Questions: Use the given data to construct a confidence interval of the requested level. x=77, n=181, confidence level 98%

Solution Steps

The sample proportion \( \hat{p} \) is calculated as follows:

\[ \hat{p} = \frac{x}{n} = \frac{77}{181} \approx 0.4254 \]

The significance level \( \alpha \) for a confidence level of \( 98\% \) is given by:

\[ \alpha = 1 - \text{confidence level} = 1 - 0.98 = 0.02 \]

Using the formula for the confidence interval for a single population proportion:

\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

where \( z \) is the z-score corresponding to the \( 98\% \) confidence level, which is approximately \( 2.326 \).

Substituting the values:

\[ 0.4254 \pm 2.326 \cdot \sqrt{\frac{0.4254(1 - 0.4254)}{181}} \]

Calculating the margin of error:

\[ \text{Margin of Error} = 2.326 \cdot \sqrt{\frac{0.4254 \cdot 0.5746}{181}} \approx 0.0854 \]

Thus, the confidence interval is:

\[ (0.4254 - 0.0854, 0.4254 + 0.0854) = (0.3400, 0.5110) \]

The calculated confidence interval \( (0.3400, 0.5110) \) matches one of the provided options:

• \( 0.340 < p < 0.511 \)

Final Answer