is the calculated sample mean, then
the standard normal test statistic is:Hypothesis testing forms an important part of statistical inference. As stated previously, statistical inference refers to the process of estimating results for the population based on measurements from a sample. In the next sections, statistical inference for a single sample is discussed briefly.
This section is divided into the following subsections:
The test statistic used in this case is based on the standard normal
distribution. If is the calculated sample mean, then
the standard normal test statistic is:
(9)
where 
is the hypothesized population mean,
is the population standard deviation
and 
is the sample size.
Example 3.2
Assume that an analyst wants to know if the mean of a population, 
, is 100. The population variance,
, is known to be 25. The hypothesis
test may be conducted as follows:
It is a clear that this is a two-sided hypothesis. Thus the critical region will lie in both of the tails of the probability distribution.
. The significance level determines
the critical values of the test statistic. Here the test statistic is
based on the standard normal distribution. For the two-sided hypothesis
these values are obtained as: [Note]
and
These values and the critical regions are shown in Figure 3.12. The analyst would fail to reject 
if the test statistic, 
, is such that:
or
|
Figure 3.12: Critical values and rejection region for Example 3.2 marked on the standard normal distribution. |
, is 25 and the sample mean is obtained
as 
.
Since this value does not lie in the acceptance region 
, we reject 
at a significance level of 0.05.
In the previous example the null hypothesis was rejected at a significance
level of 0.05. This statement does not provide information as to how far
out the test statistic was into the critical region. At times it is necessary
to know if the test statistic was just into the critical region or was
far out into the region. This information can be provided by using the
value.
The 
value is the probability of occurrence
of the values of the test statistic that are either equal to the one obtained
from the sample or more unfavorable to 
than the one obtained from the sample.
It is the lowest significance level that would lead to the rejection of
the null hypothesis, 
, at the given value of the test statistic.
The value of the test statistic is referred to as significant when 
is rejected. The 
value is the smallest 
at which the statistic is significant
and 
is rejected.
For instance, in the previous example the test statistic was obtained
as 
. Values that are more unfavorable
to 
in this case are values greater than
3. Then the required probability is the probability of getting a test
statistic value either equal to or greater than 3 (this is abbreviated
as 
). This probability is shown in Figure
3.13 as the dark shaded area on the right tail of the distribution and
is equal to 0.0013 or 0.13% (i.e. 
). Since this is a two-sided test
the 
value is:
Therefore, the smallest 
(corresponding to the test static
value of 3) that would lead to the rejection of 
is 0.0026.
|
Figure 3.13: P value for Example 3.2. |
When the variance, 
, of a population (that can be assumed
to be normally distributed) is unknown the sample variance, 
, is used in its place in the calculation
of the test statistic. The test statistic used in this case is based on
the 
distribution and is obtained using
the following relation:
(10)
The test statistic follows the 
distribution with 
degrees of freedom.
Example 3.3
Assume that an analyst wants to know if the mean of a population, 
, is less than 50 at a significance
level of 0.05. A random sample drawn from the population gives the sample
mean, 
, as 47.7 and the sample standard deviation,
, as 5. The sample size, 
, is 25. The hypothesis test may be
conducted as follows:
It is clear that this is a one-sided hypothesis. Here the critical region will lie in the left tail of the probability distribution.
. Here, the test statistic is based
on the 
distribution. Thus, for the one-sided
hypothesis the critical value is obtained as:
This value
and the critical regions are shown in Figure 3.14.
The analyst would fail to reject 
if the test statistic 
is such that:
, corresponding to the given sample
data is: 
Since 
is less than the critical value of
-1.7109, 
is rejected and it is concluded that
at a significance level of 0.05 the population mean is less than 50.
P value
In this case the 
value is the probability that the
test statistic is either less than or equal to 
(since values less than 
are unfavorable to 
). This probability is equal to 0.0152.
|
Figure 3.14: Critical value and rejection region for
Example 3.3 marked on the |
The test statistic used in this case is based on the Chi-Squared distribution.
If 
is the calculated sample variance
and 
the hypothesized population variance
then the Chi-Squared test statistic is:
(11)
The test statistic follows the Chi-Squared distribution with 
degrees of freedom.
Example 3.4
Assume that an analyst wants to know if the variance of a population
exceeds 1 at a significance level of 0.05. A random sample drawn from
the population gives the sample variance as 2. The sample size, 
, is 20. The hypothesis test may be
conducted as follows:
This is a one-sided hypothesis. Here the critical region will lie in the right tail of the probability distribution.
. Here, the test statistic is based
on the Chi-Squared distribution. Thus for the one-sided hypothesis the
critical value is obtained as:
This value and the critical regions are shown in Figure 3.15. The
analyst would fail to reject 
if the test statistic 
is such that:
|
Figure 3.15: Critical value and rejection region for Example 3.4 marked on the Chi-Squared distribution. |
corresponding to the given sample
data is:
Since 
is greater than the critical value
of 30.1435, 
is rejected and it is concluded that
at a significance level of 0.05 the population variance exceeds 1.
P value
In this case the 
value is the probability that the
test statistic is greater than or equal to 38 (since values greater than
38 are unfavorable to 
). This probability is determined to
be 0.0059.
See Also:
Statistical Inference for Two Samples
distribution.