Let Y1, Y2, . . . , Yn denote a random sample

Chapter 10, Problem 102E

(choose chapter or problem)

Let \(Y_{1}, Y_{2}, \ldots Y_{n}\) denote a random sample from a Bernoulli-distributed population with parameter  That is,

                          \(p\left(y_{1} \mid p\right)=p^{y_{1}\left(1-p^{1-y_{i}}, y_{i}\right.}=0,1\)

a Suppose that we are interested in testing \(H_{0}: p=p_{0}\) versus \(H_{a}: p=p_{a}\), where \(p_{0}<p_{a}\).

i Show that

\(\frac{L\left(p_{0}\right)}{L\left(p_{a}\right)}=\left[\frac{p_{0}\left(1-p_{a}\right)}{\left(1-p_{0}\right) p_{a}}\right]^{\sum y}\left(\frac{1-p_{0}}{1-p_{a}}\right)^{n}\)

ii Argue that \(L\left(p_{0}\right) / L\left(p_{a}\right)<k\) if and only if \(\sum_{i=1}^{n} y_{i}>k^{*}\) for some constant .

iii Give the rejection region for the most powerful test of  versus .

b Recall that \(\sum_{=1}^{n} y_{i}\) has a binomial distribution with parameters  and  Indicate how to determine the values of any constants contained in the rejection region derived in part [a(iii)].

c Is the test derived in part (a) uniformly most powerful for testing \(H_{0}: p=p_{0}\) versus \(H_{a}: p>p_{0}\)? Why or why not?

Equation transcription:

Text transcription:

Y_{1}, Y_{2}, \ldots Y_{n}

p\left(y_{1} \mid p\right)=p^{y_{1}\left(1-p^{1-y_{i}}, y_{i}\right.}=0,1

H_{0}: p=p_{0}

H_{a}: p=p_{a}

p_{0}<p_{a}

\frac{L\left(p_{0}\right)}{L\left(p_{a}\right)}=\left[\frac{p_{0}\left(1-p_{a}\right)}{\left(1-p_{0}\right) p_{a}}\right]^{\sum y}\left(\frac{1-p_{0}}{1-p_{a}}\right)^{n}

L\left(p_{0}\right) / L\left(p_{a}\right)<k

\sum_{i=1}^{n} y_{i}>k^{*}

\sum_{=1}^{n} y_{i}

H_{0}: p=p_{0}

H_{a}: p>p_{0}

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