Step 1 :Given values are the margin of error \(E = 0.079\), the proportion of residents without health insurance \(p = 0.72\), and the Z-score for a 97% confidence interval \(Z = 2.17\).
Step 2 :Use these values in the formula for sample size in a proportion estimation: \(n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}\).
Step 3 :Substitute the given values into the formula: \(n = \frac{(2.17)^2 \cdot 0.72 \cdot (1-0.72)}{(0.079)^2}\).
Step 4 :Calculate the value of \(n\).
Step 5 :Round up \(n\) to the nearest whole number to get the required sample size.
Step 6 :The sample size required to limit the margin of error to within 0.079 of the population proportion for a 97% confidence interval is \(\boxed{153}\).