Question

9. A contractor estimates the probabilities for the number of days required to complete a certain type of construction project as follows: Table 1: Distribution of days of completion Time (Days) 1 2 3 4 5 Probability 0.05 0.20 0.35 0.30 0.10 (a) What is the probability a randomly chosen project will take less than 3 days to complete? (b) Find the expected time to complete. (c) Find the variance of time required to complete a project. (d) The contractor’s cost is made up of two parts – a fixed cost of £10,000 plus £1,000 for each day taken to complete the project. Find the mean and standard deviation of total project cost. (e) If 3 projects are undertaken, what is the probability that at least 2 of them will take at least 4 days to complete, assuming independence of individual project completion times.

Answer 1

(a) What is the probability a randomly chosen project will take less than 3 days to complete?

Values ( X )	Frequency(f)	∑ fx	( X^2)	∑ f x^2
1	0.05	0.1	1	0.05
2	0.2	0.4	4	0.8
3	0.35	1.1	9	3.15
4	0.3	1.2	16	4.8
5	0.1	0.5	25	2.5

P(X<3 ) = P(x=0)+P(x=1)+P(x=2) = 0 + 0.05 + 0.2 = 0.25

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(b) Find the expected time to complete.

∑ f= 1

∑ fx = 3.2

expected time = Mean = ∑ fx / ∑ f = 3.2

Mean square =∑ f x^2 / ∑ f = 11.3

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(c) Find the variance of time required to complete a project.

Variance = (Mean square) - (Mean)^2

Variance = ∑ f x^2 - Mean^2 = 1.06

Stadard Dev= √ Var = 1.03

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(d) The contractor’s cost is made up of two parts – a fixed cost of £10,000 plus £1,000 for each day taken to complete the project. Find the mean and standard deviation of total project cost.

when considering a fixed cost of £10,000 plus £1,000 for each day taken to complete the project,
the numbers are,

Values ( X )	EXPECTED COST
1	11000
2	12000
3	13000
4	14000
5	15000

Mean = Sum of observations/ Count of observations
Mean = (11000 + 12000 + 13000 + 14000 + 15000 / 5) = 13000
Variance
Step 1: Add them up
11000 + 12000 + 13000 + 14000 + 15000 = 65000
Step 2: Square your answer
65000*65000 =4225000000
…and divide by the number of items. We have 5 items , 4225000000/5 = 845000000
Set this number aside for a moment.
Step 3: Take your set of original numbers from Step 1, and square them individually this time
11000^2 + 12000^2 + 13000^2 + 14000^2 + 15000^2 = 855000000
Step 4: Subtract the amount in Step 2 from the amount in Step 3
855000000 - 845000000 = 10000000
Step 5: Subtract 1 from the number of items in your data set, 5 - 1 = 4
Step 6: Divide the number in Step 4 by the number in Step 5. This gives you the variance
10000000 / 4 = 2500000
Step 7: Take the square root of your answer from Step 6. This gives you the standard deviation
1581.1388

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P( atleast 4 days to complete the project) = P(X=4) + P(X=5) = 0.30 + 0.10 = 0.40

when 3 projeccts are under taken, it is considered to be a binomial distribution X ~ B ( 2, 0.40)

P( X < 2) = P(X=1) + P(X=0)   
= ( 3 1 ) * 0.4^1 * ( 1- 0.4 ) ^2 + ( 3 0 ) * 0.4^0 * ( 1- 0.4 ) ^3   
= 0.648
P( X > = 2 ) = 1 - P( X < 2) = 0.352