Exercise 1 (To be returned on Tuesday, October 5)
Question 1
A mechanical jar ﬁller is used to ﬁll jars with coﬀee. The ﬁller is set so that the mean jar ﬁll is 205 grams. The standard deviation of the jar ﬁlls is 2.5 grams. If the population of jar ﬁlls is normally distributed, what percentage of jar ﬁlls will be (i) greater than 202.5 grams, (ii) between 201.25 and 208.75 grams, and (iii) greater than 212.5 grams? If coﬀee jars have a capacity of 200 grams, (iv) what percentage are underﬁlled by the jar ﬁller and, (v) how would you reset the mean jar ﬁll so that only 0.4% are underﬁlled? Question 2

(i) If random variables Y1 , Y2 , · · · , Yn are independent with a common mean µ but 2
2
2
have variances σ1 , σ2 , · · · , σn (which need not be the same) show that the sample mean ¯
Y is an unbiased estimator of µ with standard deviation
1
n

1
2

n
2
σi

.

i=1

(ii) A student takes independent measurements of an instrument but, owing to fatigue, 2
the variance of his measurements increases linearly, i.e., σr = σ 2 + rc where σ 2 > 0 ¯
c > 0 (r = 1, · · · , n). Use the result of (i) to show that the variance of Y is never less 1
¯ is not a consistent estimator of µ in this case.
than 2 c. Hence show that Y
Question 3
Eight international athletes ran a 400 meter race at sea level (London, UK). A month later they raced at an altitude of 5220 feet above sea-level (Denver, USA). athlete
1
2
3
4
5
6
7
8
London UK 47.79 46.64 45.67 44.61 45.20 47.11 46.04 48.22
Denver USA 47.62 45.93 46.20 44.82 44.24 45.42 46.21 47.55
Would you say that the athletes ran quicker at 5000 feet?

...surfaces as one body moves with uniform motion and establish the relationship between the angle of repose (𝜽) and µ. The rules of this experiment are to keep clean the surfaces of the wooden block and plane by wiping them with a piece of scratch paper or tissue to remove dust and other particles and to make sure not to touch the surfaces that you will use in this experiment to avoid contamination. The materials for this experiment as shown in Figure 4 are string, meter stick,...

...
The case between Beauty and Stylish involves concept of a valid contract, pre-contractual statements, express term and misrepresentation.
A valid contract is established between Beauty and Stylish when an offer is accepted and there is intention for both parties to create legal relations. An offer refers to the expression of willingness of the offerer to be contractually bound by an agreement if his or her offer is properly accepted. It has to be clear and certain in terms. It must also...

...Mathematicians & Electrical Engineers. It has limitations that it is only available for a small domain and it is challenging to find the nth term of the derivative.
RESOURCES
http://en.wikibooks.org/wiki/Calculus/Taylor_series
https://www.efunda.com/math/taylor_series/taylor_series.cfm
http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/series/appsTaylor.html
...

...whether the answer is justiﬁed. There might be ﬂaws in the data or the
analysis that make the conclusion unreliable. In that case you could perform
a different analysis of the same data, or look for a better source of data.
If you ﬁnd a published paper that addresses your question, you should be
able to get the raw data. Many authors make their data available on the
web, but for sensitive data you might have to write to the authors, provide
information about how you plan...

...Sing Yin Secondary School First Term Examination, 2009 – 2010 Mathematics 1 Form 3 Full marks: 100 Answer ALL questions. Unless otherwise specified, all working must be clearly shown. The diagrams in this paper are not necessarily drawn to scale. Unless otherwise specified, numerical answers should either be exact or correct to 3 significant figures. 1.
2.
Time allowed: 1.5 hours
Simplify
2 8 + 24 .
x 2 − 5x − 6 , x 3 − 27 .
(
)
(4 marks)
Factorize (a)...

...questions, beginning your answer to each question on a separate sheet.
5.
Candidates should show calculations where this is appropriate.
Graph paper is required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the
Formulae and Tables and your own...

...Lisle
Candidates should remove the formula sheet and answer sheet from the end of the paper.
Write your number and teacher’s name on the answer sheet and this question paper
immediately.
General Instructions
Total marks – 100 marks
• Reading time – 5 minutes
• Working time – 2½ hours
• Write using black or blue pen
• Calculators may be used
• A formulae sheet is provided at the
back of this paper
Section I
•
•
•
Pages 2–7...