Question

a. Find the value of Beta, when the Null Hypothesis assumes a population mean of Mu = 827, with a populaiion standard deviation of 241, the sampel size is 7and the true mean is 1035.14

b. Find the power of the test, when the Null Hypothesis assumes a population mean of Mu = 555, with a populaiion standard deviation of 157, the sampel size is 6and the true mean is 647.59

Answer 1

a)

true mean , µ = 1035.14

  

hypothesis mean,µo = 827

significance level,α = 0.05

sample size,n =7

std dev,σ = 241

δ=µ - µo = 208.14

std error of mean,σx = σ/√n = 91.0894

Zα/2= ±1.960(two tailed test)

We will fail to reject the null (commit a Type II error) if we get a Z statistic between-1.960 and1.960

these Z-critical value corresponds to some X critical values ( X critical), such that

-1.960≤(x̄ - µo)/σx≤1.960

648.468≤ x̄ ≤1005.532

now, type II error is ,ß = P (648.468≤ x̄ ≤1005.532)

Z = (x̄-true mean)/σx

Z1 =-4.245

Z2 = -0.325

so,      P(-4.245≤ Z ≤-0.325) = P ( Z ≤-0.325) - P ( Z ≤-4.245)

=0.373-0.000=0.3726 (answer)

b)

true mean , µ = 647.59

  

hypothesis mean,µo = 555

significance level,α = 0.05

sample size,n =6

std dev,σ = 157

δ=µ - µo = 92.59

std error of mean,σx = σ/√n = 64.0950

Zα/2= ±1.960(two tailed test)

We will fail to reject the null (commit a Type II error) if we get a Z statistic between-1.960 and1.960

these Z-critical value corresponds to some X critical values ( X critical), such that

-1.960≤(x̄ - µo)/σx≤1.960

429.376≤ x̄ ≤680.624

now, type II error is ,ß = P (429.376≤ x̄ ≤680.624)

Z = (x̄-true mean)/σx

Z1 =-3.405

Z2 = 0.515

so,      P(-3.405≤ Z ≤0.515) = P ( Z ≤0.515) - P ( Z ≤-3.405)

=0.697-0.000=0.6965

power = 1 - ß =0.3035