Question

Find the sample size n needed to achieve power 0.8 at EY = 5, for the test in question one at level ⍺ = 0.05. n = _____.

Question one for reference:

1. For testing H₀: EY = 3 vs H₁: EY > 3, based on a sample Y: 1,3,5,7,9, assume that var(Y) = 4 is known. The p-value is _____.

(Correct answer to question one: 0.01267366)

Answer 1

Let n be the sample size, Then the standard deviation of mean EY = = 2 /

For ⍺ = 0.05, Z = 1.645 for one tail test.

Critical value to reject H0 = 3 + 1.645 * 2 / = 3 + 3.29 /

Power = 0.8

=> P(Reject H0 | H0 is false) = 0.8

=> P(Reject H0 | EY = 5) = 0.8

=> P(Y > 3 + 3.29 / | EY = 5) = 0.8

=> P[Z > (3 + 3.29 / - 5) / (2 / )] = 0.8

=> (3.29 / - 2) / (2 / ) = -0.8416

=> 3.29 / - 2 = -1.6832 /

=> 4.9732 / = 2

=> = 4.9732 / 2

=> n = (4.9732 / 2)² = 6.18   6

So, n = 6