3. Simple Testing

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Let X1. .... Xn bei.i.d. >(8, 1) . Consider testing

Ho : 0 =0 v.s. H1 :0=1.

(a)

2 points possible (graded)

What would a Type 1 error be in this test?

ORejecting Ho when 8=0

ONot Rejecting Ho when 0=0

ORejecting Ho when 8 = 1

Not rejecting Ho when e = 1

What would a Type 2 error be in this test?

ORejecting Ho when 0=0

ONot Rejecting Ho when 0=0

ORejecting Ho when 0 = 1

Not rejecting Ho when e = 1

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(b)

1 point possible (graded)

Suppose that the rejection region of a test w has the form R = (X, : Xn > c) . Find the smallest

c such that w has level a.

(If applicable, type abs(x) for |x|, Phi(x) for @(x) = P(Z x) where Z ~ N(0, 1) , and q(alpha)

for qu, the 1 - a quantile of a standard normal variable.)

c2

STANDARD

NOTATION

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(c)

2 points possible (graded)

Suppose that the test w has level a = 0.05 . What is the power of w?

(If applicable, type abs(x) for [x|, Phi(x) for @ (x) = P(Z x) where Z ~ N(0, 1) , and q(alpha)

for qu, the 1 - a quantile of a standard normal variable, e.g. enter q(0.01) for 90.01-)

Power of w:

What does the power of w approach as n - co ?

lim Power =