t-Values Calculator

Compute t-values from p-values with the Inverse CDF (ICDF).

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Instructions

• This tool computes t-values for Student's t hypothesis testing.
• To use the tool, just submit a p-value and its degrees of freedom, υ.
• This tool comes handy when you don't have around statistical t-tables or are working with p and t values or degrees of freedom not available from such tables. Avoid pausing a problem for annoying linear interpolation workarounds!

What is computed?

• About t-values
  A t-value is used for null hypothesis (H₀) significance testing and as follows.

  We first define H₀, a critical p-value (p_critic), and the degrees of freedom υ. With p_critic and υ, a critical t-value (t_critic) is determined and then compared against a sample-calculated t-value (t_sample), such that

  • H₀ is rejected if t_sample > t_critic
  • H₀ is not rejected if t_sample < t_critic

  Statistical tables can be used to find a t_critic. These tables are compiled by solving the Inverse Cumulative Distribution Frequency (ICDF) of the Student's t Distribution. Solving the ICDF is done through numerical approximation algorithms (Hill, 1970; Shaw, 2005; Shaw 2006). These are precisely the types of algorithms used by our tool to generate a t_critic.

  If H₀ is rejected, the experiment was conclusive at the critical confidence level selected, meaning the data is sufficiently inconsistent with H₀. By contrast, if H₀ is not rejected, the experiment is inconclusive at the chosen confidence level, so the data is sufficiently consistent with H₀.

  Evidently, t_sample > t_critic suggests that the observed data is so sufficiently inconsistent with H₀ that it may be rejected. On the other hand, t_sample < t_critic does not prove that H₀ can be accepted. It only means that it cannot be rejected with the sample data examined. We can do better by formulating an alternative hypothesis known to be true, H₁ so if H₀ is rejected, we could accept H₁.

  However, to accept H₁, we must consistently reject H₀. This can be done by increasing the statistical power of the test, by increasing p_critic then decreasing t_critic. Thus, if one end rejecting H₀ more often, there is a greater chance of safely accepting H₁ (Trochim, 2006).

• Calculating t-values for t-testing Hypotheses
  Given a p-value and υ, t-values can be computed from the Inverse Cumulative Distribution (ICDF) of the Student's t Distribution by the following procedure (Wikipedia, 2017a; 2017b; 2017c).

  We first define the Probability Distribution Function (PDF)

  (1)

  Integrating (1), we obtain the Cumulative Distribution Function (CDF)

  (2)

  Inverting it, the Quantile Function or Inverse CDF is obtained.

  (3)

  where

  • Γ stands for a gamma function.
  • ₂F₁ stands for a hypergeometric function with two parameters in its numerator and one in its denominator.
  • I^-1 is the inverse of the regularized form of an incomplete beta function.

  In (3), n = υ and u = p.

  Abramowitz and Stegun (1972; see equations 26.7.3, 26.7.4, and 26.7.5), Hill (1970), Shaw (2005, 2006), and Culham (2004) have described these functions extensively.

  For the reverse calculations, computing p-values from t-values, you may want to use our p-values Calculator.

  In addition, our t,p & Effect Size Estimator tool allows you to easily move back-and-forth between calculations, then between CDF and ICDF transforms.

Who can use this tool?

• Anyone that need to compute t-values for t-test hypothesis testing.

Suggested Exercises

• Calculate a t_critic for a one-tailed test at the 95% level of confidence and 15 degrees of freedom.
  Hint: p_{critic, two-tailed} = 2p_{critic, one-tailed}
• Repeat previous exercise, this time for a two-tailed test and at the same confidence level. Compare results.
• From 25 pairs of measurements from a population you find a Pearson correlation coefficient equal to r = 0.466, where at the 95% confidence levels (two-tailed) the following hypotheses are formulated:

  H₀: "The correlations are not statistically different from zero."
  H₁: "A significant correlation exits in the population."

  Can we reject H₀?
  Hint. Calculate first t_sample from r.
• In the above case: Compute a range of t_critic values for safely accepting H₁.

References

• Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions.
  p. 948-949. National Bureau of Standards. See also full pdf.
• Culham, J. R. (2004).Factorial, Gamma and Beta Functions.
• Hill, G. W. (1970). Algorithm 396: Student's t-Quantiles. Communications of the ACM 13(10):619-620. ACM Press, October, 1970.
• Shaw, W. T. (2006). Sampling Student's T distribution - Use of the inverse cumulative distribution function.
  Computational Finance Notes 2011-12, JCF Volume 9/Number 4, Summer 2006. Journal of Computational Finance. 9 (4): 37-73.
• Shaw, W. T. (2005). New Methods for Simulating the Student t-Distribution - Direct Use of the Inverse Cumulative Distribution. The Mathematical Institutem University of Oxford, Eprints Archive.
• Trochim, W.M.K. (2006). Statistical Power.
• Wikipedia (2017a). p-value.
• Wikipedia (2017b). Student's t-distribution.
• Wikipedia (2017c). Quantile function.

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