1- Write a function f(n,a,b) that generates n random numbers
# from Uniform(a,b) distribution and returns their minimum.
# Execute the function with n=100, a=1, b=9.

2- Replicate the function call f(100,1,9) 100 thousand times
# and plot the empirical density of the minimum of 100 indep. Unif(1,9)'s

3-Use the sampling distribution from (b) to find 95% confidence
# interval for the minimum of 100 independent Unif(1,9)'s.

Please solve in R

Question 4:

The times that a cashier spends processing each person’s transaction are independent and identically distributed random variables with a mean of µ and a variance of σ² . Thus, if X_i is the processing time for each transaction, E(X _i) = µ and Var(X_i) = σ² .

Let Y be the total processing time for 100 orders: Y = X₁ + X₂ + · · · + X₁₀₀

(a) What is the approximate probability distribution of Y , the total processing time of 100 orders? Hint: Y = 100X, where X = 1 100 P100 i=1 Xi is the sample mean.

(b) Suppose for Z ∼ N(0, 1), a standard normal random variable:

P(a < Z < b) = 100(1 − α)%

Using your distribution from part (a), show that an approximate 100(1 − α)% confidence interval for the unknown population mean µ is:

(Y − 10bσ)/100 < µ < (Y − 10aσ)/100

(c) Now suppose that the population mean processing time is known to be µ = 1.5 minutes, and the population standard deviation processing time is known to be σ = 1 minute. What is the probability that it takes less than 120 minutes to process the 100 orders? If you use R, please provide the commands used to determine the probability. Could you show all steps in the hand written working for this question please.

Nonconstant Growth Stock Valuation Reizenstein Technologies (RT) has just developed a solar panel capable of generating 200% more electricity than any solar panel currently on the market. As a result, RT is expected to experience a 16% annual growth rate for the next 5 years. By the end of 5 years, other firms will have developed comparable technology, and RT's growth rate will slow to 8% per year indefinitely. Stockholders require a return of 13% on RT's stock. The most recent annual dividend (D0), which was paid yesterday, was $1.10 per share.

a. Calculate RT's expected dividends for t = 1, t = 2, t = 3, t = 4, and t = 5. Do not round intermediate calculations. Round your answers to the nearest cent.

D1 = $ ?? D2 = $ ?? D3 = $ ?? D4 = $ ?? D5 = $ ??

b. Calculate the estimated intrinsic value of the stock today, . Proceed by finding the present value of the dividends expected at t = 1, t = 2, t = 3, t = 4, and t = 5 plus the present value of the stock price that should exist at t = 5, . The stock price can be found by using the constant growth equation. Note that to find you use the dividend expected at t = 6, which is 8% greater than the t = 5 dividend. Round your answer to the nearest cent. Do not round your intermediate computations. $ ??

c. Calculate the expected dividend yield (D1/ ), the capital gains yield expected during the first year, and the expected total return (dividend yield plus capital gains yield) during the first year. (Assume that = P0, and recognize that the capital gains yield is equal to the total return minus the dividend yield.). Round your answers to two decimal places. Do not round your intermediate computations.

Expected dividend yield ___%?? Capital gains yield ___%?? Expected total return ___%??

Also calculate these same three yields for t = 5 (e.g., D6/ ). Round your answers to two decimal places. Do not round your intermediate computations.

Expected dividend yield ___% ?? Capital gains yield ___% ?? Expected total return ___% ??

d. If your calculated intrinsic value differed substantially from the current market price, and if your views are consistent with those of most investors (the marginal investor), what would happen in the marketplace?

I. If the price as estimated by the marginal investor differs from the market price, then investors will buy or sell until an equilibrium has been established, with the intrinsic value as estimated by the marginal investor is more than the actual market price.

II. If the price as estimated by the marginal investor differs from the market price, then investors will buy or sell until an equilibrium has been established, with the intrinsic value as estimated by the marginal investor equals the actual market price.

III. If the price as estimated by the marginal investor differs from the market price, then investors will buy or sell until an equilibrium has been established, with the intrinsic value as estimated by the marginal investor is less than the actual market price.

IV. If the price as estimated by the marginal investor differs from the market price, then investors will not buy or sell anything until a new equilibrium has been establishes.

What would happen if your views were not consistent with those of the marginal investor and you turned out to be correct?

I. If you think the stock is priced above or below its intrinsic value, then you should at least consider buying if the stock is undervalued or selling if it is overvalued. If you turn out to be correct, then you will make money, eventually if you hold on to an unpopular position long enough.

II. If you think the stock is priced above or below its intrinsic value, then you should at least consider selling if the stock is undervalued or buying if it is overvalued. If you turn out to be correct, then you will make money, eventually if you hold on to an unpopular position long enough.

III. If you think the stock is priced above or below its intrinsic value, then you should at least consider buying if the stock is undervalued or selling if it is overvalued. If you turn out to be correct, then you will lose money, eventually if you hold on to an unpopular position long enough.

IV. If you think the stock is priced above or below its intrinsic value, then you should at least consider selling if the stock is undervalued or buying if it is overvalued. If you turn out to be correct, then you will lose money, eventually if you hold on to an unpopular position long enough.

