Problem 14BSC

Designing Manholes According to the Web site www.torchmate.com, “manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter.” Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder breadths that are normally distributed with a mean of 18.2 in. and a standard deviation of 1.0 in. (based on data from the National Health and Nutrition Examination Survey).

a. What percentage of men will fit into the manhole?

b. Assume that the Connecticut Light and Power company employs 36 men who work in manholes. If 36 men are randomly selected, what is the probability that their mean shoulder breadth is less than 18.5 in.? Does this result suggest that money can be saved by making smaller manholes with a diameter of 18.5 in.? Why or why not?

Problem 14BSC

Answer:

Step1 of 3:

We have “manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter.” Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder breadths that are normally distributed with a mean of 18.2 in. and a standard deviation of 1.0 in. (based on data from the National Health and Nutrition Examination Survey).

That is x = 22, 18.2 and 1.0

Step2 of 3:

We need to find,

a. What percentage of men will fit into the manhole?

b. Assume that the Connecticut Light and Power company employs 36 men who work in manholes. If 36 men are randomly selected, what is the probability that their mean shoulder breadth is less than 18.5 in.? Does this result suggest that money can be saved by making smaller manholes with a diameter of 18.5 in.? Why or why not?

Step3 of 3:

a).

Consider the z statistics when x = 22

P(x = 22) = P(Z = )

= P()

= P(3.80)

Probability of 3.8 is calculated by using standard normal table(area under normal curve).In area under normal curve we have to see row 3.8 under column 0.00

P(3.80) = 0.9998

Therefore,99.98% of men will fit into the manhole.

b).

Consider the z statistics when x = 18.5

P(x = 22) = P(Z = )

= P()

= P(1.80)

Probability of 1.80 is calculated by using standard normal table(area under normal curve).In area under normal curve we have to see row 1.8 under column 0.00

P(1.80) = 0.9641

No, this result does not suggest that money can be saved by making smaller manholes with a diameter of 18.5 in. because, when considering the diameters of manholes, we should use a design based on individual men, not samples of 36 men.