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Statistics Assignment: Normal Distribution, Binomial Distribution, Poisson Distribution, and Decisio

## Testing for Normality

1.Please explain why you would or would not think each of the following variables to be normally distributed. [Hint: For normally distributed variable, most of the data are around the middle value and frequency of observations keeps falling as you move away from the middle in both directions].

a.Retirement age of the U.S workers.

b.Household income in the U.S. in 2019.

c.Height of U.S. adults in 2019.

d.Birthweight of boys in the U.S. in 2019.

e.Please state an example each of left-skewed and right-skewed variables and justify your answer. [Note: your examples should not be from the in-class discussion.]

f.Please calculate the Pearson’s coefficient of Skewness for the data on weekly  returns you analyzed in homework 1 for Amazon.com, Apple, and Google. Based on your results how would you characterize (symmetric, left-skewed or right-skewed) the data on monthly returns of the three companies?

2.Selam is a gambler and regularly plays several rounds of a gamble in which she wins \$6,000 if odd number of dots face up when a fair die is rolled twice and loses \$2,000 for any other outcome from the two rolls of the die. In other words, the outcome of the gamble is determined by rolling a die twice on each round of the gamble and Selam wins only if odd number of dots show up on top in both rolls of the die and loses if any other outcome occurs. Selam is considering playing 20 rounds of such a gamble next week hoping that she will win back the \$2,000 she lost in a similar gamble last week. If we let X represent the number of wins for Selam out of the next 20 rounds of the gamble, X will have the binomial probability distribution.

a. Please calculate the probability of success for Selam on each round of the gamble. Show how you arrived at your answer. [Hint: See the joint probability rule for independent events on slide 14 of PPT_SLIDES_SECOND_WEEK)

## Calculating Pearson's coefficient of skewness

b. What is the probability that Selam will lose all of the 20 rounds of the gamble?

c. What is the probability that Selam will win not more than half of the 20 rounds of the gamble? Show your work.

d. What is the probability that Selam will win fewer 5 out of the 20 rounds of the gamble? Show your work.

e. What is the probability that Selam will lose at least 12 out of the 20 rounds of the gamble? Show your work.

f. What is the probability that Selam will lose more than 14 out of the 20 rounds of the gamble?    Show your work.

g. How much money is Selam expected to win in the 20 rounds of the gamble?  How much money is she expected to lose? Given your results, do you think Selam is playing a smart gamble? Please show how you arrived at your results and explain your final answer.

h. Calculate and interpret the standard deviation for the number of wins for Selam in the next 20 rounds of the gamble. Show your work.

3.Vehicles have been arriving at a toll bridge at an average rate of 156 an hour. Only one toll booth is currently open and can process arrivals (collect tolls) at a mean rate of 25 seconds per vehicle.

a.How many vehicles should be expected at the toll bridge in a 20-minute period? Please show how you arrived at your answer.

b.Define X to be the number of vehicles arriving at the toll bridge in a 20-minute period and assume that X has a Poisson Probability distribution. Please calculate the probability that exactly 40 vehicles will arrive at the toll bridge in a 20-minute period.

c.Please calculate the probability of at least 50 vehicles arriving at the bridge within 20-minute period.

d.Please calculate the probability of less than 40 vehicles arriving at the bridge within 20-minute period.

e.Please calculate the probability of more than 60 vehicles arriving at the bridge within a 20-minute period.

f.Please calculate the probability of at most 45 vehicles arriving at the bridge within a 20-minute period.  Show your work.

g.Please calculate the probability of no vehicle arriving at the bridge within a 20-minute period.  Show your work.

h.Calculate and interpret the standard deviation of X.

i.For what purpose can the kind of information you have calculated above be used? If the state department of transportation wants to reduce the average wait time for the drivers to less than 26 seconds at the toll bridge, do you think they should open anther toll booth at the bridge? Please carefully justify your answer.

4.  Winta Frontier Software Solutions, Inc. (WFSI) has been investigating the possibility of developing a simulation software that can be used for analyzing risk and return from alternative lines of investment. The company is currently trying to decide between four proposed versions of the software: Simple (SS), Moderately Sophisticated (MSP), Sophisticated (SP), and Highly Sophisticated (HSP) versions of the software at respective development costs of \$100,000, \$200,000, \$250,000, and \$400,000.

• If the performance of the prototype developed by WFSI is better than existing software (a success), WFSI believes that it could sell the rights to its simulation software to a larger software developer for \$300,000, \$500,000, \$750,000, and \$1000,000 for the SS, MSP,SP, and HSP versions, respectively.

• If the performance of the prototype developed by WFSI does not exceed the performance of the existing software (a failure), WFSI believes that it could still sell the rights to its simulation software to a larger software developer for \$100,000, \$150,000, \$180,000, and \$200,000 for the SS, MSP,SP, and HSP versions, respectively.

• WFSI estimates that the probability of a success for the software is 45%. However, the company also has the opportunity to hire a consultant who specializes in the study of markets for software. The consultant uses a market survey information to predict success or failure for software. In the past the consultant predicted a success for a prototype software in 84% of cases where the eventual product was a success. However, the consultant also predicted a success for the prototype in 25% of cases where the eventual product was a failure. The consultant’s fee for the service is \$20,000.

a.Please summarize the estimated net profit from each version of the proposed software under each state of nature using the following table. Assume that there are no anticipated costs or revenues associated with the software except those reported above.

b. What decision would maximize expected profit for WFSI? Show the necessary steps.

c.What is the maximum amount WFSI should be willing to pay for a perfect forecast of success or failure for their software? Show the necessary steps.

d.   What is the maximum amount WFSI should be willing to pay for the consultant’s market survey information (EVSI)? On the basis of your findings, should the company hire the consultant? Show all the necessary steps.

e.   Please calculate efficiency of the consultant’s survey and briefly state what it means.