Region of Acceptance
In this lesson, we describe how to find the
region of acceptance for a hypothesis test.
OneTailed and TwoTailed Hypothesis Tests
The steps taken to define the region of acceptance will vary,
depending on whether the
null hypothesis and the
alternative hypothesis
call for one or twotailed hypothesis tests. So we begin with a
brief review.
The table below shows three sets of hypotheses. Each makes a statement about how
the population mean μ is related to a specified
value M. (In the table, the symbol ≠ means " not equal to ".)
Set

Null hypothesis 
Alternative hypothesis 
Number of tails 
1

μ = M 
μ ≠ M 
2 
2

μ > M 
μ < M 
1 
3

μ < M 
μ > M 
1 
The first set of hypotheses (Set 1) is an example of a
twotailed test, since an extreme value on either side of the
sampling distribution would cause a researcher to reject the null
hypothesis. The other two sets of hypotheses (Sets 2 and 3) are
onetailed tests, since an extreme value on only one side of the
sampling distribution would cause a researcher to reject the null hypothesis.
How to Find the Region of Acceptance
We define the region of acceptance in such a way that the chance of making a
Type I error
is equal to the
significance level. Here is how that is done.
 Define a test statistic. Here, the test statistic is the sample measure
used to estimate the population parameter that appears in the
null hypothesis. For example, suppose the null hypothesis is
H_{0}: μ = M
The test statistic, used to estimate M, would be m.
If M were a population mean, m would be the
sample mean; if M were a population proportion, m
would be the sample proportion; if M were a difference
between population means, m would be the difference between
sample means; and so on.

Given the significance level α , find the upper
limit (UL) of the region of acceptance. There are three possibilities,
depending on the form of the null hypothesis.

If the null hypothesis is μ < M: The upper
limit of the region of acceptance will be equal to the value for which the
cumulative probability of the
sampling
distribution is equal to one minus the
significance level. That is, P( m < UL ) =
1  α .

If the null hypothesis is μ = M: The upper limit of
the region of acceptance will be equal to the value for which the cumulative
probability of the sampling distribution is equal to one minus the significance
level divided by 2. That is, P( m < UL ) =
1  α/2 .

If the null hypothesis is μ > M: The upper
limit of the region of acceptance is equal to plus infinity,
unless the test statistic were a proportion or a percentage.
The upper limit is 1 for a proportion, and 100 for a percentage.

In a similar way, we find the lower limit (LL) of the range of acceptance.
Again, there are three possibilities, depending on the form of the null
hypothesis.

If the null hypothesis is μ < M: The lower
limit of the region of acceptance is equal to minus infinity,
unless the test statistic is a proportion or a percentage.
The lower limit for a proportion or a percentage is zero.

If the null hypothesis is μ = M: The lower limit of
the region of acceptance will be equal to the value for which the cumulative
probability of the sampling distribution is equal to the significance level
divided by 2. That is, P( m < LL ) = α/2 .

If the null hypothesis is μ > M: The lower
limit of the region of acceptance will be equal to the value for which the
cumulative probability of the sampling distribution is equal to the
significance level. That is, P( m < LL ) = α .
The region of acceptance is defined by the range between LL and UL.
Test Your Understanding
In this section, two hypothesis testing examples illustrate how to define the
region of acceptance. The first problem shows a twotailed test with
a mean score; and the second problem, a onetailed test with
a proportion.
Sample Planning Wizard
As you probably noticed, defining the region of acceptance can be complex and
timeconsuming. Stat Trek's Sample Planning Wizard can do the same job quickly,
easily, and errorfree.The Wizard is easy to use, and it
is free. You can find the Sample Planning Wizard in Stat Trek's
main menu under the Stat Tools tab. Or you can tap the button below.
Sample Planning Wizard
Problem 1
An inventor has developed a new, energyefficient lawn mower engine. He
claims that the engine will run continuously for 5 hours (300 minutes)
on a single gallon of regular gasoline. Suppose a random sample
of 50 engines is tested. The engines run for an average of 295
minutes, with a standard deviation of 20 minutes.
Consider the null hypothesis that the mean run time is 300 minutes
against the alternative hypothesis that the mean run time is not
300 minutes. Use a 0.05 level of significance. Find the region of
acceptance. Based on the region of acceptance, would you reject
the null hypothesis?
Solution: The process of defining a region of acceptance to test
a hypothesis takes four steps. We work through those steps below:

