Download z-test handout - Oklahoma State University

James W. Grice, Ph.D.
Oklahoma State University
Spring,2015
z-test for means
Description
Use this inferential statistical test when you wish to compare two population means, μ1 and μ2, and
you know one of the population means and its accompanying population standard deviation (or
variance). In psychology you will likely use the z-test for means when you are working with a
standardized test, such as an intelligence test or an ability test (e.g., GRE, ACT). These standardized
tests have well known normed values; for instance, most intelligence tests are normed to a mean of
100 and standard deviation of 15. These values are often treated as if they are population
parameters, μ and σ, respectively.
Assumptions
 Random sampling
 σ1 =σ2 Population standard deviations or variances are homogeneous
 Population distributions are normal (although Central Limit Theorem helps)
 Observations are independent (goes with random sampling)
 Variable for which means are computed is continuous
 Ho is true
Hypotheses
Ho: μ1 = μ2
HA: μ1 > μ2
or μ1 < μ2
Sampling Distribution and Critical Values
zcrit is taken from the standard normal distribution.
You can use +/- 1.96 when alpha (pcrit) = .05; +/- 2.58 when alpha = .01
Test Statistic
Compute and compare the following z-score to the critical z-value to determine if the result is
statistically significant; that is, determine if you reject the null hypothesis.
z obs 
x

n
If you reject the null hypothesis, be sure to determine which alternative hypothesis is supported
by comparing your computed mean with the comparison mean. State your conclusion in words;
for example, “It appears that the population mean for long-term alcoholics is lower than the
population norm mean.”
Effect Size
The ES provides the basis for judging the magnitude of the effect. Ideally, you can simply examine
the difference between your computed mean and the comparison mean, µ, and comment on the
meaning of the magnitude. Oftentimes in psychology, however, we don’t understand our scales
sufficiently enough to make such comments. Consequently, we “punt” and use a standardized
effect size and Cohen’s conventions to interpret the difference as “small, medium, or large.”
d
x

Cohen’s conventions: .2 small; .5 medium; .8 large
Confidence Interval
The CI provides the basis for judging the precision to which you have estimated the population
parameter, µ, in your study.
  
x   z crit 

 n
The width of the interval can be judged as “narrow, middling, or wide” precision using either the
possible scale range or the standard deviation, σ. If you use the standard deviation, then consider
the confidence interval as “middling” if its width is approximately equal to the standard deviation.
APA Style Example
A z-test for means was computed comparing the mean for the fifteen long-term alcoholics (M =
92.35, SD = 11.17) to the normed mean value of the Wechsler Adult Intelligence Scale (μ = 100,
σ= 15). The result was statistically significant (z = -1.98, p < .048, two-tailed), and long-term
alcoholics were estimated to have a lower average level of intelligence than the general
population. The magnitude of this effect, however, was not very large (7.65 point difference, d =
.51), and the population mean for the long-term alcoholics was not estimated precisely (CI.95:
84.76, 99.94).