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?

...weight, the angle between the horizontal and the incline is called angle of repose 𝜽, as shown in Figure 3. In the previous experiment, if we are measuring along the y-axis, the formula will be
ΣFy=0, f=Wcosθand if we are measuring along the x-axis, the formula will be
ΣFx=0, f=Wsinθ.
The coefficient of friction is equal to the tangent of the angle of repose.
µ=fN=WsinθWcosθ, µ=tanθ(Equation 3)
In this experiment, we should be able to determine the coefficient of friction (µ) between contact 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, pan, wooden block, platform balance, -5753105545455Figure 4. Materials
00Figure 4. Materials
2750185165036500-575310165036500inclined plane with pulley and weights.
The first part of the experiment is “Determination of the Coefficient of Friction” as shown in Figure 5. The first procedure is to position the wooden plane horizontally then measure the weights of the block and pan using the platform balance. Next is to tie one end of the string to the block’s hook and the other end to the pan...

...
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 be communicated to the offeree before it is being accepted. In addition, the acceptance has to be unqualified, unconditional and made by a positive act. In the case of Beauty and Stylish, a positive act refers to the signing of the contract. All terms of the offer must be accepted without any changes and cannot be subjected to any condition, taking effect only upon fulfillment of that condition. When Beauty and Stylish enter into the agreement, they must intend to bind and bound legally to each other by their agreement. This is the intention to create legal relations between two parties. In the meanwhile, this contract must possess consideration. A contract must therefore be a two-sided affair, with each side providing or promising to provide something of value in exchange for what the other is to provide.
Every contract, whether oral or written, contain terms. The terms of a contract set out the rights and duties of the parties. Terms are the promises and undertakings given by each...

...of other functions. Its uses are:
1. Evaluating definite Integrals, for functions without antiderivative.
2. Understanding asymptotic behaviour, how a function behaves in an important part of its domain.
3. Estimating approximate values, such as sin x and e.
The Taylor Series is also used in power flow analysis of electrical power systems (Newton-Raphson method). It is widely used in calculators to estimate approximate values. Thus, the Taylor Series is used by occupations such as 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
...

...research is often freely available2 .
Another way to ﬁnd data is Wolfram Alpha, which is a curated collection of
good-quality datasets at http://wolframalpha.com. Results from Wolfram
Alpha are subject to copyright restrictions; you might want to check the
terms before you commit yourself.
Google and other search engines can also help you ﬁnd data, but it can be
harder to evaluate the quality of resources on the web.
If it seems like someone has answered your question, look closely to see
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 to use the data, or agree to certain terms
of use. Be persistent!
1.6
Glossary
anecdotal evidence: Evidence, often personal, that is collected casually
rather than by a well-designed study.
population: A group we are interested in studying, often a group of people,
but the term is also used for animals, vegetables and minerals3 .
cross-sectional study: A study that collects data about a population at a
particular point in time.
longitudinal study: A study that follows a...

...Faculty of Actuaries
Institute of Actuaries
EXAMINATION
12 April 2005 (am)
Subject CT3
Probability and Mathematical Statistics
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1.
Enter all the candidate and examination details as requested on the front of your answer
booklet.
2.
You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3.
Mark allocations are shown in brackets.
4.
Attempt all 13 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 electronic calculator.
CT3 A2005
Faculty of Actuaries
Institute of Actuaries
1
Calculate the sample mean and standard deviation of the following claim amounts (£):
534
671
581
620
401
340
980
845
550
690
[3]
2
Suppose A, B and C are events with P ( A) = 1 , P ( B ) = 1 , P(C ) = 1 , P ( A
2
2
3
P( A
C ) = 1 , P( B
6
(a)
(b)
Determine whether or not the events A and B are independent.
Calculate the probability P( A B C ).
C) =...

...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)
(b)
(4 marks) (4 marks)
3.
Write FB116 in the expanded form and convert it to a decimal number.
4.
&& Express 0.8964 in the form
a where a and b are positive integers. b
(4 marks)
5.
Peter deposits $P in a bank at a rate of 2% p.a. compounded monthly. 2 years is $208964, find, correct to the nearest integer, the value of P.
If the expected amount after (4 marks)
6.
(a) (b)
Expand ( x + 5)( x − 3) . Solve the inequality ( x + 5)( x − 3) ≤ x 2 − 2 x + 5 and represent the solution graphically. (6 marks)
7.
Rationalize
6 3+ 2
and express the answers with the simplest surd form.
(5 marks)
8. 9.
Solve 27 2 x −1 = 9 x . In a class, there are 20 boys and n girls. are 71, 62 and 67 respectively. (a) (b) Find the value of n. After mark adjustment, each of the mark of x students is increased by 1. If the new mean mark is 67.25, find the value of x.
(5 marks) In a test, the mean mark of boys, girls and the whole class
(7 marks)...

...
Student Number ....................................... .....
Teacher’s Name ....................................... .....
MORIAH COLLEGE
Year 12
MATHEMATICS PRE-TRIAL
General Mathematics
Date: 28 MARCH 2008
Time Allowed:
Examiner:
2½ hours plus 5 minutes reading time.
D Dembo, R 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 questionpaper
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
22 marks
Attempt Questions 1 – 22
Allow about 30 minutes for this section
Answers are to be marked on the answer sheet
provided.
Section II
Pages 8–14
78 marks
• Attempt Questions 23 – 28
• Allow about 2 hours for this section
• Start each question in a new booklet.
Section I
22 marks
Attempt Questions 1–22
Allow about 30 minutes for this section
Use the multiple-choice answer sheet.
Select the alternative A, B, C or D that best answers the question. Fill in the response oval
completely.
Sample: 2 + 4 =
(A) 2
A
(B) 6
B
(C) 8
C
(D) 9
D
If you think you have made a mistake, put a cross through the incorrect answer and fill in the
new...