7. Quiz: Composite Hypotheses for Bernoulli models

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(a)

1 point possible (graded)

Let X1. .... Xn bei.i.d. Bernoulli random variables with unknown parameter pe (0, 1) .

Find a function In, ((Xn) , which depends on Xn, n, and p , such that

In.p( Xn)

N (0, 1).

71-+00

by

using the Central Limit Theorem on X'n and

.

substituting any occurrence of p in the variance by a plug-in estimator for p .

(a)

Note: If Th,p- N(0, 1) , then so does -Th,p. For this problem and the next part, use the

expression for Th,p( Xn) that is of the form (Xn - p) f (n, Xn) where f (n. Xn) is always

positive. (Or very loosely speaking, use (Xn -p ) and not (p - Xn) where applicable. )

(Enter barX_n for xn).

In, p(Xn) =

STANDARD

NOTATION

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(b)

3 points possible (graded)

(This is a quiz, hence only 1 attempt.)

Select a test with asymptotic level a , in terms of the function Th,p( Xn) , for each of the

following pairs of hypotheses:

(Choose one for each column. Note the absolute values in the first 2 rows.)

Ho : p =0.5 vs H1 : p # 0.5

:

Ho : p 5 0.5 vs H1 : p > 0.5

:

Ho : p 2 0.5 vs H1 : p <0.5

1 (In.0. { XM) >9012)

1 ( In,0. ( XM) >9012)

1 ( In,0. ( XM) >9012)

1 ( m.O.{ Xn) >qa)

1 ( In.0.{ Xn) >qa)

1 ( m.O.{ XM) >qa)

1 (In,0.{ Xn) >qa/2)

1 (In,0.5 Xn) >qa/2)

1 (In,0.5 Xn) >qa/2)

1 (In, 0. { Xn) >qa)

1 (In,0. { Xn) >qa)

1 ( In,0. { Xn) >qa)

1 (In,0.{ Xn) <-9012)

1 (In.0.{ Xn) <-9012)

1 (In.0.5 Xn) <-qa/2)

1 ( In,0. { Xn) <-qa)

1 (In.0.{ Xn) <-qa

1 ( In.0. { Xn) <-qa)

1 (In.0. { Xn) <qa12)

1 (In.0.{ Xn) <qa/2)

1 (In.0. { Xn) <qa/2)

1 (In.0.{ Xn) <qu

1 (m.O.{ Xn) <qu)

1 (In.0.{ Xn) <qa)