Hypothesis
Testing: ZTest

Example Experiment
Prenatal exposure to alcohol on birthweight in rats.
The researcher's sample has n=16 rat pups. We assume
that
 The population of birthweights is normally distributed.
 The population has a mean birthweight of 18 grams.
 The population has a standard deviation of 4 grams.

Here are the four steps involved in the statistical hypothesis
test:
 State the Hypotheses:
 Null hypothesis: No effect for alcohol consumption
on birth weight. Their weight will be 18 grams. In symbols:
 Alternative Hypothesis: Alcohol will effect birth
weight. The weight will not be 18 grams. In symbols:
 (Classical Approach)
Set the decision criteria:
 Specify alpha, the significance level.
We specify:
 Determine the critical value of Z. We do this
for the choosen significance level. For a nondirectional
test the critical value of Z is the value that has alphapercent
of the area more extreme than Z. For alpha=.05 we look
up a Z that has .025 of the distribution beyond it.
This is a Z of +1.96 and 1.96.
(Contemporary Approach)
The computer calculates the exact probability of the result
of the experiment.
 Gather Data:
Two experimenters carried out the experiment. They got the
following two samples of results:
Experiment 1 
Experiment 2 
Sample Mean = 13 
Sample Mean = 16.5 
 Evaluate Null Hypothesis:
For each experiment we calculate the standard error of the
mean, then calculate Z for each experiment. Classically,
we would then look up the P value for the obtained Z. Contemporary
practice is to have the computer determine the P for the
obtained Z. Then we make a decision.
 Determine the standard error of the mean. The
standard error is calculated by the formula:
For these data the value is 4/sqrt(16) = 1.
 Calculate the TestStatistic. To determine how
unusual the mean of the sample we will get is, we will
use the Z formula to calculate Z for our sample mean under
the assumption that the null hypothesis is true. The Z
formula is:
Note that the population mean is 18 under the null hypothesis,
and the standard error is 1, as we just calculated. We
then can calculate Z by using the obtained sample mean.
 Make a Decision:
 Classical Approach: We look the obtained
Z up in the Z table to find P, and compare it to the
Critical Z value. If the obtained P is less than alpha,
we reject the null hypothesis.
 Contemporary Approach: The computer calculates
the P value. We report it and let the reader/listener
decide.
The P value that indicates how unusual the obtained
sample's mean is, under the null hypothesis of no
effect. In other words, it indicates how often we
would obtain the results by chance alone. Using this
P value we decide whether to reject or retain the
null hypothesis.
Here's what happens for each experiment:
Experiment 1 
Experiment 2 
Sample Mean = 13
Z = (1318)/1 = 5.0
p < .0000
Reject Ho 
Sample Mean = 16.5
Z = (16.518)/1 = 1.5
p = .1339
Retain Ho 
Here is ViSta's report for these two experiments:
ViSta Applet
Report for Univariate Analysis of Experiment 1
Data. 

ViSta Applet
Report for Univariate Analysis of Experiment 2
Data. 

Next Topic: The TTest
