The sampling distribution of the mean
The Central Limit Theorem
The Normal Deviate Test (Z for samples)
The distribution of a statistic (eg. mean,
median, standard deviation) for the set of
all possible samples from a population.
For example, if we toss an unbiased coin
repeatedly in sets of three tosses, scoring
heads as 1 and tails as 0, the possible
samples are as follows:
HHT .67 of the mean
HTH .67 Mean f p
1.00 1 .125
THH .67 .67 3 .375
.33 3 .375
TTH .33 .125
THT .33 8 1.00
Characteristics of the sampling
It includes all of the possible values of a
statistic for samples of a particular n
It includes the frequency or probability of
each value of a statistic for samples of a
Invisible vessels: n = 100
Marking means: Poker chips
In the kitty: The sampling distribution of
The null hypothesis population
The entire set of scores as they are naturally,
that is, if the treatment has not affected them.
If the treatment has had no effect, then the
null hypothesis is true: thus, the name null
If a treatment has an effect, then the mean of
the treated sample will not fit well in the null
hypothesis population: It will be weird.
The Central Limit Theorem
If random samples of the same size are
drawn from any population, then
the mean of the sampling distribution of the
mean approaches m , and
the standard deviation of the sampling
distribution, called the standard error of the
mean, approaches s / n ...
as n gets larger.
Generating a sampling
From a population of six people who are given
grape Kool-Aid, persons 1, 2, and 3 have their
IQs raised, and persons 4, 5, and 6 have their
IQs go down.
Sampling without replacement, form all of the
possible unique samples of 2 people from the
population of six. (Simplified example)
In how may of the samples does the mean IQ
The normal deviate test
The normal deviate test is the Z test
applied to sample means.
To use it, you must know the population
mean and standard deviation. You may
know these as
TQM or CQI goals
Historical sample patterns
The normal deviate test...
The only difference from the simple Z test is
that the denominator is s / n , which is
known as the standard error of the mean.
To test our grape Kool-Aid gang, take a
sample of 100 Houghton students, and
compute the mean IQ = 130. Compare that
mean to a population mean of 125, with a
population standard deviation of 15.
The critical region
You can simplify a set of decisions
about sample means by establishing the
critical region for sample means which fit
a rejection criterion for Z.
For a one-tailed test at the .05 level, the
critical value of Z from table B-1 is 1.645
For a two-tailed test at the .05 level, the
critical value of Z is 1.96
Calculating the critical region
Plug the appropriate critical value of Z
(1.645 or 1.96) into the equation for the
normal deviate test, and solve for M.
Remember that for a two-tailed test, the
critical sample mean for each tail must
be calculated by working above and
below the population mean m.
Sample size and power
Test the grape Kool-aid gang again, with
sample sizes of 4, 9, 16, 25, 36, 49, 64,
You will notice that as the sample size
increases, the obtained Z-score for the
same size difference between means
If the same difference produces a larger
Z-score, the test has more power.
When can we use the normal
For a single sample mean
When we know m and s
When the sampling distribution of the
mean is normally distributed, which we
can usually assume when n is 30 or
Notable exception: reaction time
Reporting standard error in APA
In text or in tables, report standard error
with the abreviation SE.
In graphs, indicate the size of the
standard error with error bars, bracketed
lines centered at the top of the bar of the
graph for the mean, and extending one
standard error above and below the
Error bars in graphs
Normal deviate test in APA
z = 1.98, p < .05
z = 1.95, p > .05
z = 1.40, p = .08