Question

The weight of steel pieces has been measured. The goal is to test H₀ : µ = 5 against H₁ : µ > 5. The weights follow a normal distribution with variance 16. With a sample of size of n = 25, H₀ is rejected if x > 6. If the true population mean is 5.2, what is the probability of committing an error of Type II?

Answer 1

X ~ N ( µ = 5.2 , σ = 4 )
P ( X < 6 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 6 - 5.2 ) / ( 4 / √25 )
Z = 1
P ( ( X - µ ) / ( σ/√(n)) < ( 6 - 5.2 ) / ( 4 / √(25) )
P ( X < 6 ) = P ( Z < 1 )
P ( X̅ < 6 ) = 0.8413

P ( Type II error ) ß = 0.8413