STATISTICS 2 SCHEME OF WORK 2005-2006

S2 UNIT ≈ 35 lessons. The course has been allocated 30 hours, leaving 5 hours for assessment, revision and lessons missed. (We may find that we have a lot longer than this – this year is a trial year with this module and timings will be adjusted accordingly for next).

The textbook used is Heinemann T1 Old Edition. For “Hypothesis Testing” the references are to Heinemann S2 New edition (a departmental copy of this will be purchased during the year). Other sources include Crawshaw and Chambers (if unstated, assume this is the source) and also worksheets of past examination questions for certain topics, where starred (see MC for a copy of these materials).

Homework of approximately 1 hour duration is to be given after each lesson, including revision, review exercise and past paper questions whenever possible.

The external assessment will be by examination (1½ hours, 75 marks) with approximately 7 questions. It is more than likely that there will be a question on each topic area. The formulae that candidates are expected to know are indicated in the appropriate sections in this scheme. Other formulae and tables can be found in the booklet Mathematical Formulae including Statistical Formulae and Tables – please allow the pupils to familiarise themselves with this document as early as possible in the course.

In order to assist pupils’ learning, give short, timed tests as lesson starters, using examination style questions as early on in the course as is possible.

A note on synoptic assessment: in mathematics, this will address candidates’ understanding of the connections between different elements of the subject. Hence, it is possible that on the S2 examination, candidates will have to answer (parts of) questions on S1 content e.g. when answering a question on the Binomial distribution, techniques studied for Discrete Random Variables in S1 may be required as part of the solution. Specifications that may be part of the synoptic assessment will be shown with a ☼.

Time	Specification	Notes	Page and Exercise	Other Sources
10 hours	• Discrete Distributions*: Modelling & Discrete Random Variables The Binomial Distribution :☼ Parameters and conditions for use of this model Calculation of probabilities by formula Calculation of probabilities by cumulative tables Mean and variance of the binomial The Poisson Distribution :☼ Details as above for binomial The additive property of the poisson e.g. if the number of events per minute ∼ Po(λ) then the number of events per 5 minutes ∼ Po(5λ) Selection of an appropriate distribution	Recap on the definition of a model (see S1) and the concept of a d.r.v. Knowledge of binomial coefficients is required – has this been covered in P2 yet? Derivation of these formulae not required Highlight questions involving, say, a 5 minute period versus 5 consecutive periods of 1 minute each Candidates must select which of these models is appropriate in any given instance, commenting critically on their choice	Ex7C P173 Ex7D P176 Ex7E P181	[S2 Chapter 1] Ex5a P264 Ex5b P266 Ex5f P279 Waldo Site “Binomial” Ex5i P294 Ex5m P303
3 hours	• Approximations to distributions and the conditions under which these are suitable*: The Poisson Approximation to the Binomial ☼Approximations using the Normal Distribution: the use of the normal distribution to approximate both the Binomial and the Poisson distributions with the application of the continuity correction	See conditions P185 Discuss the idea of using a continuous random variable to approximate discrete conditions. Calculations with the normal should be revisited as a homework/lesson starter	Ex7F P186 Ex8C P217	Ex5j P296 Ex7j P403 Ex7k P405
7 hours	• Continuous Random Variables*:☼ The concept of continuous random variables The probability density function for a c.r.v. Mean and Variance of a c.r.v. The cumulative distribution function for a c.r.v. Mode, median and quartiles of a c.r.v. The relationship between (probability) density and (cumulative) distribution functions	Discuss use of area to represent probability P(a<X ≤ b) = ∫ f(x)dx is required The formula used to define f(x) will be restricted to simple polynomials, which may be expressed piecewise F(x 0 )=P(X ≤ x 0 )=∫f(x)dx is needed i.e. f(x) = dF(x)/dx	Read P124-5 Ex6C P141 As above Ex6E P156 As above As above – qus11, 12	[S2 Chapter 2] Ex6a P320 Ex6c P334 Ex6d P342 As above Ex6e P347
2 hours	• Continuous Distributions*: The Continuous Uniform (Rectangular) distribution: Derivation of the mean and variance and cumulative distribution function		Ex8A P197	[S2 Chapter 3] Ex6f P353
8 hours	• Hypothesis Tests: Population, census, and sample. Sampling unit and sampling frame. Concept of a statistic and its sampling distribution Concept and interpretation of a hypothesis test and null (H 0 ) and alternative (H 1 ) hypotheses Critical regions One-tailed and two-tailed tests Hypothesis tests for the parameter p of a binomial distribution and for the mean of a Poisson distribution.	Definitions of these terms. The dis/advantages associated with a census and a sample survey Use of hypothesis tests for refinement of mathematical models Use of a statistic as a test statistic Candidates are expected to know how to use tables to carry out these tests. Questions may also be set not involving tabular values. Tests on sample proportion involving the normal approximation will not be set	N.B. NewS2Book Ch4 Ex4A P89-90 Ex4B P94-5 Read Pages 96-97 Read Page 99 Read Pages 97-99 Read Pages 100-2 Ex4C P102-4 Read Pages 105-6 Ex4D P106	See Chapter 10 as appropriate