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Chapter 5
Hypothesis Tests With
Means of Samples
Part 1: Sept. 10, 2013
The Distribution of Means
• In Ch 4, comparison distributions discussed were
distributions of individual scores
• Now, interested in mean of a group of scores
– Comparison distribution of interest will be
distribution of means
The Distribution of Means
• Consists of means of a very large number of
samples of the same size
– Each sample randomly taken from the same
population of individuals
– Each point in distribution is a group mean
• This will be your comparison distribution
when you have N>1.
The Distribution of Means
– How is it created? Example…
• Population N=100, want sample n=5 test scores
• 1st random sample = persons 6, 27, 45, 88, 91
• 2nd random sample = persons 18, 30, 56, 59, 79
• Plot 78.4, 82.5, etc.
• Should look approximately normally distributed
The Distribution of Means
• Characteristics - assuming a large N
– Its mean is the same as
the mean of the population
of individuals
– Its variance is the variance
of the population divided by
the number of individuals in
each of the samples
Mean of
of means
M  
Variance of distribution
of means
M 
The Distribution of Means
• Characteristics
– Its standard deviation is the square root of its
M  M 
SD of distribution of means also known as standard error
( How much the means of samples are ‘in error’ as
estimate of mean of the population.)
The Distribution of Means
– Shape: it is approximately normal if either
• Each sample is of 30 or more individuals or
• The distribution of the population of individuals is
– Example…
Hypothesis Testing With a Distribution
of Means
• Distribution of means will be your comparison
• 1) Find a Z score of your sample’s mean on a
distribution of means
• Z score formula conceptually same as before, but now refers
to means of sample & comparison distrib
Sample mean
(M   M )
Mean of distrib of
Std dev of distrib of
• Your sample’s mean is 220 (n=64),
distribution of means has mean=200, std
dev = 6.
Example - #9 from practice prob. (p. 182)
• 1-sample z test:
• Used 1-10 scale to indicate fault of driver in accident.
– Population distribution has  = 5.5 and  = .8
– Here, 16 students rated fault when asked how likely the
driver who crashed into other was at fault?
– This sample (n=16) had mean=5.9.
– Did the manipulation significantly increase fault results?
• that is, compare our sample mean to the pop mean
• is 5.9 significantly higher than 5.5?
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