Suppose Y is a random sample of size 1 from a population

Chapter 10, Problem 96E

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QUESTION:

Suppose  is a random sample of size 1 from a population with density function

\(f\left(y \mid \theta=\left\{\begin{array}{l}

\theta y^{0-1}, 0 \leq y \leq 1, \\

0,

\end{array}\right.\right.

\)

where \(\theta>0\)

a Sketch the power function of the test with rejection region: \(Y>.5\).

b Based on the single observation , find a uniformly most powerful test of size  for testing    \(H_{0}: \theta=1\) versus \(H_{a}: \theta>1\).

Equation transcription:

Text transcription:

f\left(y \mid \theta=\left\{\begin{array}{l}

\theta y^{0-1}, 0 \leq y \leq 1, \\

0,

\end{array}\right.\right.

\theta>0

Y>.5

H_{0}: \theta=1

H_{a}: \theta>1

Questions & Answers

QUESTION:

Suppose  is a random sample of size 1 from a population with density function

\(f\left(y \mid \theta=\left\{\begin{array}{l}

\theta y^{0-1}, 0 \leq y \leq 1, \\

0,

\end{array}\right.\right.

\)

where \(\theta>0\)

a Sketch the power function of the test with rejection region: \(Y>.5\).

b Based on the single observation , find a uniformly most powerful test of size  for testing    \(H_{0}: \theta=1\) versus \(H_{a}: \theta>1\).

Equation transcription:

Text transcription:

f\left(y \mid \theta=\left\{\begin{array}{l}

\theta y^{0-1}, 0 \leq y \leq 1, \\

0,

\end{array}\right.\right.

\theta>0

Y>.5

H_{0}: \theta=1

H_{a}: \theta>1

ANSWER:

Step 1 of 3

Given:

Suppose Y is a random sample of size 1 from a population with a density function

where  > 0.

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