Question

In a planned study, the population is known to have a mean of 500 and a
standard deviation of 100. The researchers will give the experimental treatment
to 60 people and predict the mean for those 60 will be 540. The will use the 0.05
significance level.
a. Figure the power of this study.
b. Sketch the distributions involved.
c. What is beta?

Answer 1

Given that,
Standard deviation, σ =100
Sample Mean, X =540
Null, H0: μ<500
Alternate, H1: μ>500
Level of significance, α = 0.05
From Standard normal table, Z α/2 =1.6449
Since our test is right-tailed
Reject Ho, if Zo < -1.6449 OR if Zo > 1.6449
Reject Ho if (x-500)/100/√(n) < -1.6449 OR if (x-500)/100/√(n) > 1.6449
Reject Ho if x < 500-164.49/√(n) OR if x > 500-164.49/√(n)
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Suppose the size of the sample is n = 60 then the critical region
becomes,
Reject Ho if x < 500-164.49/√(60) OR if x > 500+164.49/√(60)
Reject Ho if x < 478.7644 OR if x > 521.2356
Implies, don't reject Ho if 478.7644≤ x ≤ 521.2356
Suppose the true mean is 540
Probability of Type II error,
P(Type II error) = P(Don't Reject Ho | H1 is true )
= P(478.7644 ≤ x ≤ 521.2356 | μ1 = 540)
= P(478.7644-540/100/√(60) ≤ x - μ / σ/√n ≤ 521.2356-540/100/√(60)
= P(-4.7433 ≤ Z ≤-1.4535 )
= P( Z ≤-1.4535) - P( Z ≤-4.7433)
= 0.073 - 0 [ Using Z Table ]
= 0.073
For n =60 the probability of Type II error is 0.073
c.
type 2 error is 0.073
type 2 error = beta =0.073
a.
power of the test =1-beta
power = 1-0.073
power =0.927 =92.7%
b.
normal distribution in the given data