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Formulas: Beta (Type II Error Rate) for a Student t-Test

Beta (Type II Error Rate) for a Student t-Test Formulas

Below you will find descriptions and details for the 7 formulas that are used to compute beta (type II error rate) values for Student t-tests.

Beta function:
Beta function formula
Cohen's d effect size for a t-test:
Cohen's d effect size for a t-test formula
where x1 and x2 are the means of group 1 and group 2, and σ12 and σ22 are the variances of group 1 and group 2.
Gamma function:
Gamma function formula
Lower incomplete beta function:
Lower incomplete beta function formula
Noncentral t-distribution cumulative distribution function:
Noncentral t-distribution cumulative distribution function formula
where μ is the noncentrality parameter, v is the degrees of freedom, Γ is the gamma function and I is the regularized lower incomplete beta function.
Noncentral t-distribution noncentrality parameter:
Noncentral t-distribution noncentrality parameter formula
where d is the Cohen's d effect size, and n1 and n2 are the sample sizes for group 1 and group 2.
Regularized lower incomplete beta function:
Regularized lower incomplete beta function formula
where the numerator is the lower incomplete beta function, and the denominator is the beta function.
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