Introduction to Types of Hypothesis Tests
In statistics, hypothesis tests are methods used to make inferences or draw conclusions about a population based on sample data. In this pose, we will discuss the Basic Types of Hypothesis Tests in Statistics. There are three basic types of hypothesis tests, namely (i) Left-Tailed Test, (ii) Right-Tailed Test, and (iii) Two-Tailed Test.
Note that I am not talking about Statistical tools used under specific conditions related to the data type and distribution. I am talking about the nature of the hypotheses being tested. Therefore, I will focus in this post on the area under the curve in the tails. In hypothesis testing, the distribution of the test’s rejection region can be characterized as either one-tailed or two-tailed. The one-tailed tests include both left- and right-tailed tests.
Table of Contents
Left-Tailed Test
The left-tailed tests are used when the null hypothesis is being tested in a claim that the population parameter at least ($\ge$) a given value. Note that the alternative hypothesis then claims that the parameter is less than (<) the value. For example,
A tire manufacturer claims that their tires last on average more than 35000 miles. If one thinks that the claim is false, then one would write the claim as $H_o$, remembering to include the condition of equality. The hypothesis for this test would be: $$H_o:\mu\ge 35000$$ $$H_1: \mu<35000$$
One would hope that the sample data would allow the rejection of the null hypothesis, refuting the company’s claim.
The $H_o$ will be rejected in the case above if the sample mean is statistically significantly less than 35000. That is, if the sample mean is in the left-tail of the distribution of all sample means.
Right Tailed Test
The right-tailed test is used when the null hypothesis ($H_0$) being tested is a claim that the population parameter is at most ($\le$) a given value. Note that the alternative hypothesis ($H_1$) then claims that the parameter is greater than (>) the value.
Suppose, you worked for the tire company and wanted to gather evidence to support their claim then you will make the company's claim $H_1$ and remember that equality will not be included in the claim (H_o$). The hypothesis test will be
$$H_0:\mu \le 35000$$
$$H_1:\mu > 35000$$
If the sample data was able to support the rejection of $H_o$ this would be strong evidence to support the claim $H_1$ which is what the company believes to be true.
One should reject $H_o$ in this case if the sample mean was significantly more than 35000. That is, if the sample mean is in the right-tailed of the distribution of all sample means.
Two-Tailed Test
The two-tailed test is used when the null hypothesis ($H_o$ begins tested as a claim that the population parameter is equal to (=) a given value. Note that the alternative hypothesis ($H_1$) then claims that the parameter is not equal to ($\ne$) the value. For example, the Census Bureau claims that the percentage of Punjab’s area residents with a bachelor’s degree or higher is 24.4%. One may write the null and alternative hypotheses for this claim as:
$$H_o: P = 0.244$$
$$H_1: P \ne 0.244$$
In this case, one may reject $H_o$ if the sample percentage is either significantly more than 24.4% or significantly less than 24.4%. That is if the sample proportion was in either tail (both tails) of the distribution of all sample proportions.
Key Differences
- Directionality: One-tailed tests look for evidence of an effect in one specific direction, while two-tailed tests consider effects in both directions.
- Rejection Regions: In a one-tailed test, all of the rejection regions are in one tail of the distribution; in a two-tailed test, the rejection region is split between both tails.
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