Question 1:
The diagram below represents the distances (to the nearest km) covered by a sample of company vehicles during a given week.
1.1 For the data represented by the above diagram, estimate each of the following quantities:
1.1.1 mean.
1.1.2 middle quartile.
1.1.3 mode.
1.2 Using an appropriate scale, construct a less-than-ogive for the data represented by the given graph.
Use the ogive to estimate the mid 75% range. Show the relevant details on your ogive.
QUESTION 2:
2.1
Modern hotels and certain establishments make use of an electronic door lock system. To open a door an electronic card is inserted into a slot. A green light indicates that the door can be opened; a yellow light indicates that the door is locked from the inside. Suppose that 90% of the time when the card is inserted, the door should open and a green light will appear with probability 0.98. When the door should not open a green light may still appear (electronic error) 5% of the time. Suppose a card has just been inserted and the light is green.
What is the probability that the door will actually open. Supply a probability tree with your solution.
2.2
A retail company predicted that its turnover for next year will be $4500000 and that the costs will be $4000000. However, there are a number of factors that affect both the turnover and costs. Assuming that both turnover and costs have normal distributions, each with a standard deviation of $250000, what is the probability that the company makes a profit?
2.3
An industrial chemical is packaged and sold in 20-kg plastic bags. To minimize the number of underweight sacks, the filling process has been adjusted so that the mean is 21 kg. The weight of the sacks varies because of the nature of the product. Assume that the weight has a normal distribution with a standard deviation of 2.5 kg.
2.3.1 Determine the probability of an underweight sack.
2.3.2 If the sacks can be bought in batch quantities of 30, calculate the probability that the average weight per sack in a batch, is less than 20 kg.
QUESTION 3:
A production process uses four machines (A, B, C, D) in its three-shift operation. A random sample of breakdowns was classified according to machine and the shift in which the breakdown occurred. The details are presented in the accompanying table.
|
Shift
|
A
|
B
|
C
|
D
|
|
1
|
10
|
11
|
8
|
9
|
|
2
|
16
|
9
|
13
|
11
|
|
3
|
12
|
9
|
14
|
9
|
Establish fully, at the 5% level of significance, whether or not shift and machine breakdown are independent.
QUESTION 4:
Sales from a leading motor dealership were monitored over the past four years. The sales figures are presented in the accompanying table.
|
Year
|
Quarter 1
|
Quarter 2
|
Quarter 3
|
Quarter 4
|
|
1
|
20
|
30
|
39
|
60
|
|
2
|
40
|
51
|
62
|
81
|
|
3
|
50
|
64
|
74
|
95
|
|
4
|
55
|
68
|
77
|
96
|
Using the ratio-to-moving average method and linear regression analysis, obtain seasonally-adjusted trend estimates of the quarterly sales for year 5. Show the relevant details of your calculations.
QUESTION 5:
A government committee is considering the economic benefits of a programme of preventative flu vaccinations. If vaccinations are not introduced then the estimated cost to the government if flu strikes in the next year is $7m with probability 0.1, $10m with probability 0.3 and $15m with probability 0.6. It is estimated that such a programme will cost $7m and that the probability of flu striking in the next year is 0.75.
One alternative open to the committee is to institute an "early-warning" monitoring scheme (costing $3m) which will enable it to detect an outbreak of flu early and hence institute a rush vaccination programme (costing $10m because of the need to vaccinate quickly before the outbreak spreads).
With the aid of a decision tree, determine, quantitatively, the optimal course of action the committee should take.