Question

A company has developed a design for a high-quality portable printer. The two key components of manufacturing cost are direct labor and parts. During a testing period, the company has developed prototypes and conducted extensive product tests with the new printer. The company's engineers have developed the bivariate probability distribution shown below for the manufacturing costs. Parts cost (in dollars) per printer is represented by the random variable x and direct labor cost (in dollars) per printer is represented by the random variable y. Management would like to use this probability distribution to estimate manufacturing costs.

Parts (x)	Direct Labor (y)			Total
	43	45	48
85	0.2	0.05	0.2	0.45
95	0.25	0.2	0.1	0.55
Total	0.45	0.25	0.3	1.00

(a)

Show the marginal distribution of direct labor cost and compute its expected value (in dollars), variance, and standard deviation (in dollars). (Round your answer for standard deviation to the nearest cent.)

Marginal Distribution of Direct Labor Cost

y	f(y)	yf(y)	y − E(y)	(y − E(y))2	(y − E(y))2f(y)
43
45
48
					Var(y) =
		E(y) =			σy =

)

Show the marginal distribution of parts cost and compute its expected value (in dollars), variance, and standard deviation (in dollars). (Round your answer for standard deviation to the nearest cent.)

Marginal Distribution of Parts Cost

x	f(x)	xf(x)	x − E(x)	(x − E(x))2	(x − E(x))2f(x)
85
95
					Var(x) =
		E(x) = dollars			σx = dollars

(c)

Total manufacturing cost per unit is the sum of direct labor cost and parts cost. Show the probability distribution for total manufacturing cost per unit.

z = x + y	f(z)
128
130
133
138
140
143
Total	1.00

(d)

Compute the expected value (in dollars), variance, and standard deviation (in dollars) of total manufacturing cost per unit. (Round your answer for standard deviation to two decimal places.)

expected value dollars

variance 39

standard deviation dollars

(e)

Are direct labor and parts costs independent? Why or why not?

Since the covariance equals , which ---Select---is not equal to zero, we can conclude that direct labor cost ---Select---is not independent of parts cost.

If you conclude that direct labor and parts costs are not independent, what is the relationship between direct labor and parts cost?

There is a positive correlation between the costs of direct labor and parts.

There is a negative correlation between the costs of direct labor and parts.    

The costs of direct labor and parts are independent.

(f)

The company produced 1,500 printers for its product introduction. The total manufacturing cost was $198,450. Is that about what you would expect?

The expected manufacturing cost for 1,500 printers is $ which is ---Select---lower than higher than equal to $198,450.

If it is higher or lower, what do you think may have caused it? (Select all that apply.)

A supplier increased the cost of one of the more common printer parts this company uses in the manufacturing process.At first there was a steep learning curve, but as more printers were manufactured direct labor costs decreased.There was an increase in the cost of direct labor due to an influx of many new employees.The expected manufacturing cost is equal to $198,450.

Answer 1

a)

y	f(y)	yf(y)	y − E(y)	(y − E(y))2	(y − E(y))2f(y)
43	0.45	19.35	-2	4	1.8
45	0.25	11.25	0	0	0
48	0.3	14.4	3	9	2.7
					Var(y) =
		E(y) =45			σy =4.5

b)

x	f(x)	xf(x)	x − E(x)	(x − E(x))2	(x − E(x))2f(x)
85	0.45	38.25	-5.5	30.25	13.6125
95	0.55	52.25	4.5	20.25	11.1375
					Var(x) =
		E(x) = 90.5			σx = 24.75

c)

z = x + y	f(z)
128	0.2
130	0.05
133	0.2
138	0.25
140	0.2
143	0.1

d)

expected value (in dollars)=135.5

, variance=25.25

, and standard deviation =5.0249

e)

Since the covariance equals -2 which -is not equal to zero, we can conclude that direct labor cost -is not independent of parts cost.

There is a negative correlation between the costs of direct labor and parts.    

f) The expected manufacturing cost for 1,500 printers is $ 203250 which is higher than $198,450.

At first there was a steep learning curve, but as more printers were manufactured direct labor costs decreased.