Chapter 9.2, Problem 62E

To determine

The correct option of the $95\%$ confidence intervals to reject the null hypothesis at the significance level $5\%$ :

1. $\left(0.19,0.27\right)$
2. $\left(0.24,0.30\right)$
3. $\left(0.27,0.31\right)$
4. $\left(0.29,0.38\right)$
5. None of these

Answer to Problem 62E

The correct option is a. The $95\%$ confidence interval to reject the null hypothesis at the significance level is $\left(0.19,0.27\right)$ .

Explanation of Solution

Given:

  $\begin{array}{l}{H}_0:p=0.30\\ H_a:p\ne 0.30\\ \propto =5\%\end{array}$

In favour of the alternate hypothesis, the null hypothesis needs to reject with the significance level of the $5\%$ .

So, there is need of the correct $95\%$ confidence interval. Now check all four options for the value of $0.30$ .

From the four options, three options (b, c, d) contains the value of $0.30$ , so these are options are wrong to reject 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 $H_0:p=0.30$ .

So, only option a $\left(0.19,0.27\right)$ does not contain the value of $0.30$ . And thus it will lead us with the correct confidence interval to reject the given statement of the null hypothesis.

Conclusion:

Hence, the correct option is a.