Seattle-Pipes Co. produces pipes to be supplied to a Seattle utility company. The requirement of the utility company is that the pipes need to be 200 cm long. Longer pipes are acceptable to the utility company but any pipe less than 200 cm is rejected. Seattle-Pipes loses all its production costs on pipes that are rejected. The production process is such that it has some variability in the lengths of pipes produced, and this variability can be well approximated by a Normal distribution. Seattle- Pipes can adopt one of the following three production processes:
Process A: Produces pipes with an average length of 200 cm and a standard deviation of 0.5 cm
Process B: Produces pipes with an average length of 201 cm and a standard deviation of 1 cm
Process C: Produces pipes with an average length of 202 cm and a standard deviation of 1.5 cm
a. If Seattle-Pipes adopts the third Process (Process "C'), what is the probability it will have its pipe rejected by the utility company? Enter your answer as a decimal probability (not a percent) rounded to 4 decimal places.
b. Seattle-Pipes temporarily changes its requirements and has a new requirement that it will accept any pipe of length from 199 cm to 202 cm. That is, pipes ranging in length from 199 cm to 202 cm will be accepted, others will be rejected. With this changed requirement, which production process (out of the 3) will result in the smallest percentage of rejections?
c. Seattle-Pipes earns a revenue of $200 for every pipe that gets accepted and loses all money for any pipe that is rejected. The cost of producing the pipes is $140 per pipe if the production process "A" is used, $160 per pipe if the production process "B" is used, and $177 per pipe if the production process "C" is used. Given this information, which production process would you recommend maximizing profits (revenue minus cost) if the requirement of the utility company is that pipes need to be of 200 cm (or more and any pipe shorter than 200 cm is rejected?

It is known that this variable has a normal distribution and that the distribution parameter varies depending on the process used to manufacture the pipes.

Process A: μ= 200cm δ= 0.5cm

Process B: μ=201cm δ= 1cm

Process C: μ=202cm δ= 1.5cm

Pipes with a length of 200cm or more will be accepted by the utility company (X≥200), but pipes with less than 200cm length will be rejected (X<200)

a. Using Process C, you need to calculate the probability that the pipe will be rejected, symbolically:

P(X<200)

Using the distribution data of process C you have to standardize the value:

P(Z<(200-202)/1.5)= P(Z<-1.33)= 0.09176

b. The requirements change, accepting any pipe between 199 and 202, you have to calculate the probabilities of the pipes being between those lengths using the three process:

Process A:

P(199≤X≤202) = P(X≤202) - P(X≤199)

P(Z≤(202-200)/0.5)) - P(Z≤(199-200)/0.5))

P(Z≤4) - P(Z≤-2) = 1 - 0.02275 = 0.97725

The probability of the pipe being rejected is 0.02275

Process B:

P(199≤X≤202) = P(X≤202) - P(X≤199)

P(Z≤(202-201)/1)) - P(Z≤(199-201)/1))

P(Z≤1) - P(Z≤-2) = 0.84134 - 0.02275 = 0.81859

The probability of the pipe being rejected is 1-0.81859= 0.18141

Process C:

P(199≤X≤202) = P(X≤202) - P(X≤199)

P(Z≤(202-202)/1.5)) - P(Z≤(199-202)/1.5))

P(Z≤0) - P(Z≤-2) = 0.5 - 0.02275 = 0.47725

The probability of the pipe being rejected is 1-0.47725= 0.52275

The pipes manufactured using process A has fewer chances of being rejected.

c.

Process A costs $140

P(X≥200)= 1 - P(X<200)= 1 - P(Z<0)= 1 - 0.5= 0.5

Process B costs $160

P(X≥200)= 1 - P(X<200)= 1 - P(Z<-1)= 1 - 0.15866= 0.84134

Process C costs $177

P(X≥200)= 1 - P(X<200)= 1 - P(Z<-1.33)= 1 - 0.09176= 0.90824

If they where to make 100 pipes:

Using process A: 100*0.5= 50 pipes will be accepted, so they'll win 50*($200-$140)= $3000

Using process B: 100*0.84134= 84.134≅ 84 pipes will beaccepted, so they'll win 84*($200-$160)= $3360

Using the process C: 100*0.90824= 80.824≅ 90 pipes will be accepted, so they'll win 90*($200-$177)= $2070

As you can see, using process B will maximize the profits.

I hope it helps!