Download templar_AQA_AS_stats_5831 - Hertfordshire Grid for Learning

Scheme Of Work For AQA AS Statistics (5381)
Subject Content
1. a) Collection of data
b) Presenting data
c) Simple random sampling
2) Measures Of Average and
Spread
Learning Objectives
1) To distinguish between: Primary and secondary data
 Qualitative and quantitative data
 Discrete and continuous data
 Raw data and grouped data
2)Bar diagrams of discrete data
Module
Time in
lessons
S1
Resources
S1 Chap 1
Ex 1a, 1b, 1c
S2
1
1
Key Maths or
Edexcel GCSE Stats
text
3) Pie charts
S2
2
4) Histograms for continuous data
with equal and unequal class
intervals
5) Cumulative Frequency
S2
2
Crawshaw and
Chambers Ex 1c
Crawshaw and
Chambers Ex 1b
S2
1
6) Box plots
S2
1-2
7) Understand the concept of a simple
random sample.
S1
Notes for teaching and learning
Much of the terminology could be written on a sheet for the students.
Don’t spend too much time on this first section.
To encourage independent learning students could be given the topics of
box plots, cumulative frequency, pie charts and possibly median by
interpolation of grouped data to prepare and present to the class.
Line diagrams, component bar charts and multiple bar charts.
Quick recap of constructing a pie chart and calculation of appropriate radii for
comparison of data – (need to ring AQA to check if this is still needed)
Crawshaw and
Chambers Ex 1n
Craw Ex 2c
Only a quick recap of GCSE Maths content needed
1
S1 Pages 5 – 7.
Quick recap of GCSE work. In addition formally identify outliers as a value more
than 1.5 times the IQR distant from the nearest quartile. Also mention skewness.
Students need to understand the concept of a simple random sample but it will not
be tested in the exam.
Measures needed for ungrouped and grouped data.
Spend time on the idea of bounds, mid-values of groups etc with both discrete and
continuous data
Introduce 60 – notation
Medians of grouped data must be found by interpolation but only modal class of
grouped data needs to be identified.

Calculate mode, median and mean
S1
3
S1 Ch 2 Ex 2a, 2b.
(Crawshaw and
Chambers Ex 1d,
1m and some of 1n)

Calculate range, IQR, variance and
standard deviation
S1
4
S1 Ex 2c,2d,2e
(Craw Ex 1e, 1g)
Make sure students can efficiently use the statistical functions on their calculators.
Teach the difference between s and  but questions requiring the knowledge of
which to use will not be set.

Change of scale
S1
1
S1 Ex 2f
(Craw Ex 1h)
Questions requiring linear scaling will not be set but students still need to be aware
of the effect on numerical measures. I.e. the effect on the mean and standard
deviation of adding a constant a to a set of data or multiplying a set of data by a.

Choose numerical measures
appropriate to a given context
S1
1
S1 Ex 2g

Draw and interpret scatter
diagrams
Evaluation and interpretation of the
product moment
S1
0.5 – 1
S1
3-4
Evaluate and interpret Spearmans
Rank Coefficient Of Rank
Correlation
Use tables to test a hypothesis
about a Spearmans Rank
correlation coefficient or product
moment correlation coefficient.
S3
2
S3
2
TEST
3) Correlation and regression



S1 Chap 7 and Ex
7a
X axis is the independent or explanatory variable and the y axis is the dependent
or response variable.
Emphasise product moment is a measure of linear correlation only.
Teach the fact that r = 0 does not necessarily imply that there is no relationship
between variables & that an association does not necessarily imply cause & effect.
Do some examples where students calculate xy etc and substitute into the
formula but it’s to be encouraged that r should be found directly from the calculator
Slot in some Past A level questions for practice
S3 Chap 3 and Ex
3a
Crawshaw Ex 12d
S3 Ex 3b + mixed
exercise
Teach how to use the formula but again once the data is ranked pupils should be
encouraged to use the r button directly on the calculator.
Informally you’ll have to talk about H0 and H1 and a significance test.




To find the equation of regression
lines using the method of least
squares
Interpret the values obtained for the
gradient and intercept of the
regression line
Plot a regression line on a scatter
diagram and use it for predictive
purposes
Calculate residuals and use them
to check the fit of a regression line
or to improve predictions.
S1
To identify mutually exclusive and
independent events.
To apply the addition law
P(AB)=P(A)+P(B)- P(AB)
To apply the multiplication law
P(AB)=P(A)P(B/A) =P(B)P(A/B)
and to independent events to apply
the rule P(AB)=P(A)P(B).
Solve simple probability problems
using tree diagrams or laws of
probability.
To recognise when to use the
binomial distribution
To state any assumptions
necessary to use the binomial
distribution
To apply the binomial distribution to
a variety of problems.
To know the mean, variance and
standard deviation of a binomial
distribution
To understand the concepts of a
continuous distribution
Understand the properties of
normal distributions
S1
6
S1 Chap 8
Only y on x line needs to be looked at.
Encourage the use of the calculator to obtain the gradient and intercept.
Emphasise that extending a line to forecast in the future (extrapolation) is not
always reliable.
2
Give several A level questions for practice.
TEST
4. Probability




