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NZ-SIMSS
2006 Parallel NCEA Scholarship Statistics and
Modelling Exam
All questions are based on biscuits, produced by a company named NZQ Foods. This
company produces two types of biscuit, X and Y, in 250 g packets.
Each type of biscuit is packaged in boxes containing six packets.
(1) (a) The number of boxes of each type produced per day needs to satisfy the following
constraints:
1. The total number of boxes produced is not to exceed 500, and there must be at least
200 boxes of Y produced.
2. The number of boxes of Y produced is to be at least one and a half times the number
of boxes of X produced.
3. No more than ten boxes of Y are to be produced for every two boxes of X produced.
A box of X produces a higher profit than a box of Y .
Find the number of boxes of each type that should be produced per day in order to make
the largest profit.
(b) Constraints 2 and 3 in (a) are changed so that the number of boxes of X produced is
between 30% and 45.0% of the number of boxes of Y produced. Assume that constraint 1
in (a) still applies and that each box of X now produces less profit than a box of Y.
Find the number of boxes of each type that should be produced now to produce the
largest profit.
(c) A machine seals the packets and another machine makes the boxes. On average, one
in every 200 packets has a faulty lid and one in every 100 boxes is faulty.
For a day’s production of 500 boxes, find the expected number of boxes that would be
faulty or would have a packet with a faulty lid.
State any assumptions made.
(2) (a) The 250g packets of Y are filled during production. Weights (in g) of packets are
modelled by a normal distribution with a standard deviation of 10g.
(i) Calculate the mean weight so that only 4 % of the packets have a fill volume less than
the labelled weight of 250g..
(ii) A random sample of 30 packets yielded a mean fill weight of 230g..
Construct a 90% confidence interval for the mean fill volume. With reference to your
answer to (i), what do you conclude? Justify your answer.
(iii) How many packets are needed in a sample so that the sample mean weight is within
1g of the population mean, with 95% confidence?
(b) Two filling machines, A and B, are used independently to fill packets of X. It is
suspected that the two machines are giving different mean weights.
How you could check this statistically.
© Rory Barrett 2008
(3) All incoming ingredients for making the biscuits are checked for defects. An
automatic scanner is used twice. The scanner removes 95% of defects on the first scan
and removes 90% of any remaining defects on the second scan. The company knows that
5% of incoming batches of one of the ingredients contains defects.
(a) Calculate the percentage of batches containing defects that are removed by the second
scan.
(b) Calculate the probability that a batch that was found to have defects was found on the
first scan
(c) A random sample of batches is taken before scanning.
What is the largest sample size in order to have a 40% chance or less of finding at least
two batches containing defects in the sample?
(d) A random sample of 20 batches had two with defects. Based on this result, is it likely
that the percentage of defective incoming strawberries is now not 5%? Justify your
answer.
(4) The quality of biscuits is scored on a scale of 0 (very poor) to 100 (excellent).
Testing has shown that, after a particular period, the quality Q of the biscuits starts to
decrease, and that it may be modelled by the function:
Q(t) = 50 for 0
t<D
Q(t) = A(t-D+1)B for t
D
with Q(D + 10) = 40 where t is the number of days after baking of the biscuits.
Q(t) is the quality score at time t, D is the number of days from manufacture before the
quality Q(t) starts to decrease, and A and B are constants.
(a) Suppose that D = 13.
(i) Find the values of A and B.
(ii) Sketch a graph of Q(t), for 0
t
30.
(iii) A use-by date is calculated by finding out when the model indicates when the quality
score will drop below 37.5. Find the use-by date for biscuits produced on 1st December
2008.
(b) Now suppose that D is normally distributed with a mean of 13.5 and a standard
deviation of 2.1.
When t = 30, what percentage of biscuits would have a quality score below 37.5?
