Final answer:
To estimate a population proportion with a 95% confidence level and a given margin of error, use the sample size formula with the appropriate Z-score and estimated population proportion. A confidence interval for the population mean is constructed around the point estimate with the given margin of error. Variations in the confidence level or margin of error directly affect the required sample size and the width of the confidence interval.
Step-by-step explanation:
Sample Size Calculation and Confidence Intervals
When estimating a population proportion with a confidence level of 95% and a margin of error (E) of 0.092, the formula for the required sample size (n) is n = (Z² * p * (1-p)) / E², where Z is the Z-score for the 95% confidence level, and p is the estimated population proportion. If prior research suggests that the population proportion is 0.6, then you would use that as your p-value in the formula.
To construct a confidence interval for the population mean, you use the formula: (point estimate - error bound, point estimate + error bound). If the mean is twenty-five percent (0.25), and the margin of error is 0.040, the confidence interval would be (0.25 - 0.040, 0.25 + 0.040), which simplifies to (0.210, 0.290).
The sample size needed for a 95 percent confidence interval within three percentage points of a 0.5 population proportion can be calculated using the same sample size formula mentioned above, with p set to 0.5 and E to 0.03.
To determine the minimum sample size for a 90 percent confidence level with an E of 0.05, you would use the corresponding Z-score for the 90% confidence level in the sample size formula.