Question

A normal population has mean "µ" and standard deviation 12. The hypotheses to be tested are H0: µ = 40 versus H1: µ > 40.

1. Which would result in the highest probability of a Type II error?
   1. µ = 42; n = 100
   2. µ = 42; n = 10
   3. µ = 41; n = 100
   4. µ = 41; n = 10
   5. µ = 40.9; n = 15
2. If a random sample has 100 observations, the true population mean is 42, and the significance level is 0.01 then the probability of a Type II error is approximately:
   1. 0.0
   2. 0.25
   3. 0.35
   4. 0.75
   5. 0.85
3. If you wish the power of the test in the previous question to be at least 97.5% the sample size should be at least:
   1. 662
   2. 573
   3. 437
   4. 371
   5. 215

Answer 1

1)for closer the true mean to hypothesized mean and lower the sample size, higher the type II error

option D is correct: µ = 41; n = 10

2)

std error ='σx=σ/√n=

1.2000

P(Type II error) =P(Xbar<42.791|μ=42)=P(Z<(42.7912-42)/1.2)=P(Z<0.66)=

0.7454~ 0.75

option D is correct

3)

Hypothesized mean μo=		40
true mean μa=		42
std deviation σ=		12.0
0.01 level critical Z=		2.326
0.025 level critical Zβ=		1.96
n=(Zα/2+Zβ)2σ2/(μo-μa)2=		662

option A is correct