suppose the null hypothesis is rejected state the coSolution

Solution

1. (a)H0 is rejected , but true proportion not equal to 0.57.....

2) z-score at 90% = - 1.644854 ( for 2 tailed test)

so, - 1.644854 = ( X_{bar(critical )} - 0.57 ) / sqrt ( 0.55 * 0.45 / 300 )  
so, X_{bar(critical )} = 0.6172448.....

now, P [ Z > ( 0.6172448 - 0.55) /  sqrt ( 0.55 * 0.45 / 300 ) ] = P[ Z > 2.341163] = 0.9903881

beta = 0.9903881

power = 1 - beta = 0.009611886


3)

z-score at 90% = - 1.644854 ( for 2 tailed test)

so, - 1.644854 = ( X_{bar(critical )} - 0.57 ) / sqrt ( 0.52 * 0.48 / 300 )  
so, X_{bar(critical )} = 0.6174448.....

now, P [ Z > ( 0.6174448 - 0.55) /  sqrt ( 0.52 * 0.48 / 300 ) ] = P[ Z > 2.338228 ] = 0.9903123..

beta = 0.9903123..

power = 1 - beta =  0.009687712...........