Transcript Review and further practice

Review and further
practice
(Session 10)
SADC Course in Statistics
Learning Objectives
At the end of this session you will be able to:
• compute probabilities associated with Venn
diagrams, and the binomial, Poisson and
normal distributions
• have greater confidence in explaining the
differences and links between the normal,
binomial and Poisson distributions
• Identify data arising in applications as
being suited to the binomial, Poisson or
normal models
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Review of the three distributions
• Binomial distribution is appropriate when
there is a sequence of Bernoulli trials, i.e.
trials with just 2 possible outcomes, and
interest is in looking at the number of
successes (say r) in n trials
• Poisson distribution is suitable with data in
the form of counts
• The normal distribution applies with many
naturally occurring variables of a
continuous nature, e.g. heights, weights,
etc
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Some general features
• The binomial and Poisson are discrete
distributions, while the Normal corresponds
to a continuous distribution
• Binomial takes values from 0 to n, Poisson
can take values 0, 1, 2, …. etc., while a
normally distributed variable can any value
from - to +
• For large sample sizes, the Binomial and
Poisson can be approximated by the
Normal distribution
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Parameters of the distributions
• The binomial is described by two
parameters, namely n, the number of trials,
and p, the probability of a success.
• The Poisson distribution is described by a
single parameter . This is also the mean of
the distribution.
• The normal distribution is described by two
parameters, i.e.  and . Here  is the
mean of the distribution, while  is the
standard deviation.
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Poisson approxn to Binomial
• A binomial distribution with parameters n
and p can be approximated by a Poisson
distribution with parameter  = np if n is
“large” and p is “small”.
• The approximation gets better the bigger n
becomes and smaller p becomes, but
generally good when n>50 and p<0.1.
• This result is useful because for the normal
approximation to hold, n has to be very
large to compensate for small p.
Approximation of binomial by normal is
best around p=0.5 because of symmetry.
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Identifying the distribution
In statements below, what is the likely
distribution for the key variable?
• An asset based poverty index has been
produced for classifying each HHs as very
poor or not. In a given rural area with 2475
HHs, the number of poor HHs is recorded.
• In a health survey, a record is made of the
number of episodes of diarrhoea in the HH
over a period of 3 months.
• Also recorded in this survey is the weight of
children under 1 year and mother’s age.
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Practice with probabilities and distns
The remainder of this session is devoted to
practical work concerning probability
concepts and probability distributions.
You are encouraged to ask questions and
ensure that ideas and concepts covered in
the previous have been understood.
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Practical work concerning all
three distributions follows …
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9