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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 &gt; 0.5
:
Ho : p 2 0.5 vs H1 : p &lt;0.5
1 (In.0. { XM) &gt;9012)
1 ( In,0. ( XM) &gt;9012)
1 ( In,0. ( XM) &gt;9012)
1 ( m.O.{ Xn) &gt;qa)
1 ( In.0.{ Xn) &gt;qa)
1 ( m.O.{ XM) &gt;qa)
1 (In,0.{ Xn) &gt;qa/2)
1 (In,0.5 Xn) &gt;qa/2)
1 (In,0.5 Xn) &gt;qa/2)
1 (In, 0. { Xn) &gt;qa)
1 (In,0. { Xn) &gt;qa)
1 ( In,0. { Xn) &gt;qa)
1 (In,0.{ Xn) &lt;-9012)
1 (In.0.{ Xn) &lt;-9012)
1 (In.0.5 Xn) &lt;-qa/2)
1 ( In,0. { Xn) &lt;-qa)
1 (In.0.{ Xn) &lt;-qa
1 ( In.0. { Xn) &lt;-qa)
1 (In.0. { Xn) &lt;qa12)
1 (In.0.{ Xn) &lt;qa/2)
1 (In.0. { Xn) &lt;qa/2)
1 (In.0.{ Xn) &lt;qu
1 (m.O.{ Xn) &lt;qu)
1 (In.0.{ Xn) &lt;qa)

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