Question

2. Suppose we have the hypothesis test

H0 : µ = 200

Ha : µ > 200

in which the random variable X is N(µ, 10000). Let the critical region C = {x : x ≥ c}.

Find the values of n and c so that the significance level of this test is α = 0.03 and the power of µ = 220 is 0.96.

Answer 1

H0 : µ = 200

Ha : µ > 200

N( µ, 10000/ )

let a random sample mean is Y

To reject a null hypothesis when true mean is 200

P ( Y > c) =0.03

P( (Y- 200) / (100/ ) > (c- 200) / (100/ ) ) =0.03

P( Z> (c- 200) / (100/ )) =0.03

P(Z (c- 200) / (100/ )) =0.97

((c- 200) / (100/ )) =0.97

(c- 200) / (100/ ) = 1.88 [ from standard normal table]

(c-200) * = 188--------- 1st eq

To reject a null hypothesis when true mean is 200 ( this is power)

P ( Y > c) =0.96

P( (Y- 220) / (100/ ) > (c- 220) / (100/ ) ) =0.96

P( Z> (c- 220) / (100/ )) =0.96

P(Z (c- 220) / (100/ )) =0.04

((c- 220) / (100/ )) =0.04

(c- 220) / (100/ ) = -1.75 [ from standard normal table]

(c-220) * = -175 ---------2nd equation

1st - 2nd eqation

20 = 363

or, n= 329.42 . We can select sample size of 329 or 330 ( better choice).

putting value of n in equation 1 we get

c = 210.358

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