Stats assignment

Assignment 2 – Part 2

Due October 16, 23:59 EST

Marks for each part of each question are indicated in parentheses. Please see the course outline for

Part 2 Marking Scheme for penalties. There are no bonus marks for this assignment. Show your

calculations for each question.

Total Marks: 20

Question 1 – Investment Outcomes (5 Marks)

Alan and his financial advisor Bob discuss risk measurement for investment options. Bob presents

an investment with the possible outcomes and their associated probabilities given in the table

below. He indicates that these probabilities are calculated based on past experience, industry

rations and trends, and sophisticated simulation techniques.

Table 1 Probability Distribution of Outcomes for an Investment

Outcome Probability of Outcome Assumptions

$300 20% Pessimistic

$600 60% Moderately successful

$900 20% Optimistic

a) Use the probability distributing on the table to calculate the mean (expected value) and the

standard deviation of the investment outcomes. What does the expected value imply here? (1

mark)

b) Alan interprets the standard deviation of the investment outcomes as follows: “the larger the

standard deviation (or spread of outcomes), the greater is the risk”. Explain why his interpretation

of standard deviation makes sense. (1 mark)

c) Alan compares two investments with the following means and standard deviations of the

investment outcomes. Bob explains that investment A appears to have a high standard deviation,

but not when related to expected value of the distribution. A standard deviation of $600 on an

investment with an expected value of $6,000 may indicate less risk than a standard deviation of

$190 dollars on an investment with an expected value of only $600. Calculate measure of

coefficient of variation for investment A and B. Determine which investment carries the greater

risk and explain why coefficient of variation is the appropriate measure to compare these

investment opportunities? (1 mark)

Investment A Investment B

µ = $6,000 µ = $600

σ = $600 σ = $190

d) Alan compares three new investments with the following means and standard deviations of the

investment outcomes. Calculate the coefficient of variation for investments 1, 2 and 3. Based on

the coefficient of variation, which investment involves the most risk and the least risk? (1 mark)

Do we obtain the same results by comparing standard deviations of these investments? explain

why? (1 mark)

Investment 1 Investment 2 Investment 3

µ = $600 µ = $600 µ = $600

σ = $20 σ = $190 σ = $300

Question 2 – Face Mask (5 marks)

Due to Covid-19 pandemic, the demand for disposable medical face mask has increased

drastically. A particular brand of medical face mask making machine is used to produce disposable

medical face masks in mass production. This machine is capable of producing 80 to 100 industry

standard surgical style face masks every minute. This particular machine has a 2% defective rate.

The production of medical face masks on this machine is considered a random process where each

produced mask is independent of the others. Using this information calculate the following

probabilities:

a) What is the probability that the 10th medical face mask produced is the first with a defect? (1 mark)

b) What is the probability that the face mask making machine produces no defective mask in a batch of 100? (1 mark)

c) On average, how many face masks would you expect to be produced before the first defective mask? What is the standard deviation? (1 mark)

d) Another brand of this machine that also produces surgical style face masks has a 5% defective rate where each mask is produced independently of the others. On average how

many face masks would you expect to be produced with this machine before the first with

a defect? What is the standard deviation? (1 mark)

e) Based on your answers to parts (c) and (d), how does increasing the probability of an event (defect in this case) affect the mean and standard deviation of the wait time until the event’s

occurrence (defect in this case)? (1 mark)

Question 3 - Blood Transfusion (5 marks)

A blood transfusion is a routine medical procedure in which donated blood is provided to you

through a narrow tube placed within a vein in your arm. Blood types are important when it comes

to transfusions. If you get a blood transfusion that that is not compatible with your blood type,

your body’s immune system response will be triggered to fight the donated blood. For instance, a

patient with Type O- blood can only receive Type O- blood, but a patient with Type O+ blood can

receive either Type O+ or Type O-. Ethnic background is also important in blood transfusions. If

a blood donor and recipient are of the same ethnic background, the chance of an adverse reaction

may be reduced. According to a 10-year donor database of a particular country, 0.37 of white

donors in the country are O+ and 0.08 are O-.

a) Consider a random sample of 15 white donors in this country. What is the expected value of the number of individuals who could be a donor to a patient with blood type O+, and

what is the standard deviation? (2 marks)

b) Calculate the probability that 3 or more of individuals in the sample from part (a) could donate blood to a patient with O- blood type. (1 mark)

c) On average, how many donors would need to be randomly sampled for the first type O+ donor to be identified? Assume that only while individuals are being tested. (1 mark)

d) What is the probability that exactly 4 donors must be sampled to identify the first type O+ blood? Assume that only while individuals are being tested. (1 mark)

Question 4 – Poisson Distribution (5 marks)

Suppose that random variable x has a Poisson distribution with mean value = 2. (show your

detailed answer for each part of questions below).

a) Write down the Poisson formula and describe the possible values of x. (0.5 mark) b) Starting with the smallest possible value of x, calculate p(x) for each value of x until p(x)

becomes smaller than 0.001. (1 mark)

c) Create probability distribution graph for this Poisson distribution using your results from part (b). (0.5 mark)

d) Calculate P(x = 2) (0.5 mark) e) Calculate P(x <= 4) (0.5 mark) f) Calculate P(1<= x <=4) (1 mark) g) Calculate P(2 < x < 5) (0.5 mark) h) Calculate P(2 <= x <6) (0.5 mark)