In statistical hypotheses testing procedure, an important step is to determine whether to reject the null hypothesis. The step is to compute/find the critical values and rejection region.
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Rejection Region and Critical Values
A rejection region for a hypothesis test is the range of values for the standardized test statistic which would lead us to decide whether to reject the null hypothesis. The Critical values for a hypothesis test are the z-scores which separate the rejection region(s) from the non-rejection region (also called the acceptance region of $H_0$). The critical values will be denoted by $Z_0$.
The rejection region for a test is determined by the type of test (left-tailed, right-tailed, or two-tailed) and the level of significance (denoted by $\alpha$) for the test. For a left-tailed test, the rejection region is a region in the left tail of the normal distribution, for a right-tailed test, it is in the right tail, and for a two-tailed test, there are two equal rejection regions in either tail.
Once we establish the critical values and rejection region, if the standardized test statistics for a sample data set fall in the region of rejection, the null hypothesis is rejected.
Examples: Critical Values and Rejection Region
Example 1: A university claims that the average SAT score for its incoming freshmen is 1080. A sample of 56 freshmen at the university is drawn and the average SAT score is found to be $\overline{x}=1044$ with a sample standard deviation of $s=94.7$ points.
In the above SAT example, the test is two-tailed, so the rejection region will be the two tails at either end of the normal distribution. If we again want $\alpha=0.05$, then the area under the curve in both rejection regions together should be 0.05. For this purpose, we will look up $\frac{\alpha}{2}=0.025$ in the standard normal table to get critical values of $Z_0 = \pm 1.96$. The rejection region thus consists of $Z \le 1.96$ and $Z\ge 1.96$. Since the standardized test statistic $Z=-2.85$ falls in the region, the university’s claim of $\mu = 1080$ would be rejected in this case.
Example 2: Consider a left-tailed Z test. For a 0.05 level of significance, the rejection region would be the values in the lowest 5% of the standard normal distribution (5% lowest area under the normal curve). In this case, the critical value (the corresponding) Z-score will be $-1.645$. So the critical value $Z_0$ will be $-1.645$ and the rejection region will be $Z\le -1.645$.
Note that for the case of right-tailed the rejection region would be the values in the highest 5% of the standard normal distribution table. The Z-score will be $1.645$ and the rejection region will be $Z\ge 1.645$.
Exercise: Critical Values and Rejection Region
- Find the critical values and rejection regions(s) for the standardized Z-test of the following:
- A right-tailed test with $\alpha = 0.05$
- A left-tailed test with $\alpha = 0.01$
- A two-tailed test with $\alpha = 0.10$
- A right-tailed test with $\alpha = 0.02$
- Mercury levels in fish are considered dangerous to people if they exceed 0.5mg mercury per kilogram of meat. A sample of 50 tuna is collected, and the mean level of mercury in these 50 fishes is 0.6m/kg, with a standard deviation of 0.2mg/kg. A health warning will be issued if the claim that the mean exceeds 0.5mg/kg can be supported at the $\alpha=0.10$ level of significance. Determine the null and alternative hypotheses in this case, the type of the test, the critical value(s), and the rejection region. Find the standardized test statistics for the information given in the exercise. Should the health warning be issued?
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