2 the sample is small n 30 3 the value of the

2) The sample is small (n  30).
3) The value of the population standard deviation

is unknown.
4) The sample values come from a population
with a distribution that is approximately
normal.

Test Statistic
for a Student
t
-distribution
Critical Values
 Found in Table A-3
 Degrees of freedom (df) = n -1
 Critical t values to the left of the mean are
negative
t =
x -µ
x
s
n

Important Properties of the
Student t Distribution
1. The Student t distribution is different for different sample sizes (see
Figure 6-5 in Section 6-3).
2. The Student t distribution has the same general bell shape as the
normal distribution; its wider shape reflects the greater variability that
is expected with small samples.
3. The Student t distribution has a mean of t = 0 (just as the standard
normal distribution has a mean of z = 0).
4. The standard deviation of the Student t distribution varies with the
sample size and is greater than 1 (unlike the standard normal
distribution, which has a

 = 1).
5. As the sample size n gets larger, the Student t distribution get closer to
the normal distribution. For values of n > 30, the differences are so
small that we can use the critical z values instead of developing a much
larger table of critical t values. (The values in the bottom row of Table
A-3 are equal to the corresponding critical z values from the normal
distributions.)

Figure 7-11 Choosing between the Normal and Student
t- Distributions when Testing a Claim about a Population Mean µ
Is
n > 30
?
Is the
distribution of
the population essentially
normal ? (Use a
histogram.)
No
Yes
Yes
No
No
Is

known
?
Use normal distribution with
x - µ
x

/ n
Z 
(If

is unknown use s instead.)
Use nonparametric methods,
which don’t require a normal
distribution.
Use normal distribution with
x - µ
x

/ n
Z 
(This case is rare.)
Use the Student t distribution
with x - µ
x
s
/ n
t 

Start

The larger Student
t
critical value
shows that with a small sample,
the sample evidence must be more
extreme before we consider the
difference is significant.

A company manufacturing rockets claims to use an
average of 5500 lbs of rocket fuel for the first 15 seconds of
operation. A sample of 6 engines are fired and the mean fuel
consumption is 5690 lbs with a sample standard deviation of
250 lbs. Is the claim justified at the 5% level of significance?
2. Two tail t test, n < 30, unknown
population standard deviation
3. t critical for 5% for a two tail test with 5 d.f. is 2.015
862 1
6 250
5500 5690
4 .
n s
-μ x
. t 

 
5. Fail to reject H
O
, there is no evidence at the .05 level
that the average fuel consumption is different from µ =
5500 lbs
2.015 -2.015 1.862
1. H
O
: µ = 5500 H
A
: µ  5500

P
-Value Method
 Table A-3 includes only selected values of 

Specific
P
-values usually cannot be found
 Use Table to identify limits that contain the
P
-value
 Some calculators and computer programs
will find exact
P
-values

P-Value Method
of Testing Hypotheses
 very similar to traditional method
 key difference is the way in which we decide to
reject the null hypothesis
 approach finds the probability ( P -value) of getting
a result and rejects the null hypothesis if that
probability is very low