Final answer:
To calculate the sample size for a 95% confidence interval, we need to first determine the margin of error. Assuming no information is available regarding the population proportion, a sample size of 100 with 8 defectives results in a preliminary sample proportion of 0.08. Using the safe approach, the sample size can be calculated as 1068. Computing the 95% confidence intervals using both methods, the intervals are (0.0194, 0.1406) and (0.0514, 0.1086) respectively.
Step-by-step explanation:
In order to calculate the sample size for a 95% confidence interval, we first need to determine the margin of error, which is the maximum difference between the sample proportion and the true population proportion. Let's assume that no information is available regarding the population proportion:
- For a preliminary sample with a sample size of 100 and 8 defectives, the sample proportion is 8/100 = 0.08.
- To calculate the sample size using the safe approach, we use the formula n = (z^2 * p * (1-p)) / (E^2), where E is the margin of error. Since we don't have any information about the population proportion, we can use the worst-case scenario which is p = 0.5. Assuming a margin of error of 3%, the formula becomes n = (1.96^2 * 0.5 * (1-0.5)) / (0.03^2) = 1067.11. Rounded up, the sample size should be 1068.
Computing a 95% confidence interval using both methods:
- Using the preliminary sample, the interval is given by p ± Z * sqrt(p * (1-p) / n), where Z is the z-score corresponding to the desired confidence level. For 95%, Z is 1.96. Substituting the values, the interval is 0.08 ± 1.96 * sqrt(0.08 * (1-0.08) / 100) = 0.08 ± 0.0606. Therefore, the interval is (0.0194, 0.1406).
- Using the safe approach, the interval is given by p ± Z * sqrt(p * (1-p) / n). Substituting the values, the interval is 0.08 ± 1.96 * sqrt(0.5 * (1-0.5) / 1068) = 0.08 ± 0.0286. Therefore, the interval is (0.0514, 0.1086).