Question 1  

A large shipping company recorded the number of tons shipped weekly across the pacific for 50 weeks.

The data is presented in the table below.

1.1   Arrange the above data in a less-than cumulative distribution using seven classes of equal width.

1.2   Use the distribution in 1.1 to determine the mean, median and mode.

1.3   Draw to scale, the less-than ogive of the above distribution.

1.4   Use the graph drawn in 1.3 to determine:

         1.4.1  the 70^th percentile.

         1.4.2  the mid 70% range.

Question 2                                      

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 25.30 mm and a standard deviation of 0.45 mm.  The customer will only buy those pins with lengths in the interval 25.00 ± 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 (25.30 mm) cannot be adjusted but the standard deviation can be reduced.

         Calculate the maximum reduction in the standard deviation that would make 85% 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 $150x² to decrease the standard deviation by (x/40) mm.

         Find the cost of reducing the standard deviation to the values found in 2.3.

Question 3                                                                                                            

The president of a company that manufactures air conditioners is considering switching his supplier of condensers.  Supplier A, the current producer of condensers for the manufacturer, prices its product 5% higher than supplier B does.  Because the president wants to maintain his company's reputation for quality, he wants to be sure that supplier B's condensers last at least as long as supplier A's condensers.  After a careful analysis, the president decided to retain supplier A if there is sufficient statistical evidence that supplier A's condensers last longer, on average than supplier B's condensers.  In an experiment, 15 randomly-selected midsize cars were equipped using type-A condensers while another 15 randomly-selected midsize cars were equipped with type-B condensers.  The number of kilometers (in thousands) driven by each car before the condenser broke down is displayed in the table below.

Carry out a hypothesis test, at 5% significance level, to determine whether the president should retain supplier A.

Question 4

The quarterly earnings (in $m) of a large soft-drink manufacturer has been recorded for the years 2008 - 2011.

The data is listed in the table below.

4.1 Plot a time-series graph to represent the above data.  What can you deduce from the graph?

4.2 Use the ratio-to-moving average method to calculate the adjusted seasonal indices for the four quarters and interpret the results.

4.3 Using the zero-sum method, derive the trend line equation representing the above data.

4.4 Obtain seasonally-adjusted trend estimates for the third and fourth quarters of 2012.

Question 5 

A company is considering whether it should tender for two contracts (C1 and C2) on offer from a government department for the supply of certain components. If tenders are submitted, the company will have to provide extra facilities, the cost of which will have to be entirely recouped from the contract revenue. The risk, of course, is that if the tenders are unsuccessful then the company will have to write off the cost of these facilities.

The extra facilities necessary to meet the requirements of contract C1 would cost $50,000. These facilities would, however, provide sufficient capacity for the requirements of contract C2 to be met also. In addition the production costs would be $18,000. The corresponding production costs for contract C2 would be $10,000.

If a tender is made for contract C2 only, then the necessary extra facilities can be provided at a cost of only $24,000. The production costs in this case would be $12,000.

It is estimated that the tender preparation costs would be $2,000 if tenders are made for contracts C1 or C2 only and $3,000 if a tender is made for both contracts C1 and C2.

For each contract, possible tender prices have been determined. In addition, subjective assessments have been made of the probability of getting the contract with a particular tender price as shown below. Note here that the company can only submit one tender and cannot, for example, submit two tenders (at different prices) for the same contract.

In the event that the company tenders for both C1 and C2 it will either win both contracts (at the price shown above) or no contract at all. With the aid of a decision tree prepare a report advising the company on the best course of action.