Question

Which of the following $95\%$confidence intervals would lead us to reject $H_0:p=0.30$in favor of $H_a:pnotequalto0.30$at the $5\%$significance level?

a. $(0.19,0.27)$

b.$(0.24,0.30)$

c. $(0.27,0.31)$

d. $(0.29,0.31)$

e. None of these

Short answer

The required correct option is $\left(a\right)$

The $95\%$confidence interval to reject the null hypothesis at the significance level is$(0.19,0.27)$

Step-by-step solution

Given information

$$\begin{gathered} H_0:p=0.30 \\ {H}_a:pnotequalto0.30 \\ \alpha =5\% \end{gathered}$$

The objective is to determine the correct option of the 95%confidence intervals to reject the null hypothesis at the significance level 5%

The null hypothesis must be rejected with a $5\%$significance level in favor of the alternate hypothesis.

As a result, the correct $95\%$confidence interval is required. Examine all four options for the value $0.30$

Three of the four options $(b,c,d)$have a value of $0.30,$indicating that they are incorrect for rejecting the null hypothesis.

Need to choose the option which does not contain the value of $0.30,$as the null hypothesis statement is related to $H_0:p=0.30$

As a result, the only option $\left(a\right)(0.19,0.27)$lacks the value $0.30.$As a result, we will have the correct confidence interval to reject the given null hypothesis statement.

Therefore, the correct option is$\left(a\right)$