Q 60. - The Practice of Statistics for AP

Question

Which of the following $95\%$ confidence intervals would lead us to reject H0 : $p = 0.30$ in favor of

Ha :p not equal to $0.30$ at the $5\%$ significance level?

$$\begin{gathered} \left(a\right) (0.29, 0.38) \left(c\right) (0.27, 0.31) \left(e\right) None of these \\ \left(b\right) (0.19, 0.27) \left(d\right) (0.24, 0.30) \end{gathered}$$

Step-by-Step Solution

Answer

The correct option is (b)$(0.19, 0.27)$

Step 1: Given information

Confidence level $= 95\%$

The null and alternative hypotheses are: $$\begin{gathered} H0 : p = 0.30 \\ {H}_a : p \ne 0.30 \end{gathered}$$

Step 2: Concept

Confidence intervals give information not found in significance tests, such as a range of feasible values for the true population parameter $p$ A two-sided $H0:p = p0$ test at significance level $a$ yields a result similar to a $100(1- a)$ percent confidence interval.

Step 3: Explanation

If the confidence interval does not contain $0.30$ the null hypothesis is rejected. The confidence interval $(0.19, 0.27)$ is the only one that does not contain the value of $0.30$Confidence intervals give information not found in significance tests, such as a range of feasible values for the true population parameter $p$ A two-sided $H0:p = p0$ test at significance level a yields a result similar to a $100(1- a)$ percent confidence interval. This would result in the null hypothesis being rejected. As a result, the best solution is (b).