The management of Madeira Computing is considering the introduction of a wearable electronic device with the functionality of a laptop computer and phone. The fixed cost to launch this new product is $300,000. The variable cost for the product is expected to be between $160 and $240, with a most likely value of $200 per unit. The product will sell for $300 per unit. Demand for the product is expected to range from 0 to approximately 20,000 units, with 4,000 units the most likely. (a) Develop a what-if spreadsheet model computing profit for this product in the basecase, worst-case, and best-case scenarios. If your answer is negative, use minus sign. Best-case profit $ Worst-case profit $ Base-case profit $ (b) Model the variable cost as a uniform random variable with a minimum of $160 and a maximum of $240. Model product demand as 1,000 times the value of a gamma random variable with an alpha parameter of 3 and a beta parameter of 2. Construct a simulation model to estimate the average profit and the probability that the project will result in a loss. Round your answers to the nearest whole number. Average Profit $ Probability of a Loss % (c) What is your recommendation regarding whether to launch the product?

If you win $100 for rolling a 12, win $10 for rolling a number less than 6, and lose $4 for rolling anything else, what are your expected winnings per play?

a. $1.94         b. $4.72         c. $2.78         d. -$.68

A committee of 7 is be selected from a group of 22 people. How many such committees are possible?

a. 22,254      b. 319,770      c, 170,544     d. 101,458

A woman plans on having four children. What is the probability she will have at most 2 boys?

a. 11/16         b. 1/2         c. 9/16        d. 6/16

Thirty-nine percent of the houses in a certain neighborhood read the New Yorker, 43 percent read the Philadelphia Inquirer, and 9 percent read both. What percent read at least one of these publications?

a. 64%        b. 79%         c. 82%            d. 73%

What is the probability of getting at least one 10 in twenty rolls of a pair of dice?

a. (33/36)²⁰         b. 1-(3/36)²⁰        c. 1- (33/36)²⁰       d. 1-20(33/36)          

A plant is to be built to produce blasting devices for construction work, and the decision must be made as to the extent of automation in the plant. Additional automatic equipment increases the investment costs but lowers the probability of shipping a defective device to the field, which must then be shipped back to the factory and dismantled at cost of $10 per device. The operating costs are identical for the different levels of automation. It is estimated that the plant will operate 10 years. The interest rate is 20%, and the rate of production is 100,000 devices per year for all levels of automation. Prepare a table with a column for all possible levels of automation, a column for the expected number of defectives in 100,000 devices, a column for the expected annual cost of defectives, a column for the annual cost of investment and a column for the total expected annual cost. Using the table, find the level of automation that will minimize the expected total annual cost for the investment costs and the probabilities given below

Level of Automation, Probability of producing a defective, Cost of investment ($)
1 0.100 100,000
2 0.050 150,000
3 0.020 200,000
4 0.010 275,000
5 0.005 325,000
6 0.002 350,000
7 0.001 400,000

Horus has developed successfully and needs additional funding. An IPO seems to be a natural step forward. After the bookbuilding phase, it is decided that 1,000,000 stocks would be sold in the IPO at a per-unit price of $10. You think that there is a 0.55 probability that the price will go up to $13 (underpricing) and 0.45 probability it will go down to 8 (overpricing) . You also know that there are two types of investor in the market (in addition to you): - Uninformed (e.g., individual investors): these investors always buy, and their aggregate demand is 800,000 stocks. - Informed (e.g., hedge funds): these investors know if the issue is under- or overpriced and they have the capacity to demand up to n stocks in aggregate. If the IPO is oversubscribed (i.e., if total demand is higher than the 1,000,000 stocks offered), every investor is rationed on a pro-rata basis. For instance, if total demand is 1,250,000 stocks, every investor receives 1, 000, 000/1, 250, 000 = 0.8 times the number of stocks he asked for. You plan to buy 100 stock in this IPO, what is the largest informed investor’s capacity (i.e., largest n) such that you do not lose money in expectation?

1. We want to test a new drug and we want to ensure that all age groups are represented it to do this we split up the general population into age groups and then select randomly within each group state what type of sampling this is and support your answer.

