A building contractor claims that they can renovate a 200 sq foot kitchen and dining room in 40 work hours plus minus 5 (the mean and standard deviation respectively). The work includes plumbing, electrical installation, cabinets, flooring, painting and installing new appliances. From past experience we assume that times to complete similar projects are normally distributed. In order to gain a competitive edge the contractor wishes to guarentee a completion date. If the job is not done in a specified number of hours the contracter will reimburse 20% of the total cost to the owner. What should the number of hours be if they wish to take no more than a 6% risk of having to pay up? Advice - Draw the key elements of the solution before you compute.

Question

A building contractor claims that they can renovate a 200 sq foot kitchen and dining room in 40 work hours plus minus 5 (the mean and standard deviation respectively). The work includes plumbing, electrical installation, cabinets, flooring, painting and installing new appliances. From past experience we assume that times to complete similar projects are normally distributed. In order to gain a competitive edge the contractor wishes to guarentee a completion date. If the job is not done in a specified number of hours the contracter will reimburse 20% of the total cost to the owner. What should the number of hours be if they wish to take no more than a 6% risk of having to pay up? Advice - Draw the key elements of the solution before you compute.

You can use z-scores and the z-score formula to compute the number of hours.

Formula: z = (x - mean)/sd -->sd = standard deviation

You are given the mean and standard deviation. You can find z using a z-table representing "no more than 6% risk" of having to pay up. Once you have that z-score, all you will need to do is solve the above equation for x.

I hope this will help get you started.

Answer 1

To find the number of hours the contractor should aim for in order to take no more than a 6% risk of having to pay up, we can use the z-score formula.

1. Calculate the z-score:

z = (x - mean) / sd

In this case, the mean (μ) is 40 work hours and the standard deviation (σ) is 5 work hours. To find the z-score corresponding to a 6% risk (0.06), we can use a z-table or a statistical calculator.

2. Find the z-score corresponding to a 6% risk:

Look up the z-score in the z-table that corresponds to a cumulative probability of 0.06 (i.e., 6%). The z-score represents the number of standard deviations away from the mean.

3. Solve for x:

Now that we have the z-score, we can rearrange the equation above to solve for x:

x = z * sd + mean

Substitute the calculated z-score, the standard deviation, and the mean into the equation to find the number of hours (x) that the contractor should aim for.

By following these steps, you can determine the number of hours the contractor should target to take no more than a 6% risk of having to reimburse the owner. It is important to note that this approach assumes the distribution of completion times follows a normal distribution based on past experience.