Question

There are two ways for a concrete truck to go from a ready-mixed concrete plant to the construction site. The first way is direct, for which the mean and standard deviation of travel time are 30 min and 5 min, respectively. The second way is through Town A. The mean and standard deviation of travel time between the plant and Town A are 14 min and 5 min, respectively. The mean and standard deviation of travel time between Town A and the construction site are 14 min and 6 min, respectively. If the total travel time is above 45 min, the concrete in the truck will go bad. Calculate the probability that the concrete goes bad for both cases of choosing the direct and indirect ways. Assume travel time has a normal distribution.   

Answer 1

Answer:-

Given That:-

There are two ways for a concrete truck to go from a ready-mixed concrete plant to the construction site. The first way is direct, for which the mean and standard deviation of travel time are 30 min and 5 min, respectively. The second way is through Town A. The mean and standard deviation of travel time between the plant and Town A are 14 min and 5 min, respectively. The mean and standard deviation of travel time between Town A and the construction site are 14 min and 6 min, respectively.

for the direct way, we are given the distribution here is:

The rpobability of concrete going bad here is computed as:

P(X > 45)

Converting it to a standard normal variable, we get here:

P(Z > 3)

Getting it from the standard normal tables, we get here:

P(Z > 3) = 0.0013

Therefore 0.0013 is the probbaility of concrete going bad if we choose the direct way.

Noe let the time from plant to Town A be X and that from Town A to construction site be Y.

Then, we are given here that:

As the 2 times are independent of each other, the total time distribution here is obtained as:

Therefore, the probability if we choose indirect way is computed here as:

P(Z > 2.1766)

Getting it from the standard normal tables, we get here:

P(Z > 2.1766) = 0.0148

Therefore 0.0148 is the probability of concrete going bad if we choose the indirect way here.

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