Question

select the correct answer. the results of a study revealed a sample proportion of 0.43 for a sample with 78 respondents. what is the 99% confidence interval for this situation? a. 28.54% and 57.46% b. 33.75% and 52.25% c. 32.01% and 53.99% d. 37.39% and 48.61%

Explanation:

Step1: Identify the formula

The formula for the confidence - interval of a proportion is $\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p}$ is the sample proportion, $n$ is the sample size, and $z$ is the z - score corresponding to the desired confidence level. For a 99% confidence level, the $z$ - score $z = 2.576$. Here, $\hat{p}=0.43$ and $n = 78$.

Step2: Calculate the standard error

First, calculate the standard error $SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Substitute $\hat{p}=0.43$ and $n = 78$ into the formula:
\[

$$\begin{align*} SE&=\sqrt{\frac{0.43\times(1 - 0.43)}{78}}\\ &=\sqrt{\frac{0.43\times0.57}{78}}\\ &=\sqrt{\frac{0.2451}{78}}\\ &\approx\sqrt{0.0031423}\\ &\approx0.056 \end{align*}$$

\]

Step3: Calculate the margin of error

The margin of error $ME = z\times SE$. Substitute $z = 2.576$ and $SE\approx0.056$ into the formula: $ME=2.576\times0.056\approx0.144$.

Step4: Calculate the confidence - interval

The lower limit of the confidence - interval is $\hat{p}-ME=0.43 - 0.144 = 0.286$ or 28.6% (approx), and the upper limit is $\hat{p}+ME=0.43+0.144 = 0.574$ or 57.4% (approx).

Answer:

A. 28.54% and 57.46%