Let X have a Bernoulli distribution with pmff (x; p) =

QUESTION:

Let \(X\) have a Bernoulli distribution with pmf

\(f(x ; p)=p^{x}(1-p)^{1-x}, \quad x=0,1, \quad 0 \leq p \leq 1\).

We would like to test the null hypothesis \(H_{0}: p \leq 0.4\) against the alternative hypothesis \(H_{1}: p>0.4\). For the test statistic, use \(Y=\sum_{i=1}^{n} X_{i}\), where \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample of size \(n\) from this Bernoulli distribution. Let the critical region be of the form \(C=\{y: y \geq c\}\).

(a) Let \(n=100\). On the same set of axes, sketch the graphs of the power functions corresponding to the three critical regions, \(C_{1}=\{y: y \geq 40\}, C_{2}=\{y:\) \(y \geq 50\}\), and \(C_{3}=\{y: y \geq 60\}\). Use the normal approximation to compute the probabilities.

(b) Let \(C=\{y: y \geq 0.45 n\}\). On the same set of axes, sketch the graphs of the power functions corresponding to the three samples of sizes 10,100 , and 1000 .

Equation Transcription:

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,

Text Transcription:

X  

f(x;p)=p^x(1-p)^1-x, x=0,1, 0≤p≤1

H_0:p < or = 0.4  

H_1:p>0.4

Y=sum_i=1^n  X_i  

X_1,X_2,…,X_n  

n

C={y:y > or = c}

n=100

C_1={y:y > or = 40},C_2={y: y > or = 50}

C_3={y:y > or =60}  

C={y:y> or = 0.45n}