What is the probability that a respondent likes the Impossible burger?

Math 218-01 Quantitative Methods for Business, Midterm Exam #1A, Fall 2020

Dr. Matt Sequin

NAME (PLEASE PRINT):

Instructions :

• This is an open book, open note exam. You may also use the internet for extra background information. However, you may not work with other students, tutors, or solicit assistance from other people via the internet. Submissions will be cross-checked with other students and with Blackboard’s internet scanning software. If any plagiarism is detected, the entire exam score will be a zero.

• Please write the exam out on separate pieces of paper. It is very important that the problems appear in order in your final submission. Please submit the exam by scanning it or by taking a legible picture of each page, then submitting it through Blackboard. Alternatively, you can email it to msequin@saintpeters.edu

• If a problem asks you to explain, be sure to give a thorough explanation in English sentences. For most of these problems, the explanation will be worth as much or more than getting the right numerical answer.

• The paper that you hand in should be a final draft. It should have everything written neatly in order with no cross outs, extra sentences written in the margins or other obvious edits.

GOOD LUCK!

Question Points Score 1 25 2 25 3 25 4 25

TOTAL 100

1. (25 points) Alfonso has just purchased a food truck to sell tacos. He will charge $3 per taco. The ingredients to make a taco cost $2.40, and the monthly costs of permits, gasoline and maintenance on the truck costs around $1500.

(a) Write out the revenue function R(X), the cost function C(x) and the profit func- tion P (X) for this situation. Briefly explain your answer.

(b) How many tacos will he have to sell to make a monthly profit of $1000? Briefly explain.

(c) Graph of the revenue function R(x) and the cost function C(x) on the same axis given below. Write out the (x, y) coordinates of their intersection point. Your graph should be drawn with a computer or be a very precise sketch.

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2. (25 points) 400 people were participated in a taste test of two of the new ‘meatless’ burger patties: the Impossible Burger and the Beyond Burger. After recording their responses, the respondents were divided into the four groups in the table below.

Liked Beyond Disliked Beyond Liked Impossible 258 26

Disliked Impossible 21 95

(a) What is the probability that a respondent likes the Impossible burger? Explain.

(b) What is the probability that a respondent liked one burger but not the other? Explain.

(c) Given that someone liked the Beyond burger, what is the probability that they

also liked the Impossible burger? Explain.

[ P (I|B) = P (B ∩ I)

P (B)

] (d) Using your work above, if we know a person likes the Beyond burger, are they

more likely to like the Impossible burger than a randomly chosen respondent? Explain.

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3. (25 points) Patty has applied for a business loan at the local bank to open a new independent coffee shop in town. The loan officer deciding whether or not to approve her needs to determine the probability her shop will become profitable in the next year. Unfortunately for Patty, the bank knows that there is a 40% chance of a Starbucks opening across the street from her proposed location. The bank estimates that if the Starbucks opens, she will only have a 15% chance of being profitable, but she will have a 60% chance if the Starbucks does not open.

(For all of the formulas below, let S be the event that Starbucks opens and let P be the event that Patty is profitable.)

(a) What is the probability that both Starbucks opens and she is profitable?

P (P ∩ S) = P (P |S) · P (S)

(b) Given all of this information, what is the probability that she is profitable? Show your work.

P (P ) = P (P |S)P (S) + P (P |SC)P (SC)

(c) What is the probability that either Starbucks opens or she is profitable (or both)?

P (P ∪ S) = P (S) + P (P )− P (P ∩ S)

(d) A year later, the loan officer hears that Patty’s coffee shop is profitable. What is the probability that the Starbucks opened across the street? Show your work.

P (S|P ) = P (S) · P (P |S) P (P )

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4. (25 points) Banana is an electronics company. Most people who walk into the Banana store do not buy the most expensive phone: the MyPhone XX. The company did some data collection, and determined that one of their salesmen, Steve, can convince 20% of customers to buy a MyPhone XX. The store averages 100 customers on a typical day. (The formulas for the binomial and Poisson distributions are given below.)

P ({X = k}) = n! (k!)(n− k)!

(pk)(1− p)n−k

P ({X = k}) = λ k · e−λ

k!

(a) Say that 100 customers enter the store during Steve’s shift. Use the binomial distribution to determine the probability that exactly 15 of them by a MyPhone XX. Show your work.

(b) Steve gets fired if he does not sell at least 2 MyPhone XXs on his shift. If 100 customers enter the store during his shift, use the binomial distribution to find the probability that he sells at least 2 MyPhone XXs. Explain what you did.

(c) Use the Poisson distribution to find the probability that exactly 100 customers enter the store. Explain what you did.

(d) Steve sometimes gets stressed about not selling his two phones because not enough customers come into the store. Use the Poisson distribution to estimate the probability that at least 2 customers enter the store on any given day. (There is no need to simplify your answer to a decimal.)

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