Lower-Tailed Hypothesis Test

A Lower-Tailed Hypothesis Test is a statistical hypothesis test where the region of rejection is on the left side of the sampling distribution.

AKA: Left-Sided Hypothesis Test.
Context:
- It can be defined by an alternative hypothesis ([math]\displaystyle{ H_A }[/math]) which states that population parameter or measure ([math]\displaystyle{ \mu }[/math]) is less than 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 \lt M }[/math]
2	[math]\displaystyle{ H_0: \quad \mu \geq M }[/math]	[math]\displaystyle{ H_A: \quad \mu \lt M }[/math]

Set 1 is most used expression for the null and alternative lower-tailed hypothesis test while Set 2 is less used.

Example(s):
- One Tailed One-Sample t-Test (lower-tailed test)
- One Tailed Two-sample t-Test (lower-tailed test).
- …
Counter-Example(s):
- Two-Tailed Hypothesis Test.
- Upper-Tailed Hypothesis Test.
See: Statistical Hypothesis Testing Task, Region of Rejection, Sampling Distribution.

References

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=One_tailed_test
- A test of a statistical hypothesis , where the region of rejection is on only one side of the sampling distribution , is called a one-tailed test.
  For example, suppose the null hypothesis states that the mean is less than or equal to 10. The alternative hypothesis would be that the mean is greater than 10. The region of rejection would consist of a range of numbers located on the right side of sampling distribution; that is, a set of numbers greater than 10.

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

(Wayne W. LaMorte, 2017) ⇒ Retrieved on 2017-03-12 from http://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_hypothesistest-means-proportions/bs704_hypothesistest-means-proportions3.html
- QUOTE: The research or alternative hypothesis can take one of three forms. An investigator might believe that the parameter has increased, decreased or changed. For example, an investigator might hypothesize:
  - H1: μ > μ 0 , where μ0 is the comparator or null value (e.g., μ0 =191 in our example about weight in men in 2006) and an increase is hypothesized - this type of test is called an upper-tailed test;
  - H1: μ < μ0 , where a decrease is hypothesized and this is called a lower-tailed test; or
  - H1: μ ≠ μ 0, where a difference is hypothesized and this is called a two-tailed test.

The exact form of the research hypothesis depends on the investigator's belief about the parameter of interest and whether it has possibly increased, decreased or is different from the null value. The research hypothesis is set up by the investigator before any data are collected.

2017F

(2017) ⇒ Retrieved on 2017-03-12 from http://people.richland.edu/james/lecture/m170/ch09-typ.html
- The type of test is determined by the Alternative Hypothesis (H1 )
  - Left Tailed Test H1: parameter < value. Notice the inequality points to the left. Decision Rule: Reject H0 if t.s. < c.v.
  - Right Tailed Test H1: parameter > value. Notice the inequality points to the right. Decision Rule: Reject H0 if t.s. > c.v.
  - Two Tailed Test H1: parameter not equal value. Another way to write not equal is < or >. Notice the inequality points to both sides. Decision Rule: Reject H0 if t.s. < c.v. (left) or t.s. > c.v. (right)