Question One:
The following graph represents the salaries (in $1000) paid to the employees of a certain company.
1.1 Extract relevant information from the above graph to estimate each of the following quantities:
1.1.1 Mean salary.
1.1.2 Modal salary.
1.1.3 Coefficient of variation.
1.2 Draw to scale an ogive of the data represented by the above graph and use it to obtain estimates of the 70th percentile and the mid 75% range.
Question Two:
A company supplies specialized, high-tensile pins to a customer. It uses an automatic lathe to produce the pins. Due to factors such as vibration, temperature and wear and tear, the lengths of the pins are normally distributed with a mean of 20.30 mm and a standard deviation of 0.35 mm. The customer will only buy those pins with lengths in the interval 20.00 ± 0.50 mm.
2.1 What percentage of the pins will be acceptable to the customer?
2.2 In order to improve the percentage accepted, management considers adjusting the population mean and standard deviation of the length of the pins. If the lathe can be adjusted to have any desired mean of the lengths, what should it be adjusted to? Why?
2.3 Suppose that the mean (20.30 mm) cannot be adjusted but the standard deviation can be reduced. Calculate the maximum reduction in the standard deviation that would make 80% of the pins acceptable?
2.4 The production manager then considers the costs involved. The cost of resetting the machine to adjust the population mean involves engineering costs and the cost of production time lost. The cost of reducing the population standard deviation involves, in addition to these costs, the cost of overhauling the machine and reengineering the process. Assume it costs $200x2 to decrease the standard deviation by (x/50) mm. Find the cost of reducing the standard deviation to the value found in 2.3.
Question Three:
To better understand its target market, a leading car manufacturer conducted a market survey. A random sample of respondents was chosen, and each was asked to select one of four characteristics (defensive, aggressive, enjoying and prestigious) that best described him or her as a driver. Each respondent was then asked to choose one of three models of its most popular car. The survey results are summarized in the table below.
|
Mod
e
|
Defensiv
e
|
Aggressiv
e
|
Enjoyin
g
|
Prestigiou
s
|
|
A
|
27
|
26
|
39
|
61
|
|
A
|
44
|
50
|
47
|
73
|
|
C
|
82
|
94
|
101
|
85
|
Investigate, at the 5% level of significance, whether a relationship existed between the driver's self-image and choice of model.
Question Four
The quarterly earnings (in $m) of a large soft-drink manufacturer has been recorded for the past four years. The data is listed in the table below:
|
|
Year
1
|
Year
2
|
Year
3
|
Year
4
|
|
1st Quarter
|
72
|
77
|
80
|
86
|
|
2nd Quarter
|
87
|
95
|
97
|
102
|
|
3rd Quarter
|
105
|
100
k
|
104
|
118
|
|
4th Quarter
|
74
|
81
|
83
|
87
|
Assuming a linear trend in the data, and that the cyclical and irregular components of the time series are negligible, use the ratio-to-moving-average method to estimate the quarterly earnings for year S. Show details of all calculations.
Question Five
A leading pharmaceutical company in the United States is planning on developing a new drug. The development will take place in two phases. Phase I will cost $1.5m and Phase II will cost $2.5m. Any new drug will have to be approved by the Federal Drug Administration (FDA) before it can be marketed. If the drug is approved, then a profit contribution of $7m can be realized by marketing the drug. The only fixed costs to be subtracted from this contribution are the development costs. If the drug is not approved, then all of the development cost has to be written off as a loss.
The managers estimate a 75% chance that the FDA will approve the drug. This still leaves a 25% chance of losing the developmental costs. Because of the risk involved, management proposes that a test be commissioned at the end of Phase I to determine the chances of FDA approval. The test itself will cost $500000. If the test result is positive, then the company will continue with Phase II, otherwise, the project will be aborted. The motivation for the test is that, in case the chances of FDA approval are slim, at least Phase II costs can be saved by aborting the project.
The researcher who conducts the test says that the test is not 100% accurate in predicting whether the FDA will approve the drug. He estimates the following probabilities.
P (Test positive I FDA will approve) = 0.85
P (Test negative I FDA will not approve) = 0.75
With aid of a decision tree, determine the optimal decision strategy the company should adopt.
(Hint: First calculate P(FDA will approve I Test positive) and P(FDA will not approve I Test positive) for the case where the test is conducted. For the case where the test is not conducted, use the nonconditional probabilities of FDA approval and FDA nonapproval.)
Instructions
1. Start each solution on a new page.
2. A spreadsheet (eg. Microsoft Excel) may be used only for the purposes of setting up tables, drawing of graphs etc. All solutions must be clearly shown through the use of appropriate formulae and substitution.
3. Solutions generated by software packages will not be marked.
4. No marks will be awarded if only the final answers are given. All relevant working must be shown.
5. The assignment must be typed using 11pt text font Arial or Times New Roman, spacing of VA lines.
6. It is not necessary to provide references or a bibliography. Number each solution according to the numbering in assignment handbook. NOTE: This instruction applies only to Quantitative Methods.