Assume that a ramdom sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confinence level. n=550, x=165, 95% confindence

To find the margin of error (E), we can use the formula:

E = Z * sqrt((p * (1 - p)) / n)

Where:
- E is the margin of error
- Z is the Z-score corresponding to the desired confidence level (in this case, 95%)
- p is the estimated population proportion
- n is the sample size

First, we need to find the Z-score for a 95% confidence level. The Z-scores can be looked up in a standard normal distribution table or calculated using a statistical calculator.

For a 95% confidence level, the Z-score is approximately 1.96.

Next, we substitute the values into the formula:

E = 1.96 * sqrt((165/550) * (1 - 165/550)) / 550)

Calculating the expression inside the square root:

E = 1.96 * sqrt((0.3 * 0.7) / 550)

E = 1.96 * sqrt(0.0003590909090909091)

E = 1.96 * 0.018940513

E = 0.037037

Therefore, the margin of error (E) is approximately 0.037 or 3.7%. This means that we can be 95% confident that the true population proportion falls within ±3.7% of the estimated proportion based on the random sample.