Transcript Sampling Distributions

Sampling Distributions
Statistics 2126
Introduction
• Let’s assume that the IQ in the
population has a mean () of 100 and a
standard deviation () of 15
• In fact the tests are designed that way
so we are in luck, we know the
parameters
A Little Thought experiment
• Let’s randomly select 20 people and
measure their IQs
• Let’s calculate the mean for each group
sampled
• What would you expect to get?
• What would you really get?
• What would the curve look like?
Another thought experiment
• Assigning the value of 0 to women and
1 to men for the variable ‘maleness’
• Let’s select 20 adults, randomly
• What value should we get for maleness
• What value would we get
• What would the curve look like?
No way..
• Way…
• Think about it
• You will get, usually,
the same number of
males and females
• Sometimes very few
of either
Sampling Distributions
• These two distributions are called
sampling distributions
• In this case, the sampling distribution of
the mean
• All the possible values a statistic can
take with a given sample size
The Central Limit Theorem
• Given a population distribution with a
mean () and a standard deviation ()
the sampling distribution of the mean
will have a mean  =  and a variance
of 2/n and will approach normal as n
increases no matter what the shape of
the parent population distribution
Pretty cool
• So now we can combine this knowledge with
what we know about the z distribution and
figure out how likely a given mean is
• Well we know the shape, the mean and the
standard deviation so now finding the
likelihood of a given mean for a variable is the
same as doing it for a given value
Whaaaa?
• Well before we could find out how likely
an IQ of 108 was.
• OK, so let’s do that with a mean of 108
• n=9
• It is IQ so  = 100
• And  = 15
Do the plusses and
takeaways…
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z = (x- ) / 
But we have a sampling distribution so
z = (x - ) / ( /n)
z = (108-100) / (15/3)
z=8/5
z = 1.6
p(z > 1.6) = .055
So how can we use this
knowledge?
• Say you flip a coin
• At some point you say well that is a
fixed coin, not a fair coin
• But at what level?
• Well when the probability that it is a fair
coin in < some value
• (usually .05)
Hypotheses
• We set up two, mutually exclusive
hypotheses
• H0 and HA
• The null is that there is no effect
• The alternative is that there is an effect
• If the p(H0 is true) < .05 the we reject H0
So in our example
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p(z > 1.6) = .055
Darn, too high
We fail to reject H0
Not enough evidence
Pretty darned close though
Now we can make mistakes
• False positives
• False negatives
Errors in hypothesis testing
Reality H0 true
Ha true
Decision
Do not reject
H0
Correct
decision
Type II error
Reject H0
Type I error
Correct
decision