Beta error in t tests

#1
TYPE II or β error is the probability of accepting Ho when Ho is false.

Ho is false if µ β 1 ≠ µ 1 OR µ β 2 ≠ µ 2.

What if µ 1 were some other value, µ β1; what percent of the t tests would result in accepting Ho when Ho was false?

The data in a t test can be solved for x̄ α.

µ β 1 or µ β 2 are the substitutes for µ 1 or µ 2 that make Ho false.

t test β, (x̄ α – (µ β 1 or µ β 2)) / (s /√n) gives us P (Accept Ho when false) = Type II or β error.
 
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#2
Calculate x̄ α, the x̄ value in units such as pints or inches or…, at t α.

Since t test = (x̄ 1 - x̄ 2) / (s /√n),

then t α = (x̄ α - x̄ 2) / (s /√n),

(x̄ α - x̄ 2) = t α * (s /√n),

and x̄ α = (t α * (s /√n)) + x̄ 2.

What is P (x̄ <= x̄ α) in a t distribution with mean = µ β 1?

t test = (x̄ 1 - x̄ 2) / (s /√n),

t test β= (x̄ α - µ β 1) / (s /√n),

P (t <= t test β) = P (x̄ <= x̄ α) in a t distribution with mean = µ β 1 = Type II or β error.