question &: A cireular bnes route operates brtworm and Town B and Mall respectiv

Question

question &: A cireular bnes route operates brtworm and Town B and Mall respectively. Let Ti represeat the tr A, Th represent the travel time bet wwen Town A and Town 13,T between Town B and Mall. Assume that the travel times are Mall sndl Town A. Tow A sted Town B, s represent the travel t indegeanlent noemal variables with expected value and standard devintions sown Travel Tinmse Expected Value Std Dev 7, Ta 7, 30 10 e schduled time for ose round trip of the shuttle bes is 2 bours. Determine the probabality that a round trip will not be coupleted on schelale? The An additional lane is added to the freeway connecting Town B and the mall. The new travel time bet ween Town B and the mall, 73, is related to the old travel time Ts as T 1/27 Determine the probability that a round trip will not be completed on schedule (assuming the scbeduled time is 2 hours)?

Explanation / Answer

Answering both the parts of Q5.

Part 1:

The travel time between Mall to Town A follows a Normal distribution with mean = 30, sd = 9

The travel time between Town A to Town B follows a Normal distribution with mean = 20, sd = 4

The travel time between Town B to Mall follows a Normal distribution with mean = 40, sd = 12

Sum of normal distributions is normal

with new mean = sum of means

and new variance = sum of variances

Hence, the travel time of the round trip will follow a normal distribution with

mean = 30 + 20 + 40 = 90

and variance = 9*9 + 4*4 + 12*12 = 241

i.e. standard deviation (sd) = 15.52417

So, the travel time of the round trip follows Normal distribution with mean = 90, sd = 15.52417

Probability that round trip will not be completed on schedule (in 2 hours)

= Probability that the travel time of the round trip is greater than 120

= 1 - pnorm(120, mean = 90, sd = 15.52417)

= 0.02665073

Part 2:

The new travel time between town B and mall is 1/2 * earlier travel time between town B and mall.

So, the mean and standard devaition of the new travel time between town B and mall will be half of the mean

and standard devaition of the new travel time between town B and mall earlier.

Earlier,

The travel time between Town B to Mall follows a Normal distribution with mean = 40, sd = 12

Now,

The travel time between Town B to Mall follows a Normal distribution with mean = 20, sd = 6

Sum of normal distributions is normal

with new mean = sum of means

and new variance = sum of variances

Hence, the travel time of the round trip will follow a normal distribution with

mean = 30 + 20 + 20 = 70

and variance = 9*9 + 4*4 + 6*6 = 133

i.e. standard deviation (sd) = 11.53256

So, the travel time of the round trip follows Normal distribution with mean = 70, sd = 11.53256

Probability that round trip will not be completed on schedule (in 2 hours)

= Probability that the travel time of the round trip is greater than 120

= 1 - pnorm(120, mean = 70, sd = 11.53256)

= 7.269779 * 10-6

The other image regarding rainfall at loactions A and B is irrevelant here and not a part of this question.

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