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 321 with 212 successes at a confidence level of 99%.
M.E. =
To find the margin of error (M.E.), we first need to determine the critical value for a 99% confidence level.
Since the confidence level is 99%, the alpha value (α) is 1 - confidence level = 1 - 0.99 = 0.01.
Half of the alpha value is 0.01 / 2 = 0.005, which represents the area in the tails of the distribution.
To find the critical value, we need to find the z-score that corresponds to an area of 0.005 in the tails of the standard normal distribution. Using a table or calculator, we find this to be approximately 2.576.
The margin of error (M.E.) is calculated using the formula:
M.E. = z * sqrt(p * (1 - p) / n),
where z is the critical value, p is the sample proportion (212/321), and n is the sample size (321).
Substituting the values into the formula:
M.E. = 2.576 * sqrt((212/321) * (1 - 212/321) / 321)
Calculating this expression:
M.E. ≈ 2.576 * sqrt(0.6604 * 0.3396 / 321)
≈ 2.576 * sqrt(0.0006972266)
≈ 2.576 * 0.0264131
≈ 0.068007
Therefore, the margin of error (M.E.) that corresponds to a sample of size 321 with 212 successes at a confidence level of 99% is approximately 0.068 or 6.8%. This means that we can be 99% confident that the true population proportion falls within the range of 212/321 +/- 0.068.