To determine the 95% confidence interval for the proportion of club members planning to vote for Talia, we can outline the step-by-step process as follows:
1. Determine the sample proportion:
The sample proportion, denoted as [tex]\(\hat{p}\)[/tex], is calculated by dividing the number of members planning to vote for Talia by the total number of members surveyed.
[tex]\[
\hat{p} = \frac{38}{50}
\][/tex]
2. Convert to decimal:
[tex]\[
\hat{p} = 0.76
\][/tex]
3. Margin of Error:
The margin of error [tex]\(E\)[/tex] is given as 12%, which in decimal form is:
[tex]\[
E = 0.12
\][/tex]
4. Calculate the confidence interval:
The confidence interval is calculated using the formula:
[tex]\[
C = \hat{p} \pm E
\][/tex]
Substituting the values, we get:
[tex]\[
C = 0.76 \pm 0.12
\][/tex]
5. Find the lower bound:
[tex]\[
\text{Lower bound} = 0.76 - 0.12 = 0.64
\][/tex]
6. Find the upper bound:
[tex]\[
\text{Upper bound} = 0.76 + 0.12 = 0.88
\][/tex]
7. Convert to percentages:
To convert the decimal values into percentages, multiply by 100:
[tex]\[
\text{Lower bound percentage} = 0.64 \times 100 = 64\%
\][/tex]
[tex]\[
\text{Upper bound percentage} = 0.88 \times 100 = 88\%
\][/tex]
8. Conclusion:
The 95% confidence interval for the proportion of club members planning to vote for Talia is between 64% and 88%.
Thus, the correct answer is:
[tex]\[
\boxed{\text{between } 64 \% \text{ and } 88 \%}
\][/tex]
