One-Sample t-Test Formulas

Below you will find descriptions and details for the 5 formulas that are used to compute a one-sample t-test.

Beta function:

Cumulative distribution function (CDF) for the t-distribution:

Variable definitions:
v = degrees of freedom
t = upper limit of integration
I = regularized lower incomplete beta function

Lower incomplete beta function:

Regularized lower incomplete beta function:

Variable definitions:
the numerator = lower incomplete beta function
the denominator = beta function

t-value and associated degrees of freedom:

Variable definitions:
x = sample mean
μ = hypothesized mean
s = sample standard deviation
n = sample size
df = degrees of freedom