1b. Describe what type of distribution this is and support your answer

2. We roll 5 six-sided die. What is the probability of obtanining exactly two 1s?

3. How many license plates can we have of 3 digits followed by 3 letter? We allow repeated digital and letters.

4. What is sampling error and how would you distinguish it from non sampling error?

5. What types of inferences can we make?

6. In normal distribution what is the percentage of data having a z-score less than -1?

7. An experiment involves rolling a six sided die 480 times and recording the number of 3s. What is the mean and standard deviation?

8. What type of probability involves repeating an experiment and recording the relative frequency of the outcome?

The management of Madeira Computing is considering the introduction of a wearable electronic device with the functionality of a laptop computer and phone. The fixed cost to launch this new product is $300,000. The variable cost for the product is expected to be between $160 and $240, with a most likely value of $200 per unit. The product will sell for $300 per unit. Demand for the product is expected to range from 0 to approximately 20,000 units, with 4,000 units the most likely. (a) Develop a what-if spreadsheet model computing profit for this product in the basecase, worst-case, and best-case scenarios. If your answer is negative, use minus sign. Best-case profit $ Worst-case profit $ Base-case profit $ (b) Model the variable cost as a uniform random variable with a minimum of $160 and a maximum of $240. Model product demand as 1,000 times the value of a gamma random variable with an alpha parameter of 3 and a beta parameter of 2. Construct a simulation model to estimate the average profit and the probability that the project will result in a loss. Round your answers to the nearest whole number. Average Profit $ Probability of a Loss % (c) What is your recommendation regarding whether to launch the product? blank

Currently, among the 20 individuals of a population, 2 have a certain infection that spreads as follows: Contacts between two members of the population occur in accordance with a Poisson process having rate ?. When a contact occurs, it is equally likely to involve any of the possible pairs of individuals in the population. If a contact involves an infected and a non-infected individual, then, with probability p the non-infected individual becomes infected. Once infected, an individual remains infected throughout. Let ?(?) denote the number of infected members of the population at time t. Considering the current time as t = 0, we want to model this process as a continuous-time Markov chain.

(a) What is the state space of this process?

(b) What is the probability that an infected person contacts a non-infected person?

(c) What is the rate at which an infected person contacts a non-infected person (we denoted this type of contact by I-N contact) when there are X infected people in the population?

(d) Is the inter-contact time between two I-N contacts exponentially distributed? Why?

(e) Compute the expected time until all members of the considered population are infected.

True or False.

1. In order to use results from the Central Limit Theorem for calculating that the sample mean is greater than some number, the population you are sampling from must have a normal distribution.
2. If a variable has a binomial distribution with the probability of success equal to 0.82, then the distribution is skewed to the left.
3. For calculating the probability that a sample mean is greater than some given value, in solving this problem, one of the first things you would do is to apply the continuity correction.
4. If you took many samples (size >50) from a population and calculated the proportion for each of the samples, then the distribution of these proportions would have an approximate normal distribution.
5. If a person had a z-score of 0.49, this indicates their actual raw score is close to the 50^th percentile.
6. If the sample size is equal to 40 and the true proportion is equal to 0.25, then the z-distribution can be used instead of the binomial distribution for calculating probabilities.

7.The approximation to normality for the sampling distribution of * becomes better and better as the sample size increases.

8. When sampling from a normal distribution where the mean = g and variance 6 , then the sampling distribution for R will also have a mean = g and a variance = 6² .

Consider two stocks with returns RA and RB with the following properties. RA takes values -10 and +20 with probabilities 1/2. RB takes value -20 with probability 1/3 and +50 with probability 2/3. Corr(RA,RB) = r (some number between -1 and 1). Answer the following questions

(a) Express Cov(RA,RB) as a function of r

(b) Calculate the expected return of a portfolio that contains share α of stock A and share 1−α of stock B. Your answer should be a function of α (c) Calculate the variance of the portfolio from part B (Hint: returns are now potentially dependent)

(d) What value of α* minimizes the variance of the portfolio? Your answer should be a function of r, denoted by α*(r).

(e) For what range of values for r is your α*(r) 6 1? What is the solution to the above problem if r is outside of that range? (Hint: draw a graph and find α* ∈ [0,1] that minimizes variance) (f) Is α*(r) increasing or decreading? (Hint: take the derivative with respect to r)

(g) Which r wouldtheinvestorprefertohave, positiveornegative? Whatistheintuition for that result? 3