QUESTION THREE

(a) A bag contains 10 white, 15 red and 8 green balls. A single draw of 3 balls is made.

i. What is the probability that a white, a red and green balls are drawn?

ii. What would be the probability of getting all the three white balls?

(b) A and B are two events, not mutually exclusive, connected with a random experiment E

. IfP(A) =p,P(B) = 2pand P( ) =P², and,p (= 0.65 find the following:

i. The value ofp

ii.P()

iii.P()

(c) Three persons A, B andCare being considered for the appointment as Vice Chancellor of Zambian Open University whose chances of being selected for the post are in the proportion 4 : 2 : 3 respectively. The probability that A , if selected will introduce democratization in the university structure is 0.3 ,the corresponding probabilities for B andCdoing the same are 0.5 and 0.8 respectively.

i.What is the probability that democratization is introduced in the university?

ii. The democratization is introduced in the university; what is the probability that it was introduced by

(a) A

(b) B

(c)C

QUESTION FOUR

(a) Suppose a company hires both MBA's and non - MBA's for the same kind of managerial task. After a period of employment some of each category are promoted and some are not.

The table below gives the proportion of the company's managers among the said classes:

MBANon - MBA

Promoted0.420.18

Not promoted0.280.12

If an employee is chosen at random, what is the probability that an employee is

i. MBA graduate?

ii. Promoted?

iii. MBA graduate given that is promoted?

iv. Promoted given that is MBA graduate?

v. Find out whether MBA qualification and promotion are independent events.

Explain your answer.

(b) The research unit in an organization wishes to determine whether scores on scholastic aptitude test are different for male and female applicants. Random samples of applicants file are taken and summarized below:

Applicants

FemaleMale

502.1510.5

S86.290.4

n399204

Using the above sample data, test the null hypothesis that the average score is the same for the population male and female applicants. Use 5% significance level and assume that the scores are normally distributed in each case.

(c) A manufacturer claimed that at least 95% of the equipment which he supplied to a factory conformed to specifications. An examination of a sample of 200 pieces of equipment revealed that 18 were faulty. Test his claim at a 1% significance level.

QUESTION FIVE

(a) The monthly demand for transistors is known to have the following probability

distribution:

Demand123456

Probability0.100.150.200.250.180.12

i. Determine the expected demand for transistors

ii. Determine the variance demand for transistors

iii. Suppose that the cost (C) of producing transistors is given by the equation,

C= 10, 000 + 500x. Determine the variance cost.

(b) It is known from the past experience that 80% of the students in a school do their homework. Find the probability that during a random check of ten students:

i. all have done their homework

ii. at most two have not done their homework

iii. at least one has not done the homework

iv. Find the expected number of students who do their homework.

v. Find the standard deviation of students who do their homework.

(c) In a main frame computer centre, execution time of programs follows an exponential distribution. The average execution time of the programs is 5 minutes. Find the probability that the execution time of programs is less than 4 minutes.

QUESTION SIX

(a) A certain firm uses a large fleet of delivery vehicles. Their records over a long period of time (during which their fleet size utilization may be assumed to have remained suitably constant) show that the average number of vehicles per day is 3. Estimate the probability on a given day when

i.all their vehicles will be serviceable

ii.ii. more than 2 vehicles will be unserviceable

iii.iii. exactly 4 vehicles will be unserviceable

(b) A population consists of the numbers 1, 3, 5, 7 and 9.

i. List all possible samples of size two which can be drawn from the population without replacement.

ii. Show that the mean of the sampling distribution of the sample means is equal to

the population mean.

iii. Calculate the variance of the sampling distribution of the sample mean and show that it is less than the population variance.

