Question

The number of wooden sailboats constructed per month in a small shipyard is a random variable that obeys the probability distribution given in Table below: Probability distribution of monthly Number of Sailboats /Probability( 2 is the sailboat and 0.15 is probability; likewise please consider in this way for the rest of the data) 2, 0.15; 3, 0.20; 4, 0.30; 5, 0.25; 6, 0.05; 7,0.05; Suppose that the sailboat builders have fixed monthly costs of $30,000 and an additional construction cost of $4,800 per boat. (a) Compute the mean and standard deviation of the number of boats constructed each month. (b) What is the mean and standard deviation of the monthly cost of the sailboat construction operation? 4 (c) How do your answers in part (b) change if the fixed monthly cost increases from $30,000 to $53,000? Try to compute your answer using the results of the calculation in part (b) only. (d) How do your answers in part (b) change if the construction cost per boat increases from $4,800 to $7,000, but the fixed monthly cost stays at $30,000? Try to compute your answer using the results of the calculations of parts (a) and (b) only.

Answer 1

a) Mean = μ = Σxp

Standard Deviation(X) = (Σx²p − μ²)^0.5

x	p(x)	x*p(x)	x2p(x)
2	0.15	0.3	0.6
3	0.2	0.6	1.8
4	0.3	1.2	4.8
5	0.25	1.25	6.25
6	0.05	0.3	1.8
7	0.05	0.35	2.45
Total		4	17.7

So mean, μ = 4 .  μ² = 16

S.D. = ( 17.7 - 16 )^0.5

= 1.30

b) Monthly cost = 30000 + 4800x where x is no. of boats

x	p(x)	Monthly cost (m)	m*p(x) .	m2p(x)
2	0.15	39600	5940	235224000
3	0.2	44400	8880	394272000
4	0.3	49200	14760	726192000
5	0.25	54000	13500	729000000
6	0.05	58800	2940	172872000
7	0.05	63600	3180	202248000
Total			49200	2459808000

So mean, μ = 49200 .  μ² = 2420640000

S.D. = ( 2459808000 - 2420640000 )^0.5

= 6258.43

c) So now the new monthly cost will be 53000 + 4800x

so every x will increase by 23000 ( 53000 - 30000) that means mean also increases by 23000

So new mean = 49200 + 23000 = 72200

And since every cost is increased by the same value so Standard deviation will remain the same

So new standard deviation = 6258.43

d) Now the new monthly cost will be 30000 + 7000x

since now only variable cost changing so

New Mean ( monthly cost ) = Mean for b part + Change in variable cost* mean of No. of boats

Mean for b part = 49200 Change in variable cost = 2200 Mean ( boats ) = 4

New Mean = 49200 + 2200*4

= 58000

Similarly for standard deviation

New SD ( monthly cost ) = SD for b part + Change in variable cost* SD (No. of boats)

= 6258.43 + 2200*1.304

= 9127