Chapter 10: Sampling Distributions
Each proportion is based on a different sample of the population.
The proportions vary from sample to sample because the samples comprise
10.1 Modelling Sample Proportions
p = “true proportion”
A simulation can help understand how sample proportions vary due to
̂ = proportion of success
The way in which the proportions vary sample to sample shows how the
proportions of real samples would vary.
10.2 The Sampling Distribution for Proportions
Sampling distribution: the distribution of a statistic (proportion) over many
independent samples of the same size from the same population.
o The Normal approximation can be used provided the sampled values
are independent and the sample size is large enough.
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Sampling variability: the variability we expect to see from sample to sample.
Using the Normal approximation is valuable because:
o The Normal sampling distribution model tells us what the distribution
of sample proportions would look like.
o Since the Normal model is a mathematical model, we can calculate
what fraction of the distribution will be found in any region.
Using the standard Normal table.
The Normal model becomes a better and better representation of the
distribution of the sample proportions as the sample size gets bigger.
Two assumptions in the case of the model for the distribution of sample
o Independence: sample values must be independent of each other.
To check the independence assumption, check the following
o Sampling method must not have been biased and
the data must be representative of the
I.e. in an experiment subjects are
randomly assigned to treatments.
10% Condition o The size of n must not be > 10% of the
o At least 10 successes and 10 failures must occur.
o Sample Size Assumption: the sample size, n, must be large enough.
10.3 The Central Limit Theorem – The Fundamental Theorem of Statistics
Proportions summarize categorical variables.
The sampling distribution of almost any mean becomes Normal as the
sample size grows.
o Observations need to be independent and randomly collected.
o The shape of the population distribution does not matter.
Central Limit Theorem (CLT): The CLT states that the sampling distribution
model of the sample mean (and proportion) is approximately Normal for
large n, regardless of the distribution of the population, as long as the
observations are independent.
o This is true regardless of the shape of the population distribution.
o It is an imaginary distribution since we don’t actually draw all