3.
so the pavement will not be used unless the null hypothesis is rejected
4. 5.
A test with significance level .05 rejects H when a
lower-tailed test.
6.
With , and
,
7.
Since , H
cannot be rejected. We do not have compelling evi- dence for concluding that
; use of the pavement is not justified. ■
Determination of b and the necessary sample size for these large-sample tests can be based either on specifying a plausible value of s and using the case I formu-
las even though s is used in the test or on using the methodology to be introduced shortly in connection with case III.
Case III: A Normal Population Distribution
When n is small, the Central Limit Theorem CLT can no longer be invoked to jus- tify the use of a large-sample test. We faced this same difficulty in obtaining a small-
sample confidence interval CI for m in Chapter 7. Our approach here will be the same one used there: We will assume that the population distribution is at least
approximately normal and describe test procedures whose validity rests on this assumption. If an investigator has good reason to believe that the population distri-
bution is quite nonnormal, a distribution-free test from Chapter 15 can be used. Alternatively, a statistician can be consulted regarding procedures valid for specific
families of population distributions other than the normal family. Or a bootstrap pro- cedure can be developed.
The key result on which tests for a normal population mean are based was used in Chapter 7 to derive the one-sample t CI: If
is a random sample from a normal distribution, the standardized variable
has a t distribution with degrees of freedom df. Consider testing
against by using the test statistic
. That is, the test statistic results from standardizing under the assumption that H
is true using the estimated standard deviation of , rather than
. When H is true, the
test statistic has a t distribution with df. Knowledge of the test statistic’s dis-
tribution when H is true the “null distribution” allows us to construct a rejection
region for which the type I error probability is controlled at the desired level. In par- ticular, use of the upper-tail t critical value
to specify the rejection region implies that
The test statistic is really the same here as in the large-sample case but is la- beled T to emphasize that its null distribution is a t distribution with
df rather n 2
1 5 a
5 PT t
a ,n21
when T has a t distribution with n 2 1 df Ptype I error 5 PH
is rejected when it is true t t
a ,n21
t
a ,n21
n 2 1
s 1n
X S
1n, X
T 5 X 2 m
S 1n H
a
: m . m H
: m 5 m n 2
1 T 5
X 2 m S
1n X
1
, X
2
, c
, X
n
m , 30
2 .73 . 21.645
z 5 28.76 2 30
12.2647 152 5
2 1.24
1.701 5 2.73
s 5 12.2647
n 5 52, x 5 28.76
z 2 1.645
z 5 x 2
30 s
1n H
a
: m , 30
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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.