Step 1 :We are given that the Z-score for a 90% confidence level is 1.645, the estimated proportion of the population is 0.5, and the margin of error is 0.01.
Step 2 :We use these values in the formula for calculating the sample size: \(n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}}\).
Step 3 :Substituting the given values into the formula, we get \(n = \frac{{(1.645)^2 \cdot 0.5 \cdot (1-0.5)}}{{(0.01)^2}}\).
Step 4 :Solving this equation gives us \(n = 6765.50625\).
Step 5 :Since we can't have a fraction of a person, we round up to the nearest whole number to get \(n = 6766\).
Step 6 :Final Answer: The required sample size is \(\boxed{6766}\).