Step 1 :Given that the number of respondents, \(n = 987\), and the number of respondents who said 'yes', \(x = 567\). The z-score for a 90% confidence level, \(Z = 1.645\).
Step 2 :First, calculate the sample proportion, \(p\), using the formula \(p = \frac{x}{n}\).
Step 3 :Substitute the given values into the formula to get \(p = \frac{567}{987} = 0.574468085106383\).
Step 4 :Next, calculate the margin of error, \(E\), using the formula \(E = Z \sqrt{\frac{p(1 - p)}{n}}\).
Step 5 :Substitute the values into the formula to get \(E = 1.645 \sqrt{\frac{0.574468085106383(1 - 0.574468085106383)}{987}} = 0.02588846781788074\).
Step 6 :Round the value of \(E\) to four decimal places to get \(E = 0.0259\).
Step 7 :Final Answer: The value of the margin of error \(E\) is \(\boxed{0.0259}\).