To find the margin of error (E) for estimating the population proportion (p) with a given confidence level, you can use the formula:
E = z * sqrt((p̂ * (1 - p̂)) / n)
Where:
- E is the margin of error
- z is the z-score corresponding to the desired confidence level
- p̂ is the sample proportion (x/n)
- n is the sample size
First, calculate the sample proportion (p̂) by dividing the number of successes (x) by the sample size (n):
p̂ = x / n = 220 / 550 = 0.4
Next, find the z-score that corresponds to the desired confidence level. For a 90% confidence level, the corresponding z-score is approximately 1.645 (which you can look up in a standard normal distribution table or use a calculator).
Now, substitute the values into the formula:
E = 1.645 * sqrt((0.4 * (1 - 0.4)) / 550)
Simplifying:
E = 1.645 * sqrt((0.24) / 550)
E ≈ 1.645 * 0.0309
E ≈ 0.0508 (rounded to four decimal places)
Therefore, the margin of error (E) that corresponds to the given statistics and confidence level is approximately 0.0508.