Brutus Auto is a thriving, independently owned and operated firm located in Westerville. It was founded in 1986 by “Brutus” Cooper. Brutus was born and raised in Westerville, OH. Brutus is well-known and trusted by the residents of the town. As a teenager, he was a star football player for the high school team. In his spare time, he enjoyed working on cars and had a special talent when it came to cars. After college, this hobby led Brutus to start his own business – Brutus Auto Tire and Lube.

Brutus does a wide variety of work on vehicles of all makes and models. The shop also offers a variety of parts and supplies at retail. Brutus spends a great deal of time conversing with customers, most of whom he has known his whole life. His reputation for honest and reliable work has earned him many repeat customers, with the positive word of mouth helping him to expand his business.

His wife, Diane, runs the office, and takes special care to acknowledge customers on their birthdays with hand-written cards that include coupons for discounted services. Customers often reflect that Brutus offers the kind of service you can only find from a small-town mechanic. While Brutus is slightly more expensive than the chain establishments that have recently moved into town, customers testify that the friendly environment and quality service are worth the price.

Operational details

Brutus’s shop has grown steadily over the years. Originally, Brutus maintained a modest three automobile bays at his shop, and he and his long-time buddy, Fred, conducted most of the work except oil changes which were handled by a part-time apprentice. Currently, Brutus shop has six bays (the layout of the shop is shown in Figure 1) and seven mechanics (as not all of the service activities require a bay). Each of the bays is identical so that any employee can work at any station.

Four of the employees (including Brutus and Fred) are full-time (work at least 40 hours per week), and three are interns who work part-time (work fewer than 40 hours per week). Brutus and Fred are the only two mechanics permitted to do work beyond the basics, and so more involved car services can take some time getting through the shop. Most mechanics at Brutus Auto were trained at the area vocational school, and those with particular skill are taken on as apprentices who work under Fred and Brutus on the more involved projects. Cooper provides all the tooling necessary, and is respected as a fair and caring boss. In return, Brutus’ employees are loyal and hard-workers; seldom is absenteeism a problem, and turnover is rare.

The employee schedule for a typical week is shown in Table 1. The numbers shown in Table 1 are all devoted to working on vehicles. The shop closes for one week in the winter.

Inventory

Although the shop is primarily a service provider, all of the services require materials in addition to labor, and so there are inventory considerations that Brutus must manage.

A variety of other parts and supplies (for example, oil filters and oil) must be kept available due to the regularity of use; it would be difficult to meet the high expectations of customers if these items are not available. This has become a particularly high priority ever since national service chains moved into town, as the speed of their services such as oil changes is already far superior to that of Brutus. As a result of these competitive pressures, Brutus sets a 98% target service level.

Brutus is dedicated to using a high-quality brand of parts and supplies, most of which are available from a single supplier. Note that these parts and supplies are not only used to support services but also sold as retail transactions to customers who wish to perform their own repair work.

Currently, a continuous review policy is used for all car parts and supplies, but Brutus believes this system may require too much time to maintain for all inventory items. A sample of some of the items offered, the volumes used in services and sold at retail, and the associated costs are shown in Table 2. Brutus has determined that to call the supplier, pay for transportation, and put a shipment away costs approximately $50. The stockroom is relatively small and does not require special equipment or personnel. There is virtually no theft of product, so the annual holding costs are approximately 25% of unit cost. Delivery of orders takes about one week from the local supplier.

Adding Tire Service

The owners believe that adding tire replacement would increase revenues with a negligible increase to fixed (facility) costs other than tire inventory. However, in order to be competitive with the discount chains, Brutus must target a maximum of 30 minutes total time for each customer visit specific to tires since pricing was already competitive. They estimate that the time between arrivals for cars needing such replacement would be ten minutes. A dedicated and secure parking lot with a limit of 6 spots is available specifically for cars awaiting tire service. While service requirements vary, they estimate that a dedicated technician should finish up 3 tire service jobs each hour.

1. - Identify and describe TWO recommendations you would make to

                           Brutus Cooper to improve Operations and/or to lower costs.

