One-Sample t-Test Task

A One-Sample t-Test Task is a statistical hypothesis testing task whose test statistic is a one-sample t-test.

• AKA: Single-Sample t-Testing.
• Context:
  • Task Input:
    • Input Data : [math]\displaystyle{ \{x_1,x_2,\cdots,x_n\} }[/math], sample data.
    • Input Parameters:
      • [math]\displaystyle{ \mu_0 }[/math], a hypothesized population mean value.
      • [math]\displaystyle{ \alpha_0 }[/math], a significance level value or a confidence level (a percentage).
      • (optional) [math]\displaystyle{ \sigma^2 }[/math], a population variance value.
  • output:
    • one-sample t-test statistic value,
    • P-value or Region of Acceptance
    • Region of Rejection (optional)
    • Decision Errors (optional).
  • Task Requirements
    • Hypotheses Statement: a null hypothesis and an alternative hypothesis according to one-tailed one-sample t-test or two-tailed one-sample t-test
    • Test Statistic computation: This require the calculation an one-sample t-test statistic from a sample mean and sample standard deviation.
    • P-value and/or Region of acceptance computation: these require a t-distribution calculator or t-table.
    • Decision Rule: Null hypothesis is rejected if P-value is less than [math]\displaystyle{ \alpha_0 }[/math] or if the t-test statistic value follows outside region of acceptance.
  • It can be solved by a One-Sample t-Test System (that implements a one-sample t-test algorithm).
• Example(s):
  • Let's consider the dataset corresponding Verbal Intelligent Quotient (VIQ) measure retrieved from http://www.scipy-lectures.org/_downloads/brain_size.csv and the One-Sample t-Test System based on the scipy statistical function stats.ttest_1samp() [1] then:
    • Task input: sample data = data['VIQ'], [math]\displaystyle{ \mu_0=0 }[/math]
    • Task output: t-statistic = 30.08809997084934, p-value=1.3289196468727784e-28
    • For [math]\displaystyle{ \alpha_0=0.001,0.01,0.025, 0.05 }[/math] the null hypothesis is rejected as p-value is less than significance level (because the p-value is too small, we can concluded the population mean is not 0).
  • …
• Counter-Example(s):
  • Z-Value Test.
  • Welch's t-Test.
  • Linear Regression t-Test.
  • Matched-pair t-Test.
  • Independent Two-Sample t-Test.
• See: One-Sample t-Test System, Statistical Significance, Sample Average, Sample Variance.

---

---

References

2017

• (Stat Trek, 2017) ⇒ http://stattrek.com/statistics/dictionary.aspx?definition=One-sample%20t-test
  • A one-sample t-test is used to test whether a population mean is significantly different from some hypothesized value.

    (...) The test statistic is a t statistic (t) defined by the following equation.

[math]\displaystyle{ t = \frac{(x - M )}{s \sqrt{n}} }[/math]

where [math]\displaystyle{ x }[/math] is the observed sample mean, [math]\displaystyle{ M }[/math] is the hypothesized population mean (from the null hypothesis), and [math]\displaystyle{ s }[/math] is the standard deviation of the sample.

2011

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Student%27s_t-test#One-sample_t-test
  • In testing the null hypothesis that the population mean is equal to a specified value μ₀, one uses the statistic

[math]\displaystyle{ t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}} }[/math]

where [math]\displaystyle{ \overline{x} }[/math] is the sample mean, s is the sample standard deviation of the sample and n is the sample size. The degrees of freedom used in this test are n − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means, [math]\displaystyle{ \overline {x} }[/math], is assumed to be normal. By the central limit theorem, if the sampling of the parent population is independent then the sample means will be approximately normal. (The degree of approximation will depend on how close the parent population is to a normal distribution and the sample size, n.)

---

---