# Region of Acceptance

In this lesson, we describe how to find the region of acceptance for a hypothesis test.

## One-Tailed and Two-Tailed 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 two-tailed 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 two-tailed 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 one-tailed 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

H0: 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 two-tailed test with a mean score; and the second problem, a one-tailed test with a proportion.

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Problem 1

An inventor has developed a new, energy-efficient 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 two-tailed 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.

• Specify the sampling distribution. Since the sample size is large (greater than or equal to 40), we assume that the sampling distribution of the mean 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 mean value that appears in the null hypothesis - 300 minutes.

• Compute the standard error of the sample mean. Here the standard error of the sample mean (sx) is:
sx = s * sqrt( 1/n )
sx = 20 * sqrt[ 1/50 ] = 2.83
where s is the sample standard deviation, n is the sample size, and N is the population size. (In this example, we assume that the population size is very large, so the finite population correction is equal to about one.)

• Find the lower limit of the region of acceptance. Given a two-tailed 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 divided by 2. That is, P( x < LL ) = α/2 = 0.05/2 = 0.025. To find this lower limit, we use the Normal Distribution Calculator. We input the following entries into the calculator: cumulative probability = 0.025, mean = 300, and standard deviation = 2.83. The calculator tells us that the lower limit is 294.45, given those inputs.

• Find the upper limit of the region of acceptance. Given a two-tailed hypothesis, the upper limit (UL) 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( x < UL ) = 1 - α/2 = 1 - 0.025 = 0.975. To find this upper limit, we use the Normal Distribution Calculator. We input the following entries into the calculator: cumulative probability = 0.975, mean = 300, and standard deviation = 2.83. The calculator tells us that the upper limit is 305.55, given those inputs.

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 one-tailed 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 sp   is:
sp = sqrt[ P' * ( 1 - P' ) / n ] * ( N - n ) / ( N - 1 ) ]
sp = 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 one-tailed 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 one-tailed 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.