Question

consider the research (alternative) hypothesis that less than 25 percent of U of T students watch the television show "American Idol." Suppose a random sample of 78 U of T students is selected and asked if they watch the show. With a 5% significance level, what is the probability of failing to reject the null hypothesis if in fact only 20 percent of U of T students watch the show

Answer 1

H0:p=0.25

we have to find probability of failing to reject the null hypothesis if in fact only 20 percent of U of T students watch the show. p( failling to rejecting the null under Pa=0.20)

first Find the rejection region: p(Z>Zalpha)=0.05

p(Z>1.6449)=0.05

n=78, p0=0.25

(-0.25)/sqrt(0.25*0.75/78)=1.6449

=0.25+1.6449*sqrt(0.25*0.75/78)=0.3306

Rejection region is >0.3306

probability of failing to reject the null hypothesis if in fact only 20 percent of U of T students watch the show

p(failling to rejecting the null under Pa=0.20) =p( <0.3306)

=p((-0.20)/sqrt(0.20*0.80/78) <(0.3306-0.20)/sqrt(0.20*0.80/78))

=p(Z<2.8835)

=0.9980

probability of failing to reject the null hypothesis if in fact only 20 percent of U of T students watch the show = 0.9980

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