Clarissa just graduated from High school and she is starting her new job today. Her new employer gave her a $15,000 signing bonus that she will invest today. She plans to retire 50 years from today (i.e., at the end of year 50). Once she retires, she would like to be able to withdraw from her retirement account $180,000 at the end of each year, starting the year after she retires (i.e., year 51). She expects that her retirement will last for 20 years (and the amount in her account after that should be 0). If Sofia can earn a return of 9% on her investments for the first 50 years and then 4% once she retires, how much money (in equal payments) does she need to save and invest each year in the first 50 years of her life to meet her goal (she invests ar the end of the year, i.e. this an ordinary annuity)? Round to the nearest 2 decimal places - if for example the answer is 7,345.567 then round it to 7,345.57.

3. In town A, a survey of 54 families finds a mean household size of 2.13, with a standard deviation of 1.0. In town B, a survey of 65 families finds a mean household size of 2.39, with a standard deviation of 1.2. Find an 86% confidence interval around the difference in means (3).

PLEASE HELP!

A credit reporting agency claims that the mean credit card debt in a town is greater than $3500. A random sample of the credit card debt of 20 residents in that town has a mean credit card debt of $3619 and a standard deviation of $391. At α=0.10, can the credit agency’s claim be supported?

5. A market research organization takes a simple random sample of 200 households in a town with 25,000 households. Of the 200 sampled households, 158 report having a video ondemand service like Netflix, Amazon Video, or Hulu.

(a) To answer parts (b) and (c), you need a box model. The box representing the population has a standard deviation that is (circle one) known OR estimated to be (fill in the value) _______________.

(b) The percentage of households in the town with a video on-demand service is estimated as _______________%, and this estimate is likely to be off by about _______________%.

(c) A 90%-confidence interval for the percentage of households in the town with a video ondemand service goes from _______________% to _______________%.

(d) Explain why it's ok to use the normal approximation to answer part (c). Your answer should contain the name of a theorem and one or more numerical comparisons.

(e) TRUE or FALSE (circle one): There is a 90% chance that the true percentage of households in the town with a video on-demand service falls within the interval calculated in (c).

A market research organization takes a simple random sample of 200 households in a town with 25,000 households. Of the 200 sampled households, 158 report having a video on-demand service like Netflix, Amazon Video, or Hulu.

(a) To answer parts (b) and (c), you need a box model. The box representing the population has a standard deviation that is (circle one) known OR estimated to be (fill in the value) _______________.

(b) The percentage of households in the town with a video on-demand service is estimated as _______________%, and this estimate is likely to be off by about _______________%.

(c) A 90%-confidence interval for the percentage of households in the town with a video on-demand service goes from _______________% to _______________%.

(d) Explain why it's ok to use the normal approximation to answer part (c). Your answer should contain the name of a theorem and one or more numerical comparisons.

(e) TRUE or FALSE (circle one): There is a 90% chance that the true percentage of households in the town with a video on-demand service falls within the interval calculated in (c).

Suppose that 5 percent of the population of a certain town contracted the COVID19 virus. There is a medical diagnostic test for detecting the disease. But the test is not very accurate. Historical evidence shows that if a person has the virus, the probability that her/his test result will be positive is 0.9. However, the probability is 0.15 that the test result will be positive for a person who does not have the virus.

1. Define clearly the events before you answer the following questions.

2. For a person selected randomly from the town, the test result was positive. What is the probability that the person has the Corona virus?

3. What is the difference between the probability found in the previous question and the 5%? How do we call the two probabilities?

4. For a person selected randomly from the town, the test result was negative. What is the probability that the person has the Corona virus?

5. For a person selected randomly from the town, the test result was positive. What is the probability that the person does not have the Corona virus?

A community is currently being served a single self-serve gas station with six pumps. A competitor is opening a new facility with 12 pumps across town. The table below shows the travel times in minutes from the four different areas in the community to the sites and the number of customers in each area.

(a) Using the Huff retail location model and if λ = 2, calculate the probability of a customer traveling from each area to each site.

(b) Estimate the proportion of the existing market lost to the new competitor.

Travel Times to Gas Stations (in minutes)

The twelfth paragraph states that the Chilean government’s "fiscal restraint" in the face of strong capital inflows during the 1990s left the government with "enough money...to cushion the economic downturn when foreign capital eventually turned tail in 1998." Explain.

https://www.economist.com/node/10286077