Upper-Tailed Hypothesis Test

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An Upper-Tailed Hypothesis Test is a statistical hypothesis test where the region of rejection is on the right side of the sampling distribution.

Set Null Hypothesis Alternative Hypothesis
1 [math]\displaystyle{ H_0: \quad \mu=M }[/math] [math]\displaystyle{ H_A: \quad \mu \gt M }[/math]
2 [math]\displaystyle{ H_0: \quad \mu \leq M }[/math] [math]\displaystyle{ H_A: \quad \mu \gt M }[/math]
Set 1 is the most used expression for the null and alternative upper-tailed hypothesis test while Set 2 is less used.


References

2017a

2017b

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)