Reizenstein Technologies (RT) has just developed a solar panel capable of generating 200% more electricity than any solar panel currently on the market. As a result, RT is expected to experience a 15% annual growth rate for the next 5 years. By the end of 5 years, other firms will have developed comparable technology, and RT's growth rate will slow to 7% per year indefinitely. Stockholders require a return of 13% on RT's stock. The most recent annual dividend (D₀), which was paid yesterday, was $1.10 per share.

1. Calculate RT's expected dividends for t = 1, t = 2, t = 3, t = 4, and t = 5. Do not round intermediate calculations. Round your answers to the nearest cent.

   D₁ = $

   D₂ = $

   D₃ = $

   D₄ = $

   D₅ = $

2. Calculate the estimated intrinsic value of the stock today, . Proceed by finding the present value of the dividends expected at t = 1, t = 2, t = 3, t = 4, and t = 5 plus the present value of the stock price that should exist at t = 5, . The stock price can be found by using the constant growth equation. Note that to find you use the dividend expected at t = 6, which is 7% greater than the t = 5 dividend. Round your answer to the nearest cent. Do not round your intermediate computations.
   $
   
3. Calculate the expected dividend yield (D₁/ ), the capital gains yield expected during the first year, and the expected total return (dividend yield plus capital gains yield) during the first year. (Assume that = P₀, and recognize that the capital gains yield is equal to the total return minus the dividend yield.). Round your answers to two decimal places. Do not round your intermediate computations.

   Expected dividend yield	%
   Capital gains yield	%
   Expected total return	%

   
   Also calculate these same three yields for t = 5 (e.g., D₆/ ). Round your answers to two decimal places. Do not round your intermediate computations.

   Expected dividend yield	%
   Capital gains yield	%
   Expected total return	%

4. 
   
5. If your calculated intrinsic value differed substantially from the current market price, and if your views are consistent with those of most investors (the marginal investor), what would happen in the marketplace?
   I. If the price as estimated by the marginal investor differs from the market price, then investors will buy or sell until an equilibrium has been established, with the intrinsic value as estimated by the marginal investor equals the actual market price.
   II. If the price as estimated by the marginal investor differs from the market price, then investors will buy or sell until an equilibrium has been established, with the intrinsic value as estimated by the marginal investor is more than the actual market price.
   III. If the price as estimated by the marginal investor differs from the market price, then investors will buy or sell until an equilibrium has been established, with the intrinsic value as estimated by the marginal investor is less than the actual market price.
   IV. If the price as estimated by the marginal investor differs from the market price, then investors will not buy or sell anything until a new equilibrium has been establishes.
   -Select-IIIIIIIV
   
   What would happen if your views were not consistent with those of the marginal investor and you turned out to be correct?
   I. If you think the stock is priced above or below its intrinsic value, then you should at least consider selling if the stock is undervalued or buying if it is overvalued. If you turn out to be correct, then you will make money, eventually if you hold on to an unpopular position long enough.
   II. If you think the stock is priced above or below its intrinsic value, then you should at least consider buying if the stock is undervalued or selling if it is overvalued. If you turn out to be correct, then you will make money, eventually if you hold on to an unpopular position long enough.
   III. If you think the stock is priced above or below its intrinsic value, then you should at least consider buying if the stock is undervalued or selling if it is overvalued. If you turn out to be correct, then you will lose money, eventually if you hold on to an unpopular position long enough.
   IV. If you think the stock is priced above or below its intrinsic value, then you should at least consider selling if the stock is undervalued or buying if it is overvalued. If you turn out to be correct, then you will lose money, eventually if you hold on to an unpopular position long enough.
   -Select-IIIIIIIV

Economic Analysis (The Roberta Wagner Tire Problem)

When Roberta Wagner first launched e-tire, she thought that she had a home run on her hands. It had always been hard to buy bicycle tires online: e-tire would be a onestop marketplace for bicycle tires, matching bicycle tire producers with customers. But Wagner wasn’t clear on one important detail: how to set the prices at which the producers would sell to consumers. So Wagner did some market research. She surveyed tire producers and came up with a model for supply. Given a price for bicycle tires, p, suppliers would be willing to sell 650 + 5 × p tires. Meanwhile, Wagner’s market research led to a simple equation for consumers’ demand for tires. Given a price, p, consumers would want 1,000 − 2 × p bicycle tires.

At the price of $30 per tire, how many tires are demanded and supplied?

Wagner could tell that the website was not working as well as she had planned. She took a deep breath and then decided to structure e-tire entirely differently. She eliminated the requirement for a fixed price, and instead allowed sellers and buyers to negotiate a price via public text message on the site. Suppose that the sellers decided to follow a simple rule: so long as there were unsold tires, they would cancel all transactions and try a new price, 10 dollars lower than the previous price. Conversely, as long as there were buyers without tires, they would cancel all transactions and try a new price, 10 dollars higher than the previous price. So they next try a price of 60 dollars.

What is the final price at which buyers and sellers will trade tires?

