Hi Hellblender,
Get familiar with concepts and skills that are often
assessed for all your subjects. For mathematics, be sure that you
know these:
http://wenshih.files.wordpress.com/2010/01/pure-mathematics-checklist-2010.pdf
Below is the set of skills and concepts for probability
and statistics I have posted in another thread, with some revisions
added here:
Topic 1: Permutations and combinations
- Application of multiplication, addition and mutual exclusion
principles.
- Solving problems involving arrangements related to lines, circles
and beads.
- Solving problems involving restrictions and repetitions.
- Problems that have appeared so far include: arrangement of people
in a line and circle, selection of people to form groups,
arrangement of letters, arrangement of digits (with probability
assessed at the same time).
Topic 2: Probability
- Application of multiplication, addition and mutual exclusion
principles.
- Appreciation of mutually exclusive and independent events.
- Solving problems involving conditional probabilities.
- Solving problems involving the use of tables of outcomes (e.g.
sum of two die scores), Venn diagrams and tree diagrams.
- Solving problems involving the use of an infinite geometric
progression.
For example: A and B play a game in which they each throw a die in
turn until someone throws a six. Find the probability that A wins
if he starts the game.
- Application of permutations and combinations in finding
probabilities.
Topic 3: Binomial and Poisson distributions
- Finding Binomial and Poisson probabilities.
- Solving problems involving conditional probabilities.
- Commenting on the appropriateness of using Binomial/Poisson
distribution for a random variable.
- Finding unknowns (n, p for Binomial or mean for Poisson) based on
given probabilities.
- Finding probabilities involving the sum of Poisson
variables.
- Solving problems involving a combination of Binomial and Poisson
distributions.
- Using the Poisson distribution to approximate the Binomial
distribution.
Topic 4: Normal distribution
- Finding Normal probabilities.
- Solving problems involving conditional probabilities.
- Commenting on the appropriateness of using Normal distribution
for a random variable.
- Finding unknowns (mean, standard deviation or variance) based on
given probabilities.
- Finding unknowns like the value a that satisfies P(X > a) =
given probability.
- Solving problems involving the linear combination of independent
Normal variables, i.e. aX + bY.
- Appreciating the difference between nX and X_1 + X_2 + ... +
X_n.
- Using the Normal distribution to approximate the Binomial
distribution with continuity correction.
- Using the Normal distribution to approximate the Poisson
distribution with continuity correction.
Topic 5: Sampling
- Commenting on the appropriateness of sampling methods (random,
systematic, stratified, quota) for given contexts.
- Solving problems involving the sampling distribution from a
normal distribution.
- Application of the Central Limit Theorem.
Topic 6: Hypothesis testing
- Finding of unbiased estimates of the population mean and
variance from a sample.
- Appreciation of the meaning of p-value and level of
significance.
- Solving problems involving the t-test and z-test.
- Obtaining inequalities involving the level of significance,
mu-zero (value of the population mean), sample size or population
mean.
Topic 7: Regression and correlation
- Sketching a scatter diagram and using it to comment on the
relationship between variables or to identify a data pair which
should be regarded as suspect.
- Appreciation of the independent and dependent variables.
- Appreciation of the cases where the product moment correlation
coefficient (r) is negative, positive, zero.
- Finding the equations of regression lines and r from the given
data.
- Finding unknowns, e.g. pair of x, y values, given the data.
- Using equations of regression lines to estimate values and
commenting on their reliability (based on
interpolation/extrapolation or suitability of the linear
model).
- Application of a square, reciprocal or logarithmic transformation
to achieve linearity for non-linear relationships between
variables.
- Effect on r when there is a change of units in a variable or a
translation of values of a variable.
When you practise, be smart by being very selective of
what you want to reinforce of your concepts and skills, since time
is limited and you have to study other subjects too. For example:
if you are practising for Maths you do not need to spend three
hours to do a full paper, instead you may select questions that
cover all five core areas of pure maths (functions and graphs,
sequences and series, calculus, vectors and complex numbers) with
marks adding up to 50 and then spend 1.5 hours to complete them.
Take SAJC 2010 prelim paper 1 for instance, attempt these
questions:
Q5 (which covers functions) [9]
Q8 (which covers AP/GP) [10]
Q11 (which covers integration techniques) [13]
Q7 (which covers plane problems) [11]
Q4 (which covers, for the most part, De Moivre's theorem)
[8]
which add up to 51 marks, just slightly over 1.5
hours.
This strategy can easily be applied to other school
papers, based on my experience.
Jiayou, the best is yet to be!
Cheers,
Wen Shih