The
P
value of the test is the
associated with the critical region. Any smaller
value for
expands the critical region and the test fails to reject the null
hypothesis when
.
The
P
value is easy to compute after the test statistic is observed.
3
.
51
x
3
.
51
x
038
.
0
962
.
0
1
)
08
.
2
08
.
2
(
1
16
/
5
.
2
50
3
.
51
16
/
5
.
2
50
7
.
48
1
)
3
.
51
7
.
48
(
1
value

Z
P
Z
P
X
P
P
Figure 96
P
value is area of shaded region when
.
3
.
51
x
91 Hypothesis Testing
91.5 Connection between Hypothesis Tests and
Confidence Intervals
A close relationship exists between the test of a hypothesis for
, and
the confidence interval for
.
If [
l
,
u
] is a 100(1
)
confidence interval for the parameter
, the
test of size
of the hypothesis
H
0
:
0
H
1
:
0
will lead to rejection of
H
0
if and only if
0
is
not
in the 100(1
)
CI [
l
,
u
].
91 Hypothesis Testing
•
CI provides
a range of likely values
for
𝜇
at a stated confidence level
•
Hypothesis testing is a framework for displaying the risk levels such as
the
pvalue
associated with a specific decision.
91.5 Connection between Hypothesis Tests and
Confidence Intervals
Recall of the terms
•
Type1 error
•
Type2 error
•
Power of the test
•
Significance level
•
Size of a test
•
pvalue
•
Confidence level
91 Hypothesis Testing
91.6 General Procedure for Hypothesis Tests
1. Identify the parameter of interest.
2. Formulate the null hypothesis,
H
0
.
3. Specify an appropriate alternative hypothesis, H
1
.
4. Choose a significance level,
.
5. Determine an appropriate test statistic.
6. State the rejection criteria for the statistic.
7. Compute necessary sample quantities for calculating the test statistic.
8. Draw appropriate conclusions.
92 Tests on the Mean of a Normal Distribution, Variance
Known
92.1 Hypothesis Tests on the Mean
Consider the twosided hypothesis test
The
test statistic
is
:
H
0
:
0
H
1
:
≠
0
n
X
Z
/
0
0
(91)
92 Tests on the Mean of a Normal Distribution, Variance
Known
92.1 Hypothesis Tests on the Mean
Reject
H
0
if the observed value of the test statistic
z
0
is either
:
z
0
>
z
/2
or
z
0
< 
z
/2
Fail to reject
H
0
if the observed value of the test
statistic
z
0
is

z
/2
<
z
0
<
z
/2
92 Tests on the Mean of a Normal Distribution, Variance Known
•
The
reference distribution
for this test is the standard normal
distribution. The test is usually called a
z
test
.
EXAMPLE 92
Propellant Burning Rate
Air crew escape systems are powered by a solid propellant. The
burning rate of this propellant is an important product characteristic.
Specifications require that the mean burning rate must be 50
centimeters per second and the standard deviation is
2
centimeters per second. The significance level of
0.05 and a
random sample of
n
25 has a sample average burning rate of
centimeters per second. Draw conclusions.
The sevenstep procedure is
1.
Parameter of interest:
The parameter of interest is
, the mean
burning rate.
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 Summer '20
 Normal Distribution, Null hypothesis, Statistical hypothesis testing