Two-Tailed Hypothesis Test
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A Two-Tailed Hypothesis Test is a statistical hypothesis test where the region of rejection lies on both sides of the sampling distribution.
- AKA: Two-Sided Test.
- Context:
- It can range from being a One-Sample Two-Tailed Hypothesis Test to being a Two-Sample Two-Tailed Hypothesis Test.
- It can be defined by an alternative hypothesis ([math]\displaystyle{ H_A }[/math]) which states that population parameter or measure ([math]\displaystyle{ \mu }[/math]) differs from (i.e. it is not equal to) a specific value ([math]\displaystyle{ M }[/math]) stated in the null hypothesis ([math]\displaystyle{ H_0 }[/math]).
Set Null Hypothesis Alternative Hypothesis 1 [math]\displaystyle{ H_0: \quad \mu=M }[/math] [math]\displaystyle{ H_A: \quad \mu \neq M }[/math] 2 [math]\displaystyle{ H_0: \quad \mu=M }[/math] [math]\displaystyle{ H_A: \quad \mu \gt M \; or\; \quad \mu \lt M }[/math]
- Set 1 is the most used expression for the null and alternative two-tailed hypothesis test while Set 2 is less used.
- Example(s):
- a Two-Tailed t-Test, such as a Two-Tailed One-Sample t-Test or a Two-Tailed Two-sample t-Test.
- Counter-Example(s):
- See: Statistical Hypothesis Testing, Sampling Distribution.
References
2022
- Ronny Kohavi. (2022). “Use Two-Sided Tests or Two One-Sided Tests (TOST) in A/B Testing." 16 Feb 2022
- QUOTE: ... As a rule of thumb, organizations doing A/B tests, where successful experiment results are shared for learning purposes (e.g., in the organization or in publications), and/or accounted towards key metrics, should use two-sided tests with alpha in the range of 0.005-0.05 or two one-sided tests (TOST) with independently set alphas (so not necessarily a test for equivalence). ... In the most common use of two-sided test, also called two-tailed test, the null hypothesis is that the delta between the means of the metric of interest for the two variants equals zero. We allocate a portion of alpha (e.g., 0.05) to the two tails of the distribution, usually alpha/2 to each. ...
2017a
- (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/One-_and_two-tailed_tests
- In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic. A two-tailed test is used if deviations of the estimated parameter in either direction from some benchmark value are considered theoretically possible; in contrast, a one-tailed test is used if only deviations in one direction are considered possible. Alternative names are one-sided and two-sided tests; the terminology "tail" is used because the extreme portions of distributions, where observations lead to rejection of the null hypothesis, are small and often "tail off" toward zero as in the normal distribution or "bell curve", pictured above right.
2017b
- (QCT, 2017) ⇒ Retrieved on 2017-03-12 from https://www.quality-control-plan.com/StatGuide/sg_glos.htm#transformation
- The null hypothesis for a statistical test is the assumption that the test uses for calculating the probability of observing a result at least as extreme as the one that occurs in the data at hand. An alternative hypothesis is one that specifies that the null hypothesis is not true. For the one-sample t test, the null hypothesis is that the population mean equals a specific value. or a two-sided test, the alternative hypothesis is that the mean does not equal that value. It is also possible to have a one-sided test with the alternative hypothesis that the mean is greater than the specified value, if it is theoretically impossible for the mean to be less than the specified value. One could alternatively perform one-sided test with the alternative hypothesis that the mean is less than the specified value, if it were theoretically impossible for the mean to be greater than the specified value. One-sided tests usually have more power than two-sided tests, but they require more stringent assumptions. They should only be used when those assumptions (such as the mean always being at least as large as they specified value for the one-sample t test) apply.
2017c
- (Stattrek,2017) ⇒ http://stattrek.com/statistics/dictionary.aspx?definition=Two_tailed_test
- Two-Tailed Test - A test of a statistical hypothesis , where the region of rejection is on both sides of the sampling distribution , is called a two-tailed test.
For example, suppose the null hypothesis states that the mean is equal to 10. The alternative hypothesis would be that the mean is less than 10 or greater than 10. The region of rejection would consist of a range of numbers located on both sides of sampling distribution; that is, the region of rejection would consist partly of numbers that were less than 10 and partly of numbers that were greater than 10.
- Two-Tailed Test - A test of a statistical hypothesis , where the region of rejection is on both sides of the sampling distribution , is called a two-tailed test.
2017D
- (Stattrek, 2017) ⇒ http://stattrek.com/hypothesis-test/region-of-acceptance.aspx
- 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 [math]\displaystyle{ \mu=M }[/math] [math]\displaystyle{ \mu \neq M }[/math] [math]\displaystyle{ 2 }[/math] 2 [math]\displaystyle{ \mu\geq M }[/math] [math]\displaystyle{ \mu \lt M }[/math] [math]\displaystyle{ 1 }[/math] 2 [math]\displaystyle{ \mu\leq M }[/math] [math]\displaystyle{ \mu \gt M }[/math] [math]\displaystyle{ 1 }[/math]
- 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.
2017E
- http://cliffsnotes.com/study-guides/statistics/principles-of-testing/one-and-twotailed-tests
- … Figure 1.Comparison of (a) a two‐tailed test and (b) a one‐tailed test, at the same probability level (95 percent).
- … Figure 1.Comparison of (a) a two‐tailed test and (b) a one‐tailed test, at the same probability level (95 percent).