Hypothesis Testing
The decision making starts by identifying something of concern and then formulating two hypotheses about it.
Hypothesis is a statement that describes something is true.
One hypothesis is called null hypothesis,
, and the other is called
alternative hypothesis, 
.
Statistical Hypothesis Test is a process, which need test statistic, used for decision making about two hypotheses.
The possible results are as follows:
|
|
|
|
|
Fail to reject |
|
Type II error |
|
Reject |
Type I error |
|
Need to find sample region as a reject region (or called critical region).
Using 
to find reject sample region; using 
to find sample size needed (
is not used in the class
of Stat 281.)
Inference for One Population:
Population should have normal distribution for all the following cases.
Significant level for hypothesis
testing is and confident level is 
.
Inference for population mean 
In the class of Stat 281, the c
in the hypothesis is always zero.
1) Population
mean is unknown and population variance is known 
Use normal curve table. (It is also called Z test.)
a) 
vs 
(Two tail alternative
hypothesis)
Test statistic: 
and critical region is
or 
p-value for the
test is 
b) vs 
(left tail alternative
hypothesis)
Test statistic: and critical region is
p-value for the
test is 
c) vs 
(right tail
alternative hypothesis)
Test statistic: and critical region is
p-value for the
test is 
confident
interval for population mean :
2) Population mean is unknown and population variance is unknown
For sample size
, use T Table. (It is
also call T test.)
a) vs (Two tail alternative
hypothesis)
Test statistic: 
and critical region is
or 
p-value for the
test is 
b) vs (left tail alternative
hypothesis)
Test statistic: 
and critical region is
p-value for the
test is 
c) vs (right tail
alternative hypothesis)
Test statistic: and critical region is
p-value for the
test is 
confident
interval for population mean :
For sample size
, we can use Z test procedures.
If
is unknown, then it should be replaced by 
sample standard
deviation.
where
------ Formula (2.6) page 86
Inference for population variance
3) For
population variance
a) 
vs 
(Two tail alternative
hypothesis)
Test statistic: 
and critical region is
or 
p-value for the
test is 
. Where 
is evaluation of statistic.
b) vs 
(left tail alternative
hypothesis)
Test statistic: and critical region is
p-value for the
test is 
. Where is evaluation of statistic.
c) vs 
(right tail
alternative hypothesis)
Test statistic: and critical region is
p-value for the
test is 
. Where is evaluation of statistic
confident
interval for population variance
Inference for one population proportion
Let 
be population proportion which is unknown and 
is sample proportion
with sample size 
.
Assume that sample size is large enough.
(Thumb of
rule says 
)
a) 
vs 
(Two tail alternative
hypothesis)
Test statistic: 
and critical region is
or
p-value for the
test is
b) vs 
(left tail alternative
hypothesis)
Test statistic: and critical region is
p-value for the
test is
c) vs 
(right tail alternative
hypothesis)
Test statistic: and critical region is
p-value for the
test is
confident
interval for is:
Inference for Two Populations
Both populations should have normal distributions for all the following cases.
Significant level for hypothesis
testing is and confident level is. 
and
are values from normal curve table.
is sample mean of random sample of
size from the population which has 
. 
is sample mean of random sample of
size 
from the population which has 
Inference for two population means 
Independent test for two population means:
In the class of Stat281, c is always zero.
1) Population
means, are unknown and population variances are known 
. Using normal curve
table
a) 
vs 
(Two tail alternative
hypothesis)
Test statistic: 
and critical region is
or
p-value for the
test is 
b) vs 
(left tail alternative
hypothesis)
Test statistic: and critical region is
p-value for the
test is
c) vs 
(right tail
alternative hypothesis)
Test statistic: and critical region is
p-value for the
test is
confident
interval for population mean 
:
2) Population means are unknown and population variances are unknown
The population variances will be
replaced by the corresponding sample variances
, respectively.
i) For
sample size
, use T Table.
Usually, we call independent T test for mean from two populations. (It is also called independent T test.)
a) When both populations have
different variances. Follow all the
cases from 1) of Inference for Two Populations with the test statistic replaced
by 
. It means that the
variance 
is replaced by 
, and the variance 
is replaced by 
on the test statistic from 1).
The exact sampling distribution
for has T-distribution with a complicated degree of freedom. A simple approximation for the degree of
freedom, 
, is smaller value of 
and
, which gives a conservative result.
Therefore, use T-table with degree
of freedom which is the smaller value of
andto find 
or 
to replace and .
b) When both population variances
are equal ( 
The pool estimator 
will be used to
replace common variance. Then the test
statistic is 
which has sampling
distribution T distribution with degree of freedom 
Therefore, use T-table with degree
of freedom which is to find or to replace and.
ii) For sample size
, use normal curve table
Follow all the cases from 1) of
Inference for Two Populations with the test statistic
. It means that the
unknown variances will be replaced by sample variances accordingly.
Paired test for means from two
populations
Let
, 
be pair sample from experimental units.
For example, 
, is sample from experimental units before treatment applied and 
, is sample from experimental units after treatment applied. The main purpose is to compare the population
mean after treatment and population mean before treatment.
Let
, be treated as one sample situation.
a) When, use 
as test statistics,
where 
is sample standard deviation from sample, . Refer z-test for the
one population inference above.
b) When, use 
as test statistics,
where is sample standard deviation from sample, . Refer t-test for the
one population inference above. (This is
also called pair-wise T test.)
Inference for both population variances
Use F-distribution table. 
3)
For population variances
a) 
vs 
(Two tail alternative
hypothesis)
Test statistic: 
and critical region is
or 
p-value for the
test is 
b) vs 
(left tail alternative
hypothesis)
Test statistic: and critical region is
p-value for the
test is 
c) vs 
(right tail
alternative hypothesis)
Test statistic: and critical region is
p-value for the
test is 
confident
interval for 
Inference for proportions of two populations
Let 
be the first population proportion and 
be the second population
proportion.
Select
sample of size 
from first population
and sample proportion is 
Select
sample of size 
from second population
and sample proportion is 
Assume
that sample sizes and are large enough.
a) 
vs 
(Two tail alternative
hypothesis)
Test statistic: 
where
and
critical region is or
p-value for the
test is
b) vs 
(Left tail alternative
hypothesis)
Test statistic: where
and
critical region is 
p-value for the
test is
c) vs 
(Right tail
alternative hypothesis)
Test statistic: where
and
critical region is
p-value for the
test is
confident
interval for
is :
The following example is used for you to understand the necessary procedure needed for hypothesis testing.
Example. An automobile manufacturer
who wished to advertise that one of its models achieves 30 miles per gallon
(mpg) decides to carry out a fuel efficiency test. Six non-professional drivers
are selected, and each one drives a car from
Solution: 1) Set up hypothesis
(Claim: at
least 30 mpg) 
2) Select test statistic: Since sample size is 6 < 30, use t test.
Reject
region (critical region) 
3) Evaluate
test statistic value, 
, based on data on hand.
, 
4) Conclusion: Since 
, no sufficient evidence to reject the claim: at least 30 mpg
under significant level of 0.05
5) P-value: 
.
Since p-value > 0.05, no sufficient evidence to reject the claim
under significant level of 0.05.