Chapter 7. Hypothesis Testing: Introduction to Hypothesis Testing (Critical Region Approach
Determining the Critical Region
Once the hypotheses have been formulated, the next step is to determine the criteria for a decision. Specifically, you need to determine what values of the sample statistic will lead to the rejection of the null hypothesis. Because a sample provides an incomplete picture of a population, some discrepancy between a sample statistic and its corresponding population parameter is to be expected.
How much discrepancy is reasonable to expect if the null hypothesis is true can be derived from the sampling distribution of the sample statistic. If the null hypothesis is true, it is likely that the sample statistic will be relatively close in value to the mean of the sampling distribution.
As the difference between a sample statistic and the mean of the hypothesized sampling distribution increases, the probability of the null hypothesis being true decreases. If a sample produces a statistic that is extremely unlikely to occur given that the null hypothesis is true, this should lead to the rejection of the null hypothesis.
In order to formalize what constitutes as an extremely unlikely result, the significance level of the test needs to be set.
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Alpha level
Definition
The level of significance or alpha level of a statistical test is the probability threshold that determines how unlikely a sample statistic has to be in order for the null hypothesis to be rejected.
Notation
#\alpha#
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For example, setting an alpha level of #\alpha = 0.05# means that if you find a sample statistic that has a #5\%# or less chance of occurring if the null hypothesis is true, the null hypothesis should be rejected.
Decreasing the alpha level of a hypothesis test means that stronger evidence is required in order to reject the null hypothesis. Once the alpha level of the test has been set, the next step is to determine the critical region of the sampling distribution.
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Critical region
Definition
The critical region of a sampling distribution contains the values for the sample statistic which are so unlikely to occur if the null hypothesis is true, that finding one of these values will lead to the rejection of the null hypothesis.
The boundaries of the critical region are called critical values.
Graphical representation
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When conducting a #z#-test, the Standard Normal Table is used to determine the boundaries of the critical region. For a #z#-test, the exact location of the boundaries is determined entirely by the alpha level of the test. The table below displays the most commonly used significance levels and the corresponding critical values.
\[\begin{array}{c|c}
\alpha&\text{Critical }z \text{ values}\\
\hline
0.10&\pm 1.645\\
0.05&\pm 1.96\\
0.01&\pm 2.58\\
\end{array}\]
3. Critical Region
If the null hypothesis is true, the Summer Course should have no effect on the students' grades and the population of students that participate in the Summer Course will be identical to that of the original population of students; that is, a normal distribution with #\mu = 6.5# and #\sigma=1#.
Next, it is needed to consider all possible outcomes for a sample of #n =100# students. This is the distribution of sample means for #n=100#. If the null hypothesis is true, then the distribution of sample means will have the following properties:
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- #\mu_{\bar{X}}= \mu_0 = 6.5#
- #\sigma_{\bar{X}} = \cfrac{\sigma}{\sqrt{n}}=\cfrac{1}{\sqrt{100}} = 0.1#
Finally, the distribution of sample means is used to determine the critical region of the test. The university decides to set the alpha level of the test at #\alpha = 0.05#, meaning the critical region consists of the extreme #5\%# of the sampling distribution.
The critical values for a #Z#-test with #\alpha =0.05# are #Z = \pm 1.96#. This means that finding a #Z#-statistic less than #-1.96# or greater than #1.96# should lead to the rejection of the null hypothesis.
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