During the Summer and Fall of 2020, the COVID-19 crisis led to an enormous change in the demand for bicycles. Consumers were afraid to take public transportation and so instead purchased bicycles. Describe what this would do to the prevailing price on e-tire, using both a graph and a brief paragraph.

A ride-sharing app company that matches customers who want a ride somewhere with nearby drivers who will drive them. How does Wagner’s e-tire dilemma in Parts 1 and 2 relate to a ridesharing app, which has to set prices for those rides?

Many people, sadly, require a kidney transplant from a living donor. There are typically many more people who need a kidney than there are people donating kidneys. Use any country’s rules for organ donation as an example. Explain what this problem has to do with the shortage or surplus of kidneys facing patients with kidney disease. How would your team recommend solving the kidney shortage or surplus problem? What tradeoffs did you discuss in coming up with your solution?

Economic Analysis (The Roberta Wagner Tire Problem)

When Roberta Wagner first launched e-tire, she thought that she had a home run on her hands. It had always been hard to buy bicycle tires online: e-tire would be a onestop marketplace for bicycle tires, matching bicycle tire producers with customers. But Wagner wasn’t clear on one important detail: how to set the prices at which the producers would sell to consumers. So Wagner did some market research. She surveyed tire producers and came up with a model for supply. Given a price for bicycle tires, p, suppliers would be willing to sell 650 + 5 × p tires. Meanwhile, Wagner’s market research led to a simple equation for consumers’ demand for tires. Given a price, p, consumers would want 1,000 − 2 × p bicycle tires.

At the price of $30 per tire, how many tires are demanded and supplied?

Wagner could tell that the website was not working as well as she had planned. She took a deep breath and then decided to structure e-tire entirely differently. She eliminated the requirement for a fixed price, and instead allowed sellers and buyers to negotiate a price via public text message on the site. Suppose that the sellers decided to follow a simple rule: so long as there were unsold tires, they would cancel all transactions and try a new price, 10 dollars lower than the previous price. Conversely, as long as there were buyers without tires, they would cancel all transactions and try a new price, 10 dollars higher than the previous price. So they next try a price of 60 dollars.

What is the final price at which buyers and sellers will trade tires?

During the Summer and Fall of 2020, the COVID-19 crisis led to an enormous change in the demand for bicycles. Consumers were afraid to take public transportation and so instead purchased bicycles. Describe what this would do to the prevailing price on e-tire, using both a graph and a brief paragraph.

A ride-sharing app company that matches customers who want a ride somewhere with nearby drivers who will drive them. How does Wagner’s e-tire dilemma in Parts 1 and 2 relate to a ridesharing app, which has to set prices for those rides?

Many people, sadly, require a kidney transplant from a living donor. There are typically many more people who need a kidney than there are people donating kidneys. Use any country’s rules for organ donation as an example. Explain what this problem has to do with the shortage or surplus of kidneys facing patients with kidney disease. How would your team recommend solving the kidney shortage or surplus problem? What tradeoffs did you discuss in coming up with your solution?

Ag-Coop is a large farm cooperative with a number of agriculture-related manufacturing and service divisions. As a cooperative, it pays no federal income taxes. The company owns a fertilizer plant that processes and mixes petrochemical compounds into three brands of agricultural fertilizer: greenup, maintane, and winterizer. The three brands differ with respect to selling price and the proportional content of basic chemicals.

Ag-Coop’s Fertilizer Manufacturing Division transfers the completed product to the cooperative’s Retail Sales Division at a price based on the cost of each type of fertilizer plus a markup.

The Manufacturing Division is completely automated so that the only costs it incurs are the costs of the petrochemical feedstocks plus overhead that is considered fixed. The primary feedstock costs $1.50 per pound. Each 100 pounds of feedstock can produce either of the following mixtures of fertilizer.

Production is limited to the 850,000 kilowatt-hours monthly capacity of the dehydrator. Due to different chemical makeup, each brand of fertilizer requires different dehydrator use. Dehydrator usage in kilowatt-hours per pound of product follows:

Monthly fixed costs are $90,000. The company currently is producing according to output schedule A. Joint production costs including fixed overhead are allocated to each product on the basis of weight.

The fertilizer is packed into 100-pound bags for sale in the cooperative’s retail stores. The sales price for each product charged by the cooperative’s Retail Sales Division follows:

Selling expenses are 20 percent of the sales price.

The Retail Sales Division manager has complained that the prices charged by the Manufacturing Division are excessive and that he would prefer to purchase from another supplier.

The Manufacturing Division manager argues that the processing mix was determined based on a careful analysis of the costs of each product compared to the prices charged by the Retail Sales Division.

Required:

a. Assume that joint production costs including fixed overhead are allocated to each product on the basis of weight. What is the cost per pound of each product, including fixed overhead and the feedstock cost of $1.50 per pound, given the current production schedule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b. Assume that joint production costs including fixed overhead are allocated to each product on the basis of net realizable value if sold through the cooperative’s Retail Sales Division. What is the allocated cost per pound of each product, given the current production schedule? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

c. Assume that joint production costs including fixed overhead are allocated to each product on the basis of weight. Calculate the operating profit under both Schedule A and Schedule B. (Do not round intermediate calculations. Round your answers to 2 decimal places.)