(c) Suppose that is a discrete random variable with probability mass function

(P x) =cx2 ,x= 1, 2,3, 4 .

i. Find the value ofc

ii. Find ()

iii. Find E[X (X-1)]

The proposed nine microgrids will be built in 2019 and satisfy the need for increased coastal resilience if the project passes the NPV test in a Benefit Cost Analysis. The microgrids have a construction cost of $18 million, 70% of which will be spent in year 0 and 30% in the next year. The project will cost the state $ 250,000 to maintain and operate each year starting with year 1. The benefits of the project are estimated at $3 million each year starting with year 1. The microgrids are losing efficiency with increased use and that makes the benefits decline at a constant rate of 3% each year. However, the rate of decline can be higher (5.5%) or lower (1.5%) depending on the assumption of the number of extreme weather events that will take place during the life time of the microgrids. The system is expected to last 15 years before major components need to be replaced. State regulations require the use of a nominal interest rate of 7%. The average inflation rate has been estimated at 2.5% but it may be low as 1% or as high as 3%. The state expects to incur a costof disposing safely of the microgrids' components in year 15. Many of these components contain elementsthat are hazardous to human health and the environment. The safe disposal cost was estimated at $300,000 but this estimate comes from a range of $150,000 and $1 million.

a. (30pts) Compute NPV1for the nine-microgrid project and make a recommendation to the governor of CT.

b. (10pts) Conduct a Best and Worst Case Sensitivity Analysis. How, if at all, will NPV and recommendation change due to this uncertainty? (Computations needed)

c. (2pts) Find the IRR of this project.

d. (18pts)The NPV estimated above was based on the assumption that the annual benefits are certain.

However, consider that the benefits of the microgrids are based on whether or not an extreme weather event occurs and the state uses the microgrids as a way to deal with power outages in the affected areas. The state does not trust this annual benefit estimate and wants to perform an analysis and find the NPV based on an expected value of annual benefits. The state already estimated that when there is an extreme whether event such as a Category 4 super storm, the benefits from the microgrids will be $2million annually, for a Category 3 storm, benefits will be $1.2 million annually, and for a Category 2 or less storm the benefits will be of $500,000 each year. Probabilities of the three event categories are presented below. Further, assume that there is no rate of decline in benefits but all other parameters and assumptions stay the same as described above. How does the NPV change?

Storm event Probability Category 4 0.3 Category 3 0.5 Category 2 or less 0.2

show how calculations were made

A major snack company produces variety of cakes and cookies that are sold in

various grocery stores and online stores. One of the famous products of this

company is "dark chocolate delight", which is a snack cake. The cakes are

individually packed and sold in boxes of dozens. The machinery used to produce this

product is 20 years old. Over the time, the variation in weight of cakes produced

gradually increased. The weight of individual cake listed as 50 grams and the actual

weight follows normal distribution with mean of 56 grams and standard deviation

of 2 grams. The production process is set at 56 grams to limit the percentage of

cakes that are underweight. As we know increased variation will result in increased

waste. The ingredients cost about $ 0.0032 per gram of cake. The company currently

produces 24,000 "dark chocolate cakes" (that is 2000 boxes of dozen cakes) each

day and makes a profit of $ 0.50 per box sold.

The company is considering replacing existing machinery with the new equipment,

which will increase daily production rate by 25%. With the new equipment the

standard deviation of cake weight would be 1.6 grams. Senior marketing leaders

are convinced that the demand for "dark chocolate delight" is very high and selling

more boxes would not be a problem. The cost of new machinery including

installation is $ 500,000.

As team reporting to CFO of the company, which option you would recommend

(staying with old equipment or buying new equipment). Please justify your decision

with statistical analysis.

A) The epidemiologists created this 2x2 table for their study.

Hypertension

Yes No Total

Age

60+ 815 500 1315

40-59 265 420 685

Total 1080 920 2000

Calculate and interpret the crude (overall) odds ratio for the study.

B) Next the researchers investigated the relationship between the rural/urban residence and hypertension.

Hypertension

Residence

Yes No Total

Rural 680 320 1000

Urban 400 600 1000

Total 1080 920 2000

Calculate and interpret the odds ratio for the relationship between residence and hypertension.

C) The 3^rdstep in their investigation was to measure the relationship between age and residence. The epidemiologists created this 2x2 table.

Residence

Rural Urban Total

Age

60+ 840 475 1315

40-59 160 525 685

Total 1000 1000 2000

Calculate and interpret the odds ratio for the relationship between age and residence.