Formulate hypotheses. The first step is to state the null
hypothesis and an alternative hypothesis.
Null hypothesis: μ = 300 minutes
Alternative hypothesis: μ ≠ 300 minutes
Note that these hypotheses constitute a twotailed test. The null hypothesis
will be rejected if the sample mean is too big or if it is too small.

Identify the test statistic. In this example, the test
statistic is the mean run time of the 50 engines in the sample  295 minutes.

Define the region of acceptance. To define the
region of acceptance, we need to understand the
sampling distribution of the test statistic. And we need to
derive some probabilities. Those points are covered below.
Thus, we have determined that the region of acceptance is defined by the values
between 294.45 and 305.55.

Accept or reject the null hypothesis. The sample mean in this
example was 295 minutes. This value falls within the region of acceptance.
Therefore, we cannot reject the null hypothesis that a new engine runs for 300
minutes on a gallon of gasoline.
Problem 2
Suppose the CEO of a large software company
claims that at least 80 percent of the company's
1,000,000 customers are very satisfied. A survey of 100 randomly
sampled customers finds that 73 percent are very satisfied.
To test the CEO's hypothesis, find the region of acceptance.
Assume a significance level of 0.05.
Solution: The process of defining a region of acceptance to test
a hypothesis takes four steps. We work through those steps below:

Formulate hypotheses. The first step is to state the null
hypothesis and an alternative hypothesis.
Null hypothesis: P > 0.80
Alternative hypothesis: P < 0.80
Note that these hypotheses constitute a onetailed test. The null hypothesis
will be rejected if the sample proportion is too small.

Identify the test statistic. In this example, the test
statistic is the proportion of sampled customers who say they are very
satisfied; i.e., 0.73.

Define the region of acceptance. To define the
region of acceptance, we need to understand the
sampling distribution of the test statistic. And we need to
derive some probabilities. Those points are covered below.

Specify the sampling distribution. Since
the sample size is large (greater than or equal to 40),
we assume that the sampling distribution of the proportion
is normal, based on the central limit theorem.

Define the mean of the sampling distribution.
We assume that the mean of the sampling distribution is equal
to the hypothesized population proportion, which appears in
the null hypothesis  0.80.

Compute the standard deviation of the sampling
distribution .
Here the standard deviation of the sampling distribution
s_{p}
is:
s_{p} = sqrt[ P' * ( 1  P' ) / n ] * ( N  n
) / ( N  1 ) ]
s_{p} = sqrt[ 0.8 * 0.2 / 100 ] * ( 999,900)
/ (999,999 ) ] = sqrt(0.0016) * 0.9999 = 0.04
where P' is the test value specified in the null hypothesis, n is the sample
size, and N is the population size.

Find the lower limit of the region of acceptance. Given a
onetailed hypothesis, the lower limit (LL) will be equal to the value for
which the cumulative probability of the sampling distribution is equal to the
significance level. That is, P( x < LL ) = α = 0.05. To find this lower limit, we use the
Normal Distribution Calculator. We input the following entries into the
calculator: cumulative probability = 0.05, and mean = 0.80. The calculator
tells us that the lower limit is 0.734, given those inputs.

Find the upper limit of the region of acceptance. Since we
have a onetailed hypothesis in which the null hypothesis states that the
satisfaction level is 0.80 or more, any proportion greater than 0.80 is
consistent with the null hypothesis. Therefore, the upper limit is 1.0 (since
the highest possible proportion is 1.0).
Thus, we have determined that the region of acceptance is defined by the values
between 0.734 and 1.00.

Accept or reject the null hypothesis. The sample proportion in
this example was a satisfaction level of 0.73. This value falls outside the
region of acceptance. Therefore, we reject the null hypothesis that 80 percent
of the utility's customers are very satisfied.