Example 8.9
than the standard normal z distribution. The rejection region for the t test differs from that for the z test only in that a t critical value
replaces the z critical value z
a
. Similar comments apply to alternatives for which a lower-tailed or two-tailed test is appropriate.
t
a ,n21
The One-Sample t Test
Null hypothesis: Test statistic value:
Alternative Hypothesis Rejection Region for a Level a Test
upper-tailed lower-tailed
either or two-tailed t 2t
a 2,n21
t t
a 2,n21
H
a
: m 2 m t 2t
a ,
n2 1
H
a
: m , m t t
a ,
n2 1
H
a
: m . m t 5
x 2 m s
1n H
: m 5 m
Glycerol is a major by-product of ethanol fermentation in wine production and con- tributes to the sweetness, body, and fullness of wines. The article “A Rapid and
Simple Method for Simultaneous Determination of Glycerol, Fructose, and Glucose in Wine” American J. of Enology and Viticulture, 2007: 279–283 includes
the following observations on glycerol concentration mgmL for samples of standard-quality uncertified white wines: 2.67, 4.62, 4.14, 3.81, 3.83. Suppose the
desired concentration value is 4. Does the sample data suggest that true average concentration is something other than the desired value? The accompanying normal
probability plot from Minitab provides strong support for assuming that the popu- lation distribution of glycerol concentration is normal. Let’s carry out a test of
appropriate hypotheses using the one-sample t test with a significance level of .05.
2.0 3.0
2.5 3.5
Glycerol conc Percent
4.5 4.0
5.5 Mean
StDev N
3.814 0.7185
5 RJ
P-Value 0.947
0.100
5.0 1
10 20
5 30
40 50
60 70
80 90
95 99
Figure 8.4 Normal probability plot for the data of Example 8.9
1. 2.
3. H
a
: m 2 4 H
: m 5 4 m 5
true average glycerol concentration
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
4. 5.
The inequality in H
a
implies that a two-tailed test is appropriate, which requires . Thus H
will be rejected if either or
.
6.
, from which ,
and the estimated standard error of the mean is
The test statistic value is then .
7.
Clearly does not lie in the rejection region for a significance level of
.05. It is still plausible that . The deviation of the sample mean 3.814 from
its expected value 4 when H is true can be attributed just to sampling variability
rather than to H being false.
The accompanying Minitab output from a request to perform a two-tailed one- sample t test shows identical calculated values to those just obtained. The fact
that the last number on output, the “P-value,” exceeds .05 and any other reason- able significance level implies that the null hypothesis can’t be rejected. This is
discussed in detail in Section 8.4.
Test of mu 4 vs not 4
Variable N
Mean StDev
SE Mean 95 CI
T P
glyc conc 5
3.814 0.718
0.321 2.922, 4.706
⫺0.58 0.594
■
B and Sample Size Determination
The calculation of b at the alternative value m⬘
in case I was carried out by expressing the rejection region in terms of e.g., and then subtracting m⬘ to standardize correctly. An equiva-
lent approach involves noting that when , the test statistic
still has a normal distribution with variance 1, but now the mean value of Z is given by
. That is, when , the test sta-
tistic still has a normal distribution though not the standard normal distribution. Because of this,
is an area under the normal curve corresponding to mean value
and variance 1. Both a and b involve working with nor- mally distributed variables.
The calculation of for the t test is much less straightforward. This is
because the distribution of the test statistic is quite compli-
cated when H is false and H
a
is true. Thus, for an upper-tailed test, determining involves integrating a very unpleasant density function. This must be done numeri-
cally. The results are summarized in graphs of b that appear in Appendix Table A.17. There are four sets of graphs, corresponding to one-tailed tests at level .05 and level
.01 and two-tailed tests at the same levels.
To understand how these graphs are used, note first that both b and the nec- essary sample size n in case I are functions not just of the absolute difference
but of . Suppose, for example, that
This departure from H will be much easier to detect smaller b when
, in which case m
and m⬘ are 5 population standard deviations apart, than when . The fact that b for the t test depends on d rather than just
is unfortunate, since to use the graphs one must have some idea of the true value of
s . A conservative large guess for s will yield a conservative large value of
and a conservative estimate of the sample size necessary for prescribed and .
b mr
a b
mr um
2 mr u s 5
10 s 5
2 um
2 mru 5 10. d 5
um 2 mr us
um 2 mru
b mr 5 PT , t
a ,n21
when m 5 mr rather than m T 5
X 2 m S 1n
b mr
mr 2 m s 1n
b mr
m 5 m r
mr 2 m s 1n
Z 5 X 2 m
s 1n m 5 m
r x m
1 z
a
s 1n
x m 5
4 t 5 2
.58 t 5
3.814 2 4.321 5 2.58 s
1n 5 .321. s 5
.718, x 5
3.814 gx
i
5 19.07, and
gx
i 2
5 74.7979
t 2 2.776
t 2.776
t
a 2,n21
5 t
025,4
5 2.776
t 5 x 2
4 s
1n
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Example 8.10
Once the alternative and value of are selected, d is calculated and its
value located on the horizontal axis of the relevant set of curves. The value of b is the height of the
df curve above the value of d visual interpolation is nec- essary if
is not a value for which the corresponding curve appears, as illus- trated in Figure 8.5.
