Alright, let's delve into this step by step.
1. First, we need to figure out the proportion of successes in our sample. Successes here refers to the desired or favourable outcomes in our sample size. We have a total of 99 successes out of a sample size of 219. Therefore, the sample proportion is calculated by dividing the number of successes by the sample size: 99 / 219 ≈ 0.452, or 45.2%. This means that in our sample, about 45.2% were successful.
2. Now we go for the standard error. The standard error measures the variation in the sampling distribution. We can calculate the standard error using the formula for the standard deviation of a sample proportion: sqrt[p(1 - p) / n], where sqrt[] denotes the square root, p is the sample proportion, and n is the sample size. By substituting our known values, we get: sqrt[0.452(1 - 0.452) / 219] ≈ 0.0336.
3. To proceed further, we need to find the appropriate z-score for our given confidence level (99%). The z-score is a measure of how many standard deviations an element is from the mean. For a confidence level of 99%, the z-score is approximately 2.576.
4. Now we calculate the margin of error. The margin of error is the product of the z-score and the standard error (margin of error = z-score * standard error). For our given problem, the margin of error becomes: 2.576 * 0.0336 = 0.0866.
5. The confidence interval tells us the range within which we expect our population parameter to fall a certain percentage of the time (in our case, 99% of the time). We calculate it by subtracting the margin of error from the sample proportion and by adding the margin of error to the sample proportion. Therefore, our 99% confidence interval is: (0.452 - 0.0866, 0.452 + 0.0866) = (0.365, 0.539).
So, with 99% confidence, we can claim that the true proportion of successes lies between 36.5% and 53.9%.