Question

A$95\%$confidence interval for the proportion of viewers of a certain reality television

show who are over 30 years old is $(0.26,0.35)$. Suppose the show's producers want to est the hypothesis $\backslash H_0:p=0.25$against Ha: $H_a:p\ne 0.25$. Which of the following is an appropriate conclusion for them to draw at the $\alpha =0.05$

a. Fail to reject $H_0$; there is convincing evidence that the true proportion of viewers of this reality TV show who are over 30 years old equals $0.25$

b. Fail to reject $H_0$there is not convincing evidence that the true proportion of viewers of this reality TV show who are over 30 years old differs from$0.25.$

c. Reject $H_0$; there is not convincing evidence that the true proportion of viewers of this reality TV show who are over 30 years old differs from $0.25$

. d. Reject $H_0$; there is convincing evidence that the true proportion of viewers of this reality TV show who are over 30 years old is greater than $0.25.$

e. Reject $H_0$; there is convincing evidence that the true proportion of viewers of this reality TV show who are over 30 years old differs from$0.25$.

Short answer

Reject$H_0$, there is convincing evidence that the true proportion of viewers differs from $0.25$

Step-by-step solution

Step 1:Given information

A $95\%$ confidence interval for the proportion of viewers of a certain reality television show who are over$30$ years old is$(0.26,0.35).$Suppose the show’s producers want to test the hypothesis $H_0$: $p=0.25$ against Ha: $p\ne 0.25.$

Step 2:Explaination

It is observed that the confidence interval not having$0.25,$it is showing that it is not likely that the proportion of viewers of a certain actually television show is $0.25$and therefore there is convincing proof that the proportion is different from $0.25$(since the alternative hypothesis is $H_1:p\ne 0.25$).

Hence, the correct option is (e)