Over three decades ago, four businesspeople who had experience in retailing natu

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Over three decades ago, four businesspeople who had experience in retailing natural foods through food stores believed that there was a demand for a supermarket for natural foods. As a result, in 1980 in Austin, Texas, they founded the first Whole Foods Market store in a building that had around 10,000 square feet and a staff of 19. This store was quite large compared to health food stores at the time. By 1984, the company was successful enough to expand to Houston and Dallas. In 1988, they purchased the Whole Food Company in New Orleans and expanded there. The next year, they moved into the West Coast with a store in Palo Alto, California. Even though the company has built a number of its own stores, much of the company growth has come through mergers and acquisitions, many of which came in the 1990s in such places as North Carolina, Massachusetts, Rhode Island, both Northern and Southern California, and Michigan. After the turn of the century, Whole Foods Market established a presence in Manhattan (NY), followed by a move into Canada and later into the United Kingdom.

Presently, Whole Foods Market has 297 stores in 38 U.S. states, the District of Columbia, Canada, and the United Kingdom, with 44 more in development. There are over 57,000 team members, 82% of whom are full-time employees. Existing stores now average 43,000 square feet in size, about four times as large as the original “supermarket.” Whole Foods Market is the ninth largest food and drug store in the United States with almost $2 billion in sales last year and is number 284 on the list of Fortune 500 companies.

Whole Food Markets is the largest retailer of natural and organic foods and prides itself in doing the research necessary to assure customers that offered products are free of artificial flavors, colors, sweeteners, preservatives, or hydrogenated fats. The company attempts to customize each store by stocking it with products that are most in demand in any given community. Whole Foods Market management cares about their employees, and the company has been named by Fortune magazine as one of the “100 Best Companies to Work For” in the United States every year since the list was first compiled 13 years ago. The company attempts to be a good community citizen, and it gives back at least 5% of after-tax profits to the communities in which they operate. In January 2008, Whole Foods Market was the first U.S. supermarket to commit to completely eliminating disposable plastic bags. The Core Values of the company are “Whole Foods, Whole People, and Whole Planet.” The Whole Foods Market searches “for the highest quality, least processed, most flavorful and natural foods possible …” The company attempts to “create a respectful workplace where people are treated fairly and are highly motivated to succeed.” In addition, the company is committed to the world around us and protecting the planet.

2. Suppose that, on average, in a Whole Foods Market in Dallas, 3.4 customers want to check out every minute. Based on this figure, store management wants to determine how to best staff checkout lines in order to meet demand.

- Binomial , Poisson, or Hypergeometric Distribution? -

a) What probability distribution could be used to study this problem? Write the name of the distribution and the related parameters below. (2 points)

b) Use R / R Studio to compute (all) cumulative probabilities for the probability distribution described above, attach your R output to the worksheet for submission. Store management wants to staff checkout lines such that demand for checkout can be met 99% of the time. In this case, store management would have to provide sufficient staff levels for what number of customers? Support your answer by referring to your cumulative probabilities and show any additional working below. (2 points)

c) What percentage of the time would the store have 6 or more customers who want to check out in any one minute period? Support your answer by referring to probabilities calculated using R / R Studio and show any additional working below. (2 points)

d) Based on the distribution described in part a), use R / R Studio to randomly generate 100 numbers from this distribution, how many of the 100 numbers had values of 6 or more? Attach your R output to the worksheet for submission. (1 point)

Explanation / Answer

2.a)

Let X be the number of customers per minute that want to check out, then X follows Poisson distribution with parameter = average customer per minute that want to check out = 3.4.

i.e. X~Poisson(3.4)

Therefore, Poisson distribution could be used to study this problem.

(Poisson distribution gives probability of number of events or occurrences in given time period)

b)

i) The number of customers per minute that want to check out will be 0,1, 2, …, n.

i.e. X takes values from 0 to n.

To compute (all) cumulative probabilities for X using R/R Studio, suppose n=20 and the R command will be

[cumulative probability gives P(Xx).]

ii) The number of customers for which the store management would have to provide sufficient staff levels by considering the demand for checkout met 99% of the time can be given as

output:

c) We have to find the percentage of the time would the store have 6 or more customers who want to check out in any one minute period. i.e. P(X6).

This probability can be find as

P(X6) = 1-P(X<6) = 1 - P(X5) = 1 - (cumulative probability for x=5) = 1 - 0.8705 = 0.1295

d) To generate 100 numbers from Poisson distribution, use the following command

To find how many of the 100 numbers had values of 6 or more, use the following command

Output:

[note- rpois generate random values, therefore for each run we get different results]

x=1:20
c=ppois(x,lambda=3.4)
output=data.frame(x,c)
names(output)=c("No. of customers checkout per minute","cumulative probability")
output

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