© Rory Barrett 2008
The data in the table below shows the monthly values of exports of this product to the
United States for the period January 2006 till December 2008. It gives the centred
moving means of order 12 for the period July 2006 till May 2008. The monthly transport
costs are also shown.
Month
Jan-06
Feb-06
Mar-06
Apr-06
May-06
Jun-06
Jul-06
Aug-06
Sep-06
Oct-06
Nov-06
Dec-06
Jan-07
Feb-07
Mar-07
Apr-07
May-07
Jun-07
Jul-07
Aug-07
Sep-07
Oct-07
Nov-07
Dec-07
Jan-08
Feb-08
Mar-08
Apr-08
May-08
Jun-08
Jul-08
Aug-08
Sep-08
Oct-08
Nov-08
Dec-08
Months
since
Dec 05
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
Exports
in $000
5.0
6.0
12.0
15.0
26.0
72.0
184.0
156.0
19.0
17.0
8.0
6.0
5.0
6.0
7.0
19.0
31.0
79.0
209.0
207.0
22.0
16.0
7.0
7.0
4.0
4.0
4.0
8.0
8.0
30.0
52.0
31.0
6.0
3.0
2.0
1.0
CMM
$(000)
43.8
43.8
43.6
43.6
44.0
44.5
45.8
49.0
51.2
51.3
51.2
51.2
51.2
51.1
50.9
50.3
48.9
45.9
37.3
23.4
15.4
14.2
13.5
Transport
Costs
$(000)
1.0
1.0
1.0
1.0
2.0
1.5
2.0
2.1
3.5
3.8
3.7
4.0
4.0
4.0
4.1
4.2
4.4
4.4
4.4
4.3
4.3
4.3
5.0
5.2
5.5
6.0
6.3
6.4
6.2
5.1
6.3
6.5
6.7
6.5
6.5
6.7
A graph showing the raw sales and the centred moving mean is also shown as well as a
graph of the centred moving mean and a trend line.
© Rory Barrett 2008
250.0
200.0
150.0
100.0
50.0
0.0
0
5
10
0
5
10
15
20
25
30
35
40
80.0
60.0
40.0
20.0
0.0
15
20
25
30
35
40
-20.0
-40.0
-60.0
-80.0
-100.0
y = -0.0155x 3 + 0.6112x 2 - 6.7459x + 65.825
R2 = 0.9316
-120.0
(5)By referring to the data in the table, the two graphs, and the fact that there was a
collapse of world financial markets in late 2008 write a report of no more than two pages
discussing sales of the company’s product. As part of your report, produce a prediction
and comment on the validity of your forecasts. Suggest further action.
© Rory Barrett 2008
(6)
Total Sales($000)
T
1000
800
700
Year
2006
2007
2008
Price per packet
P
$2.50
$3.00
$3.50
Number of shops
N
25
30
35
(a) The table shows the total yearly sales T ($000) of the product in Auckland , the price
per packet P ($), and the number of shops N selling it..
(i) The equation T = a + bP +cN is believed to represent this sales information.
Form equations from the data given in the table, attempt to solve them and comment.
Further data is obtained including the amount spent on marketing. The following table
shows this.
Amount
Total
Price per
Number of
spent on
Year
Sales($000)
packet
shops
marketing
T
P
N
($000)
M
2006
1000
$2.50
25
5
2007
800
$3.00
30
3
2008
1500
$3.50
38
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(ii) A new model is proposed .namely T = aM + bP +cN. Again form equations, solve and
comment.
(b) As a result of rising costs in transport and falling sales to the USA the company will
abandon sales to the USA if transport costs exceed 25% of the average monthly value of
exports to that country. When is this likely to occur ? The graph of the transport costs
below may be helpful. The trend line previously given should be used.
80
60
40
20
0
-20
0
10
20
30
-40
-60
-80
-100
-120
y = -0.0155x 3 + 0.6087x 2 - 6.7007x + 65.583
y = 0.1662x + 1.201
© Rory Barrett 2008
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