Two Tailed One-Sample t-Test

A Two Tailed One-Sample t-Test is One-Sample t-Test that is an Two-Tailed Hypothesis Test.

• Context
  • It is usually defined by the following null hypothesis and alternative hypothesis :
    • [math]\displaystyle{ H_0 :\; \mu_X = \mu_0 }[/math] and [math]\displaystyle{ H_A :\; \mu_X \neq \mu_0 }[/math]

where [math]\displaystyle{ \mu_X }[/math] is population mean value of random variable [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \mu_0 }[/math] is the hypothesized or theoretical value. The region of acceptance defined by the range between values for which the cumulative probability of the sampling distribution is equal to [math]\displaystyle{ \alpha/2 }[/math] and [math]\displaystyle{ 1- \alpha/2 }[/math], i.e.

(a) t-score value, [math]\displaystyle{ t_- }[/math], for which [math]\displaystyle{ P(T\leq t_{-}) = \alpha/2 }[/math]

(b) t-score value, [math]\displaystyle{ t_+ }[/math], for which [math]\displaystyle{ P(T\leq t_{+}) = 1 -\alpha/2 }[/math]

where α is a significance level. The null hypothesis is rejected when a t-statistic is greater than [math]\displaystyle{ t_{+} }[/math] or less than [math]\displaystyle{ t_{-} }[/math].

• Example(s):
  • Let's consider a sample size [math]\displaystyle{ n=50 }[/math], sample mean [math]\displaystyle{ \overline{x}=385 }[/math], and sample standard deviation, [math]\displaystyle{ s=30 }[/math]. Let's assume we would like to test null hypothesis [math]\displaystyle{ H_0 :\; \mu_X = 400 }[/math] and alternative hypothesis [math]\displaystyle{ H_A :\; \mu_X \neq 400 }[/math]. For a significance level [math]\displaystyle{ \alpha=0.05 }[/math], the t-score value for [math]\displaystyle{ P(T\leq t) = \alpha/2 }[/math] is [math]\displaystyle{ t=-2.01 }[/math] and for [math]\displaystyle{ P(T\leq t) = 1- \alpha/2 }[/math] is [math]\displaystyle{ t=2.01 }[/math] (note that the degrees of freedom is given [math]\displaystyle{ df=n-1=49 }[/math]). Thus, region of acceptance is defined by [math]\displaystyle{ -2.01\leq t \leq 2.01 }[/math], this implies the region of rejection lies on both sides of the sampling distribution, defined by [math]\displaystyle{ t \lt -2.01 }[/math] and by [math]\displaystyle{ t \gt 2.01 }[/math]. In this case one-sample t-statistic is [math]\displaystyle{ t=\frac{385-400}{30/\sqrt{50}}=-3.54 }[/math], thus the null hypothesis is rejected.
  • …
• Counter-Example(s):
  • One-Tailed One-Sample t-Test.
• See: Two Tailed Two-sample t-Test, One-Sample t-Test Task, Student's t-Test, Parametric Statistical Test, Statistical Hypothesis Testing Task.

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References

2017a

• (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.

2017b

• (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.

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