ASSIGNMENT: QUANTITATIVE METHODS

QUESTION 1: A large shipping company recorded the tonnage shipped weekly across the pacific for 50 weeks. The data is presented in the table below.

1.1. Using seven classes of equal width, arrange the above data in a less-than cumulative distribution. Let 350 tonnes be the lower limit of the initial class.

1.2. Use the distribution in 1.1 to calculate the lower quartile and the mode.

 1.3. Draw, to scale, the less-than ogive for the distribution obtained in 1.1.

1.4. Use the graph drawn in 1.3 to determine the mid-70% range.

QUESTION 2: The production department of a leading newspaper has embarked on a quality improvement effort and has chosen as its first project an issue that relates to the blackness of the newspaper print. Each day a determination needs to be made concerning how "black" the newspaper print is. The blackness of print is measured on a standard scale in which the target value is 1.0. Data collected over the past year indicate that the print blackness is normally distributed with an average of 1.01 and a standard deviation of 0.12. Each day, one printed spot on the first newspaper printed is chosen randomly and the blackness of the spot is measured. Suppose the blackness of print is considered acceptable if the blackness of the spot is between 0.95 and 1.05.

2.1. Calculate the probability that the blackness of the spot is:

2.1.1 between 0.95 and 1.0?

2.1.2 less than 0.95 or greater than 1.05?

2.2. If the objective of the production team is to reduce the probability that blackness is below 0.95 or above 1.05, would it be better to focus on process improvement that lowered the blackness to the target value of 1.0 or on process improvement that reduced the standard deviation to 0.08? Justify your answer quantitatively.

2.3. Each day, a random sample of 25 printed spots is chosen and the blackness of the spots is measured. Assuming that the distribution of blackness has not changed from what is was previously, what is the probability that the average blackness of the spots is less than 0.95 or greater than 1.05?

QUESTION 3: The president of a company that manufactures air conditioners is considering switching his suppliers. Supplier A, the current producer of condensers, prices its product 5% higher than supplier B. Because the president wants to maintain his company's reputation for quality, he wants to be certain that B's condensers last at least as long as A's condensers. After careful analysis, the president decides to retain supplier A if there is sufficient statistical evidence that A's condensers last longer, on average than B's condensers. In a test, 15 randomly-selected midsize cars were equipped using A's condensers while another 15 randomly-selected midsize cars were equipped with B's condensers. The number of kilometers driven by each car before the condenser broke down is displayed (in thousands) in the table below.

Establish, at 5% significance level, whether the president should retain supplier A.

QUESTION 4: The quarterly sales levels for years 2011 - 2014 of a leading tyre manufacturer are displayed in the accompanying table.

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 2015. Show details of all calculations.

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 $60000. These facilities would, however, provide sufficient capacity for the requirements of contract C2 to be met also should the tender be for both C1 and C2. . In addition, the production costs for C1 would be $20000.

If a tender is made for contract C2 only, then the necessary extra facilities can be provided at a cost of only $30000. The production costs in this case would be $15000. (The production costs for contract C2 would be $12000 if the tender is for both C1 and C2.).

It is estimated that the tender preparation costs would be $3000 if tenders are made for contracts C1 or C2 only and $4000 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 quantitative report advising the company on the best course of action.