5. Binomial Distribution




6. Normal Distribution





To transform to standardised
normal distribution and to use
tables to calculate probabilities
Use tables of percentage points of
the normal distribution
To find an unknown mean and/or
standard deviation of a normal
distribution.
8
S1 Chap 3
(Craw Exercises
3a,3b,3d,3g)
S1
S1
5
S1 Chap 4
Students doing the course who have only done Intermediate Maths at GCSE will
not be familiar with calculating with probabilities knowing when to add or subtract.
On S1 only simple probabilities will be set that can be solved by counting equally
likely outcomes and/or the use of tree diagrams or frequency tables.
n
(Craw Ex 5a,5b,5c)
Use of x notation is required
Students need to calculate probabilities using the formula and by using tables.
0.5
S1 P.81
i.e. area under the curve = 1, mustn’t take negative values etc.
0.5
S1 P.93
i.e. shape, symmetry and area properties, knowledge that approx 2/3 of the
observations lie within    and equivalent results.
S1 Chap 5
Rounding z values to 2 dp is acceptable.
7
(Craw Ex 7a – 7g)
i.e. given the probability go backwards to get the z value.
Students may need to solve two simultaneous equations.
7. Central Limit Theorem




8. Confidence Intervals



To understand the terms population
and sample
To know that unbiased estimates of
the population mean and variance
are x and S2.
To understand what is mean’t by
the distribution of the sample mean
To find probabilities involving
sample means via knowledge of
the Central Limit Theorem.
S1
To calculate a confidence interval
for the mean of a normal
distribution with a known variance
To calculate a confidence interval
for the mean of any distribution
from a large sample using a normal
distribution.
To be able to make inferences from
confidence intervals.
S1
To understand the concepts of
trend, seasonal variation, short
term and random variation.
Use moving averages to estimate
seasonal effects.
Make forecasts by extrapolating the
trend and where appropriate
applying a seasonal effect.
Modify forecasts where appropriate
to allow for short term variation.
To define simple (without
replacement) and unrestricted (with
replacement) random samples.
To use random numbers from
tables or calculators to generate
random samples.
To define stratified, cluster, quota
and systematic samples
To find the expected value,
standard deviation and variance of
discrete random variables
To apply and interpret the above in
real world situations
Knowledge of the conditions
necessary for a Poisson distribution
To model a real world situation
using a Poisson distribution
Top use tables of the Poisson
distribution
To use the mean and variance of
the Poisson distribution
To have knowledge of and use the
distribution of the sum of
independent Poisson distributions.
S2
2
S1 Ex 5j
(Craw Ex 8f)
Use the terms ‘parameter’ and ‘statistic’ when referring to population and sample.
To include the standard error of the sample mean

n
and it’s estimator
S
.
n
i.e. using the normal distribution as an approximation to the sampling distribution of
the mean of a large sample from any distribution.
3-4
S1 Chap 6
Where variances are known and unknown.
Based on whether a calculated confidence interval includes or does not include a
‘hypothesised’ mean value.
TEST
9. Time series analysis




10. Sampling



11. Discrete probability
distributions


12. Poisson Distribution





3-4
S2 Chap 1
Could set this topic as one where groups study the chapter and present it to the
class.
Students may need to use regression to estimate trend.
Students need to understand that forecasts are projections of past patterns and
should be treated with caution.
S2
2-3
S2 Chap 2
Most of the notes could be on a sheet for pupils.
Advantages and disadvantages of the different methods of sampling are needed.
S2
3
S4 Chapter 1
(More in Craw Ex 4a
– 4d)
S2
4
Use of E(X) = x. P(X=x)
E(g(X) = g(x). P(X=x) i.e. E(X2) = x2 . P(X=x)
Var(X) = E((X-E(X))2 = E(X2) – ((E(X))2.
S1 – Edition 1 Ex 6a
and 6b.
Evaluation of probabilities using the formula is not needed. Only need to teach
calculating probabilities using tables.
(Craw Ex 5i and 5m)
For the questions in both textbooks you will need to check that all questions can be
done just using tables before you set them.
Page 122
13. Hypothesis tests for the mean



14. Contingency tables




15. Distribution free methods
(Non Parametric tests)







To define a null and alternative
hypothesis and significance level of
a hypothesis test. To construct a
critical region and to understand
whether to use a 1 or 2 tailed test.
Have an understanding of the
concepts of Type 1 and Type 2
errors.
To test a hypothesis about a
population mean based on :1) a sample from a normal
distribution with known
standard deviation.
2) A large sample from an
unspecified distribution
To analyse contingency tables
using the 2 distribution
To recognise conditions under
which this analysis is valid
To combine classes in a
contingency table to ensure
expected values are over 5.
To apply Yates correction when
dealing with a 2x2 table
To understand what is meant by a
non parametric test
1. Tests of average
To carry out a sign test (for
medians)
To carry out a Wilcoxon signed
rank test for medians/means
2. Analysis of paired samples
Use sign test and Wilcoxon signed
rank test to analyse results of a
paired comparison.
Understand what is meant by
control and experimental group;
appreciate why blind and double
blind trials are used; to understand
the terms experimental error, bias,
replication and randomisation.
3. Two independent samples
To carry out a Mann Whitney U test
on data collected as two unpaired
samples
4. More than two independent
samples
Use Kruskal Wallis test to test the
hypothesis that more than 2
independent samples come from
identical populations.
S2
4
S3 Chap 2
(and S4 Chap 3)
A Type 1 error is the probability of rejecting H0 when it is actually true
A Type 2 error is the probability of accepting H0 when it is false.
Questions requiring you to calculate a Type 2 error will not be set.
S4 Chap 3
Students must appreciate the need for random samples.
S4 Chap 4 Ex 4a
S3
4
S4 Chap 5
(Craw Ex 11c)
S3
3
S3 Chap 4
2-3
S3 Chap 5
2
S3 Chap 6
2
Hopefully a handout
from the board!
(There is a question
on the Specimen
Paper)
Wilcoxon signed rank test assumes the distribution is symmetrical and
consequently that the mean and median are the same.
Questions requiring choice between the sign test, Wilcoxon signed rank test and
the z test from Module S2 may be set.
Hypothesis to be tested is that the 2 independent samples come from normal
populations