Download Inferential Statistics

Inferential Statistics
Population Curve
Mean
Mean Group of 30
Population Curve
Theoretically, a new frequency
distribution will appear representing
the means of groups of 30.
Mean
Computing the means of randomly
selected groups of 30
Sampling Distribution of the Mean
Population Mean
Population Distribution
Sampling Distribution Mean
(n = 30)
Sampling Distribution for
random groups of 30
Sample Sizes
If the sample size was really large
then it would be unlikely to get
sample group means that were very
far from the population mean.
Population Distribution
n=1000
n=100
n=30
If the sample size was really
small then sample group means
would be distributed similarly
to the population.
As the sample size increases the
Sampling Distribution of the Mean gets narrower.
Standard Error of the Mean
n=1000
n=100
Population Distribution
n=30
The standard deviation of a Sampling Distribution is called the Standard Error.
For any distribution the larger the sample size the
smaller the numerical standard error.
Standard Error
Sampling Distribution Mean
1 Standard Error
Sampling Distribution
(n = 30)
50.0
34.13
The relationship of standard error to a
sampling distribution is the same as
standard deviation to a normal distribution.
Standard Error
Sampling Distribution Mean
1 Standard Error
Sampling Distribution
(n = 30)
50.0
34.13
Remember this is not a distribution of scores on the test.
It is a distribution of the means of
randomly selected groups of 30.
Measuring Group Means Against the
Sampling Distribution
Sampling Distribution
(n = 30)
1 Standard Error
50.0 34.13
15.87% (.16)
Probability of higher mean
At any given Standard Error it is possible to compute the likelihood
that a higher or lower group mean could have occurred by chance
(remember the relationship of z scores to percentile rank).
Measuring Group Means Against the
Sampling Distribution
2 Standard Errors
Sampling Distribution
(n=30)
50.0 34.13 13.59
2.28% (.02)
Probability of higher mean
At any given Standard Error it is possible to compute the likelihood
that a higher or lower group mean could have occurred by chance
(remember the relationship of z scores to percentile rank).
Probability
• How unlikely does the occurrence of a group
mean have to be before we would say that it is
so unlikely that it couldn’t have happened by
chance? It must have occurred for some other
reason.
Significance
Probability of
higher mean
If the probability that a given group mean would occur by
chance in a sampling distribution is very small then the
occurrence of that group mean is said to be significant.
Significance
Probability of
higher mean
To use the z-score analogy when the percentile ranking of a
group mean is really high (or really low) then it is significant. It
is significant because it is unlikely to occur randomly.
Significance
5% (.05)
1% (.01)
Most social science research declares that group means
occurring by chance less than 5% of the time are significant.
Most medical research uses 1% or much less.
With a z-score we used a table to look up percentile rank. Since the normal
distribution in now a sampling distribution based on a specific group size the ztable
work.test example.
Ourwon’t
pre/post
A new table for each possible group size needs to be generated to test the
Post-test mean 84.61
“percentile rank” of the group mean comparison.
Fortunately the computer does this for you.
Pre-test mean 74.61
5% (.05)
1% (.01)
Probability (p) that this score or higher could have
appeared by chance is .010 (1% or 1 in 100 times)
substantially less than 0.5 (5 % or 1 in 20 times)
t-Critical
• The sampling distribution of the mean is a
different shape for every sample size.
• Therefore the cut-off for rejecting the null has a
different value for every sample size.
• Jackie had you look up the cut-off value by
determining the df (identifying the distribution)
and then looking up on a table of “critical values”
whether the difference was significant.
• Now you can read the p value directly to do this.