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Statistics and Research methods
Wiskunde voor HMI
Bijeenkomst 4
The Distribution of Means


Comparison distributions considered so far
were distributions of individual scores
Mean of a group of scores
–

Comparison distribution is distribution of means
Used in hypothesis tests with means of
samples
The Distribution of Means

Distribution of means
–
Distribution of the means of each of a very large
number of samples of the same size (with each
sample randomly taken from the same population of
individuals)
The Distribution of Means

Characteristics
–
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
M  

 
N
2
2
M
The Distribution of Means

Characteristics
–
Its standard deviation is the square root of its
variance
2
M  
–
2
M




N
N
Shape: it is approximately normal if either


Each sample is of 30 or more individuals or
The distribution of the population of individuals is normal
Review of the Different
Kinds of Distributions
Distribution of a population of individuals
 Distribution of a particular sample
 Distribution of means
See also Table 7-1

Z-test

Hypothesis testing with a distribution of means

Confidence intervals
(betrouwbaarheidsintervallen)
–
Example: Opgave 1 bijeenkomst 4
t Test for a Single Sample





Population mean known, variance not known
Compare mean of sample scores with
population mean
Hypothesis testing steps with population
variance estimated from sample scores
Shape of the comparison distribution:
the t distribution
Degrees of freedom: df = N-1
t Test for a Single Sample
(continued)

The t distribution
–
Varies in shape according to the degrees of freedom
t Test for Dependent Means


Unknown population mean and variance
Two scores for each person
–

Use difference scores
–

Repeated measures (within-subjects) design
Assume that the population mean is 0
Assumption: Normal population distribution
–
t tests are robust to moderate violations of this
assumption
t Test for Dependent Means

Difference scores
–
–

For each person, subtract one score from the other
Carry out hypothesis testing with the difference
scores
Population of difference scores with a mean of 0
–
Population 2 has a mean of 0