A company has developed a design for a high-quality portableprinter. The two key components of manufacturing cost are directlabor and parts. During a testing period, the company has developedprototypes and conducted extensive product tests with the newprinter. The company's engineers have developed the bivariateprobability distribution shown below for the manufacturing costs.Parts cost (in dollars) per printer is represented by the randomvariable x and direct labor cost (in dollars) per printeris represented by the random variable y. Management wouldlike to use this probability distribution to estimate manufacturingcosts.
| Parts(x) | Direct Labor(y) | Total |
|---|
| 43 | 45 | 48 |
|---|
| 85 | 0.2 | 0.05 | 0.2 | 0.45 |
| 95 | 0.25 | 0.1 | 0.2 | 0.55 |
| Total | 0.45 | 0.15 | 0.4 | 1.00 |
(a)
Show the marginal distribution of direct labor cost and computeits expected value (in dollars), variance, and standard deviation(in dollars). (Round your answer for standard deviation to thenearest 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)= dollars | | | ?y= dollars |
(b)
Show the marginal distribution of parts cost and compute itsexpected value (in dollars), variance, and standard deviation (indollars). (Round your answer for standard deviation to the nearestcent.)
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 | | | ?x= dollars |
(c)
Total manufacturing cost per unit is the sum of direct laborcost and parts cost. Show the probability distribution for totalmanufacturing 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 standarddeviation (in dollars) of total manufacturing cost per unit. (Roundyour answer for standard deviation to two decimal places.)
expected value dollarsvariancestandard deviation dollars
(e)
Are direct labor and parts costs independent? Why or whynot?
Since the covariance equals ,which ---Select--- is is not equal to zero, we canconclude that direct labor cost ---Select--- is is notindependent of parts cost.
If you conclude that direct labor and parts costs are notindependent, what is the relationship between direct labor andparts cost?
There is a positive correlation between the costs of directlabor and parts.There is a negative correlation between the costsof direct labor and parts. The costs ofdirect labor and parts are independent.
(f)
The company produced 1,700 printers for its productintroduction. The total manufacturing cost was $198,450. Is thatabout what you would expect?
The expected manufacturing cost for 1,700 printers is$ which is ---Select--- lower than higherthan 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 printerparts this company uses in the manufacturing process.At first therewas a steep learning curve, but as more printers were manufactureddirect labor costs decreased.There was an increase in the cost ofdirect labor due to an influx of many new employees.The expectedmanufacturing cost is equal to $198,450.
