Answer:
Step-by-step explanation:
Hello!
The study variable is
X: Pipe length.
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!
