Collect the sample data and compute the value of
the test statistic
◦
In the Valentine Day case, the value of the test statistic
was calculated to be z =
–
1.00
5.
Calculate the p-value by corresponding to the test
statistic value
◦
In the Valentine Day case, the area under the standard
normal curve in the left-hand tail to the left of the test
statistic value z =
–
1.00
◦
The area is 0.1587
◦
The p-value is 0.1587 · 2 = 0.3174
LO9-3

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9-52
Steps Using a p-value to Test a
“Not Equal To” Alternative
Continued
5.
Continued
◦
That is, if H
0
is true, probability is 0.3174 of obtaining a
sample whose mean is at least as extreme as 326
◦
This probability is not so low as to be evidence that
H0
is
false and should be rejected
6.
Reject H
0
if the p-value is less than a
◦
In the Valentine Day case,
α
was 0.05
◦
Calculated p-value of 0.3174 is greater than
α
This implies that the test statistic z =
–
1.00 is greater than
the rejection point
–
z
0.025
=
–
1.96
◦
Therefore do not reject H
0
at the
α
= 0.05 significance
level
LO9-3

9-53
9.3 t Tests about a Population Mean:
σ
Unknown
Assume the population being sampled is
normally distributed, or sample size n is
large (30 or more)
The population standard deviation
σ
is
unknown, as is the usual situation
◦
If the population standard deviation
σ
is
unknown, then it will have to estimated from the
sample standard deviation
Under these two conditions, have to use the
t distribution to test hypotheses
LO9-4: Use critical
values and p-values to
perform a t test about a
population mean when
σ
is unknown.

9-54
Defining the t Statistic:
σ
Unknown
Let
x
be the mean of a sample of size n with
standard deviation s; calculate the standard error
SE = s/√n.
μ
0
is the claimed value of the population mean
Define a new test statistic
If the population being sampled is normal or n is
large, and s is used to estimate
σ, then …
The sampling distribution of the t statistic is a t
distribution with n
–
1 degrees of freedom
s
n
x
n
s
x
SE
x
t
)
(
0
0
0
LO9-4

9-55
t Tests about a Population Mean:
σ
Unknown
Alternative
Reject H
0
if:
p-value
H
a
:
μ
>
μ
0
t > t
Area under t distribution to
right of t
H
a
:
μ
<
μ
0
t <
–
t
Area under t distribution to
left of
–
t
H
a
:
μ
μ
0
|t| > t
/2
*
Twice area under t
distribution to right of |t|
t
α
,
t
α
/2
, and p-values are based on
n
–
1 degrees of freedom
(for a sample of size
n
)
* either
t
>
t
α
/2
or
t
<
–
t
α
/2
LO9-4

9-56
9.4 z Tests about a Population
Proportion
Alternative
Reject H
0
if:
p-value
H
a
: p > p
0
z > z
Area under t distribution to
right of z
H
a
: p < p
0
z <
–
z
Area under t distribution to
left of
–
z
H
a
: p
p
0
|z| > z
/2
*
Twice area under t
distribution to right of |z|
Where the test statistics is
* either
z
>
z
α
/2
or
z
<
–
z
α
/2
n
p
p
p
p
ˆ
z
0
0
0
1
LO9-5: Use critical
values and p-values to
perform a large sample
z test about a
population proportion.

9-57
9.5 Type II Error Probabilities and
Sample Size Determination
(Optional)
Want the probability
β
of not rejecting a false null
hypothesis
◦
Want the probability
β
of committing a Type II error
1 -
β
is called the power of the test
Assume that the sampled population is normally
distributed, or that a large sample is taken
Test…
◦
H
0
: μ = μ
0
vs
◦
H
a
: μ < μ
0
or H
a
: μ > μ
0
or H
a
: µ ≠ µ
0
Want to make the probability of a Type I error
equal to
α