Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 9.1 | The t Statistic: An Alternative to z 269
3. To quantify our inferences about the population, we compare the obtained sample
mean (M) with the hypothesized population mean (μ) by computing a z-score test
statistic.
z 5 M 2m
s M
obtained difference between data and hypothesis
5
standard distance between M and m
The goal of the hypothesis test is to determine whether the obtained difference between
the data and the hypothesis is significantly greater than would be expected by chance.
When the z-scores form a normal distribution, we are able to use the unit normal table
(Appendix B) to find the critical region for the hypothesis test.
■ The Problem with z-Scores
The shortcoming of using a z-score for hypothesis testing is that the z-score formula requires
more information than is usually available. Specifically, a z-score requires that we know
the value of the population standard deviation (or variance), which is needed to compute
the standard error. In most situations, however, the standard deviation for the population is
not known. In fact, the whole reason for conducting a hypothesis test is to gain knowledge
about an unknown population. This situation appears to create a paradox: You want to use a
z-score to find out about an unknown population, but you must know about the population
before you can compute a z-score. Fortunately, there is a relatively simple solution to this
problem. When the variance (or standard deviation) for the population is not known, we use
the corresponding sample value in its place.
■ Introducing the t Statistic
In Chapter 4, the sample variance was developed specifically to provide an unbiased estimate
of the corresponding population variance. Recall that the formulas for sample variance
and sample standard deviation are as follows:
The concept of degrees
of freedom, df 5 n 2 1,
was introduced in
Chapter 4 (page 115)
and is discussed later in
this chapter (page 271).
sample variance 5 s 2 5
SS
n 2 1 5 SS
df
sample standard deviation 5 s 5Î SS
n 2 1
Î 5 SS
df
Using the sample values, we can now estimate the standard error. Recall from Chapters 7
and 8 that the value of the standard error can be computed using either standard deviation
or variance:
standard error 5s M
5 s Ïn
or s M
5Î s2
n
Now we estimate the standard error by simply substituting the sample variance or standard
deviation in place of the unknown population value:
estimated standard error 5 s M
5
s
Ïn
or s M
5Î s2
n
(9.1)
Notice that the symbol for the estimated standard error of M is s M
instead of σ M
, indicating
that the estimated value is computed from sample data rather than from the actual population
parameter.