Question 7 Assume that a sample is used to estimate a population proportion p. Find the margin of error M.E. that corresponds to a sample of size 235 with $40.9 \%$ successes at a confidence level of $99.9 \%$. \[ M . E .=\square \% \] Report answer accurate to one decimal place (as a number of percentage points). Answer should be obtained without any preliminary rounding (however, the critical value may be rounded to 3 decimal places).

Step 1 ：We are given a sample size of 235, a sample proportion of 40.9% or 0.409, and a z-score of 3.291 which corresponds to a confidence level of 99.9%.

Step 2 ：We can use these values to calculate the margin of error using the formula: \(M.E. = Z * \sqrt{\frac{p(1-p)}{n}}\)

Step 3 ：Substituting the given values into the formula, we get: \(M.E. = 3.291 * \sqrt{\frac{0.409(1-0.409)}{235}}\)

Step 4 ：Solving the above expression, we find that the margin of error is approximately 10.6%

Step 5 ：Thus, the margin of error that corresponds to a sample of size 235 with 40.9% successes at a confidence level of 99.9% is \(\boxed{10.6\%}\)