Time

Specification

Notes

Page and Exercise

Other Sources

½ hour

• Mathematical models in probability and statistics:

Introduce the course content and ideas

Discuss the basic ideas of mathematical modelling as applied in probability and statistics

 

 

See insert “What is Statistics all about?”

Learn the definition of a model: “A statistical procedure devised to describe or make predictions about the expected behaviour of a real world problem”.

 

 

 

P1-3 (P1-3)

 

 

5½ hours

• Probability*:

Elementary Probability, including terminology, sample space diagrams and complementary events.

 

The addition rule & Venn diagrams.

 

The multiplication rule and conditional probability.  Making use of tree diagrams.

 

 

 

 

Independent events and mutually exclusive events

 

P(A’) = 1 – P(A) is required

 

 

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

 

P(A ∩ B) = P(A) P(B/A)

Include examples that sample both with and without replacement.  Also look at situations where different orders of events result in the same outcome (e.g. P(2 People pass & 1 fails)), briefly introducing factorial notation as an alternative.

P(B/A) = P(B), P(A/B) = P(A), P(A ∩ B) = P(A)P(B) for indep.; P(A ∩ B) = 0 for exclusive.

[Overall Revision: Ex5E P92-4]

 

 

Ex5AP73; (Ex5A P99)

Ex5BP78; (Ex5B P105)

Ex5D P90-91

 

 

 

Ex5CP90; (Ex5C P112)

 

Ex3AP141-2

 

 

Ex3BP147

 

Ex3DP157

Ex3GP168-9

 

 

 

 

Ex3HP173; Ex3IP177

4 hours

• Measures of location and dispersion*:

Types of data – vocabulary

 

 

Measures of location:

-Mean

 

-Median & Quantiles : see Waldo site “Linear Interpolation to get the median”.

 

 

 

 

-Mode

 

Measures of dispersion:

-Variance & Standard Deviation:

 

 

 

 

-Range and Interpercentile ranges

 

 

 

Other:

Interpretation of measures of location and dispersion and the relative dis/advantages of each

 

 

Skewness and the concept of outliers

 

Discrete, continuous, (qualitative), (quantitative), grouped, ungrouped (define class width, boundaries etc.)

“One number to suitably represent a set of data”

-Include use of coding and combined mean examples.  Formulae not in booklet

-Suppose f is the fraction associated with the quantile and n is the total frequency.  If fn is an integer r, the quantile is the mean of the rth and the (r+1)th observation.  If fn is not an integer but lies between r and (r+1), the quantile is the (r+1)th observation

 

 

“One number to signify the spread of a data set”

Standard Deviation = √ Variance

Include examples with coding

The use of “n” as a divisor is adequate for S1

Demonstrate both formulae as Σ(x-μ)/n is useful in regression later on.

IQR = Q 3 – Q 1 is required and could be from a cumulative frequency polygon or by simple interpolation from a table.  For grouped data, interpolation at the value of fn is adequate.

 

e.g. most data should lie within one standard deviation of the mean.  It may be best to use median & IQR as opposed to mean & st.dev. if the data contains outliers

Any rule to identify an outlier will be specified in the question e.g. any value that is more than 1.5x the st.dev. away from the mean

 

Read P5-7, 16-18

 

 

Ex3AP48-52 (Ex4AP62)

CodingP52Q15; P111Q50

 

 

 

 

 

 

 

Ex4AP64-8 (Ex4BP78)

 

 

 

 

Ex1oP90

 

Coding Ex1iP49

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ex2dP131

 

 

 

SeeCh2 P98 & Ex2aP106

3 hours

• Graphical Representation of Sample data*:

-Stem & Leaf diagrams

 

 

-Boxplots

 

 

 

-Histograms: see Waldo site “Histograms”

 

Back-to-back ones will be required to compare distributions

 

Include labels and a scale and, when comparing two boxplots, draw them on the same scale.  Candidates may be asked to represent outliers with crosses past the end of the whisker

Including grouped data with unequal classes (frequency density = frequency/class width)

Include a brief look at relative frequency ones

 

Ex2AP13 (Ex3AP21q6-12)

 

Ex4AP64qus12, 15

 

 

 

Ex2BP26Qu7-11 (Ex3BP35Qu6-12)

 

 

 

 

Ex2dP131

 

 

 

Ex1bP20 –do an inverse question to find frequency

6 hours

• Discrete Random Variables*:

Concept and definition

 

The Probability function p(x) and the cumulative distribution function F(x)

 

Mean and Variance of a d.r.v.

 

 

Mean and Variance for a linear combination of random variables

 

The discrete uniform distribution

 

 

 

Use p(x) = P(X = x) and F(x 0 ) = P(X ≤ x 0 ) = Σp(x)

 

 

Use of E(X) and E(X²) for calculating the variance

 

E(aX + b) = aE(X) + b

Var(aX + b) = a²Var(X) are both needed

 

The mean and variance of this distribution

 

 

 

Ex8AP150(Ex6AP126)

 

 

Ex8BP157(Ex6BP133)

 

 

Ex8CP160 (Ex6DP149-select)

 

Ex8DP185(Ex7AP164)

 

 

 

Ex4aP223 (worksheet)

 

Ex4b, 4c, 4d P229

6 hours

• Continuous Random Variables*:

Brief introduction (more in S2)

 

Normal Distribution

Knowledge of shape, properties and symmetry

Mean and Variance

Cumulative distribution function and the use of tables

(see Waldo site “Normal Distribution”)

 

Focus on the reason we use areas to represent probabilities (link to P1 Integration)

 

Probability density function is not required

Derivation of these is not required

Derivation of this is not required.  The standardisation formula must be learned.  Pupils may interpolate but it is not required.

Questions may involve the solution of simultaneous equations – this may, therefore, involve the “reverse” normal table

 

 

 

 

Ex9AP177(Ex8BP209)

 

 

 

 

Ex7a,7b,7c,7d,7g P373 →

5 hours

• Correlation & Regression:

Scatter diagrams and correlation

 

Product Moment Correlation Coefficient – its use, interpretation and limits

Line Graphs – “explanatory” (independent-x) and “response” (dependent-y)

 

 

 

 

Linear Regression: “Least Squares Regression Line” and the interpretation of the regression line

 

Line of Best Fit through average point

 

Derivation and tests of significance will not be required

Use to make predictions within the range of the explanatory variable and discuss the dangers of extrapolation (does a line extend back or forward with meaning?).

Variables other than x, y may be used and a linear change of variable may be required

Derivations will not be required

Answers can be given to, say, 3s.f., but full values must be substituted back when, say, finding “c”

When interpreting, more than a comment such as “negative correlation” is required.  E.g. in a graph of weight against time, the y-axis intercept signifies birth weight and the gradient growth rate.

 

Ex6AP126

 

As above

 

Read Pages 131-9

 

 

 

 

 

Ex7AP139

 

Ex12aP635

 

Ex12dP669

 

 

 

 

 

 

 

Ex12bP643; Ex12cP656

 

I CROSS

 

I STAY

 

QUIET

 

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X

 

BUSY

 

X

 

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