This is a summary to the book used in Statistical Modelling in Communication Research at the University of Amsterdam. The document includes links to learning videos and a detailed step-by-step guide for SPSS.
We know that we need a sampling distribution to generalize conclusions about a random
sample to conclusions about the population.
Thousands of samples make up the sampling distribution. If we calculate the mean of the
sampling distribution – which is also called the expected value – we have the true value in the
population.
In this example, the expected value is 2. On average, 2 out of 10 candies are yellow, which is
0.2 as a proportion. This is also the proportion of yellow candies in the population.
We can come to a sampling distribution by drawing only one sample (instead of
thousands, which is very time consuming).
à nevertheless, we can never be sure that our sampling distribution that we have created
with our one sample is correct (to the true population). We have to check for requirements!
There are three possibilities of doing this:
1. Bootstrapping
2. The exact approach
3. Theoretical approximation
Bootstrapping
Bootstrapping sampling distributions is most similar to the technique of drawing thousands of
samples from the population. The difference is that we draw thousands of samples from our
original sample and NOT from the population.
1. We draw ONE sample. The units drawn have been numbered from 1-25.
2
, 2. The computer draws samples from our original sample, which are called bootstrap
samples. The bootstrap sample contains of just as many units (N) as the original
sample. (Number of candies = 25).
à we notice that the same number depicted on the candies can appear more than once
(yellow candy number 24 has been drawn 5 times). This is called sampling with
replacement.
Why are we drawing with replacement?
• If we draw a sample without replacement from our initial sample of the same size as
the initial sample, the new sample must contain all observations from the initial
sample. As a result, the new sample is identical to the initial sample. All samples that
we draw are identical. This does not provide an interesting sampling distribution.
When we're drawing with replacement, each new sample drawn from the original
sample can be different, so the proportion of yellow candies varies across these
bootstrap samples. We can create a meaningful sampling distribution from these
varying proportions of yellow candies.
Calculating probabilities without replacement
• In actual research, we sample without replacement. Because if a person would
participate twice in our research, we would not yield new information.
• If we do NOT replace a candy to the sample, the number of yellow candies in the
population is reduced by one after we have drawn the first yellow candy à we get a
NEW probability of drawing a yellow candy (less than 20%)
o However, if the population is much larger than the sample, we can ignore
these complications because the probabilities hardly differ from the sample
probabilities that we have if we sample with replacement.
3. Thousands of bootstrap samples are being drawn and those sample statistics are being
added in order to form a sampling distribution.
à we notice that the new sampling distribution (yellow) has more or less the same
shape as the true sampling distribution (grey)
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, This technique works for every sample statistic that we can think of (the other two methods
CANNOT be used for all sample statistics).
Nevertheless, bootstrapping only works if our original sample is more or less
representative of the population. NOTE there is always a chance that a random sample
does not reflect the population well. But sometimes bootstrapping is the only or best option
that we have
What happens, if the original sample would have been different? (original sample: a
sampling distribution with a population proportion of 20%)
à notice how the bootstrap sampling distribution differs from the population’s sampling
distribution! This happens because we are drawing thousands of samples FROM the original
sample and not the population!
The Exact Approach
With the exact approach we can calculate the exact probabilities of all possible sample results
(with the binominal probability formula). If we think we know the proportions of
probabilities in the population of yellow candies, we can calculate the exact probability of x
numbers of yellow candies.
Example:
We draw 3 candies (which have the probability of 20% to be yellow)
Yellow: p = .200 = 20%
Not Yellow: p = .800 = 80%
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