Question

You have an SRS of size n = 10 from a Normal distribution with s = 1.1. You wish to test

H0: µ = 0

Ha: µ > 0

You decide to reject H0 if x > 0.05 and to accept H0 otherwise.

Find the probability (±0.1) of a Type I error. That is, find the probability that the test rejects H0 when in fact µ = 0: _____

Find the probability (±0.001) of a Type II error when µ = 0.35. This is the probability that the test fails to reject H0 when in fact µ = 0.35 ____

Answer 1

Answer:

Given,

n = 10 , s= 1.1

Type I error = P ( reject H0 | H0 is true )

P ( X̅ > 0.10 | µ = 0 )

Z = ( X - µ ) / ( σ / √(n))
Z = ( 0.05 - 0 ) / ( 1.1 / √ ( 10 ) )
Z = 0.143

P ( X̅ > 0.05 ) = 1 - P ( Z < 0.143 )
P ( X̅ > 0.05 ) = 1 - 0.5557
P ( X̅ > 0.05 ) = 0.4443

part 2)

P ( Type I error ) = 0.4443

Z = ( X - µ ) / (σ/√(n)
Z = ( 0.05 - 0.35 ) / ( 1.1 / √10 )
Z = - 0.8624
P ( Z < -0.86 ) = 0.1949

Type I error = P ( Accept H0 | H0 is false )

P ( X̅ <= 0.05 | µ = 0.35 )

P ( Type II error ) = 0.1949

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