <z < 1.87) = P(Z <1.87) - - 0.9693 - = 0.8131 P(Z <-1-01) 0.1562 2. P( 670 < x < 796) = 0.81 P(X) 7) = = 1-P 1- (X< 7) P(x - 7-7.5 4.337.95 - 1-P (Z < -1.13) - ) - 0.8708 0 1292 = 0.13 P (x1) = 1. P (X<1) = 1-P CX-4 1-0.7.5 0 6.433 = 1- P(Z < -1.50) - - 0.0 668 = 0.9332 Let y = No. of wait time that yo Bin (n=95. p=0.9332) exceed 1 minute

Data P147,92) = I-PCV 291) 1 - 885588 = 0.114412 obtained using R software pbinom (91,950.9332) d p (x>5) - PCX- 5-7.5 4.33) = P(Z > 0.58) - - P(Z (0:58) - 1-0.7190 let y = No. of wait time that exceeds 5 min. YN Bin (n=95, p = 0.281) Mean = np - = 95 X 0.281 26.695 variance = npc-p) : 95 x 0.291 % (-0.281) = 19.194 ? YON (26.695, 19.194) P ( 47,56) = P(Y> 55.5) ... using continuity coy yechon -Pryl 55.5-26.695 o sig.194 = P(276.57) = 1-PCZL6.571