Apr 25’10
HYPOTHESIS TESTING:
ILLUSTRATIVE
#15b
EXAMPLES
INVOLVING
t
DISTRIBUTION
As we have already noted, the population variance is usually
un
known
in actual
situations involving statistical inference about a population mean. When sampling is from
a normally (or approximately normally) distributed population with an
unknown
variance
, the
relevant test statistic for testing a null hypothesis
H
0
is
_
_
t = (x – µ
0
)/(s/√n)
(15b.1)
_
where
s
is a
sample
(not population!) deviation,
n
is a sample size,
x
is a sample mean,
and
µ
0
is the hypothesized value of the mean. When
H
0
is true, the statistic (15b.1) is
distributed as Student’s
t
with
n – 1
degrees of freedom. The hypothesis testing procedure
resembles the tenstep procedure that have been described in the handouts #15, 15a for the
situation when a p
opulation va
riance is known, but now we deal with
t
statistic (15b.1) instead of
z
statistic
z =
(x – µ
0
)/(σ/√n )
(15a.1). The following example illustrates the hypothesis testing
procedure involving the
t
statistic.
Example 1c
(problem 7.2.13 on pg. 234 of the textbook)
Can we conclude that the mean
maximum voluntary ventilation for apparently healthy college seniors in not
110
liters
per minute?
Let
α
= 0.01 .
A sample of
20
individuals yields the following values:
132, 33, 91, 108, 67, 169, 54, 203, 190, 133, 96, 30, 187, 21, 63, 166, 84, 110, 157, 138
Solution
. Researchers can conclude that the mean maximum voluntary ventilation is different
from
110
liters if they can reject the
null hypothesis that the mean
maximum voluntary
ventilation is equal to 110 liters
. Let us follow the tenstep decisionmaking procedure of
two

sided
hypothesis test which is similar to the procedure described earlier in handouts #15, 15a.
1. Data
. The available data on the values of
maximum voluntary ventilation
of
20
individuals
randomly selected from the population of interest allows us to compute a sample
mean of
111.6
liters and a sample deviation of
56.30313
liters [use the MS Excel functions =
AVERAGE()
and =
STDEV()
]
2. Assumptions
. It is assumed that the population in question is approximately normally
distributed with an
unknown
variance.
3. Hypotheses
. The null hypothesis
H
0
to be tested is that the mean age of the population is equal
to
110
liters. The alternative hypothesis
H
A
is that the mean age of the population is
not
equal to
110
liters. Symbolically these hypotheses may be written as
H
0
: µ = µ
0
and
H
A
: µ ≠ µ
0
or