Final answer:
The margin of error (E) calculation involves the critical z-value corresponding to the confidence level, the sample proportion (p), and the sample size (n). To calculate the necessary sample size, one uses an altered formula incorporating the error bound of the margin (EBM). Changes in confidence levels will alter the width of the confidence interval.
Step-by-step explanation:
Understanding Margin of Error and Sample Size Calculation
To calculate the margin of error (E) for a given confidence level, we use the formula
E = za/2 * sqrt[ p(1-p) / n ], where za/2 is the critical z-value that corresponds to the chosen confidence level, p is the point estimate of the population proportion, and n is the sample size.
For example, if a survey of 2,000 adults found that a sample proportion (p') was 0.78 and we want a 90% confidence interval, we need to find the z-score (za/2) corresponding to the 90% confidence level. This z-score is often found in statistical tables or by using statistical software.
Once the z-score is known, we can use the formula to compute the margin of error. Solving for n, the equation for sample size becomes n = (za/2)^2 * p'(1-p') / EBM^2, where EBM is the error bound of the margin. Always make sure to round up the value of n to the nearest whole number to ensure the sample size is large enough.
When constructing a confidence interval with standard normal distribution Z~ N(0, 1), we use the population standard deviation (o) to calculate the error bound and the confidence interval is given by the formula:
x ± za/2 * o/sqrt(n).
Factors such as sampling method, response rate, and question wording might affect survey outcomes and are not accounted for by the margin of error.
Changing the confidence level will affect the width of the confidence interval; for instance, decreasing the confidence level from 99% to 90% would generally result in a narrower interval, assuming all other factors remain constant.