Biology Reference
In-Depth Information
B. Z-Test and Student's t-Test
The Z-test and Student's t-test 3 are used to compare the means of two
samples. Assume, as above, that we want to evaluate the claim
H 0 : m A m B (or, equivalently,
m A m B
0) versus the alternative
H
: m A <
m B (or, equivalently
m A m B < 0). In this case, the sampling
distribution of x B
x A , will be approximately normal, and we would use
either a Z-test or a t-test. We use a t-test when the variance of either
group is unknown [in which case the unknown variance(s) are estimated
by the sample variance(s)], leading to the need to use a t-distribution as
the sampling distribution. In fact, in most situations, the group variances
are unknown, necessitating the use of the t-test. We should note,
however, that for large samples, the resulting t-distribution is closely
approximated by a standard normal distribution. It is clear from Figure 4-2
(B) that the density graphs of the t-distribution and the standard
normal distribution in Figure 4-1(A) are very similar. If the degrees of
freedom in the t-distribution exceed 30, the two distribution densities are
virtually indistinguishable.
To evaluate the probability of making a type I error, we compute a test
statistic as we did above. We first present the simpler procedure of using
a Z-test.
Z-test: Let's go back to our corn example and use the data from Trial 2 to
test the one-sided alternative hypothesis:
H: Average yield of corn B is superior to the average yield of corn A (that is,
m A <
m B ),
against the null hypothesis:
H 0 : Corn B has the same or inferior average yield to corn A (that is,
m A m B ).
To apply the Z-test, we need to have information about the population
variance. For purposes of the illustration, let's assume that we
know that the population variance of corn is equal to 0.16 (i.e., the
standard deviation is 0.4). If we use Eq. (4-3) with the d ata from
Trial 2, we find the empirical mean yi el d of corn A is x A
¼
2
:
44,
whil e the empirical mean of corn B is x B
84. The variance of x A
and x B is equal to the sum of variances of their components divided by
10 (the numbe r of p l ants in each sample). If the null hypothesis is true,
the difference x B
¼
2
:
x A will have a normal distribution, with mean
m ¼ m B m A
0 and variance 0.032 (equal to (0.16
þ
0.16)/10). Thus,
the difference x B
x A has a normal distribution with parameters
m
p
0
0 and standard deviation
s ¼
:
032
¼
0
:
1789
:
Following Figure 4-4(B),
3. The t-test was developed by W. S. Gossett (1876-1937), who worked in Dublin,
Ireland, at the Guinness Brewery and published under the pen name Student.
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