Sample Size for Proportion Find the sample size required to estimate the percentage of statistics students who take their statistics course online. Assume that we want 95% confidence that the proportion from the sample is within two percentage points of the true population percentage.

Verified step by step guidance

1. Determine the formula for calculating the required sample size for a proportion. The formula is: $n=\frac{z^2}{}$, where $n$ is the sample size, $z$ is the z-score corresponding to the confidence level, $p$ is the estimated proportion, and $E$ is the margin of error.
2. Identify the values given in the problem: the confidence level is 95%, so the z-score corresponding to this level is approximately 1.96. The margin of error is 2%, or 0.02. Since no prior estimate of the proportion is given, use $p=0.5$, which maximizes the required sample size.
3. Substitute the known values into the formula: $n=\frac{{1.96}^2}{}$.
4. Simplify the numerator: calculate ${1.96}^2$ and multiply it by $0.5(1-0.5)$.
5. Simplify the denominator: calculate ${0.02}^2$, then divide the simplified numerator by this value to find the required sample size.

Key Concepts

Sample Size Determination

Sample size determination is the process of calculating the number of observations or replicates needed to obtain a desired level of precision in estimating a population parameter. In this context, it involves using statistical formulas to ensure that the sample accurately reflects the population, particularly when estimating proportions.

Confidence Level

The confidence level represents the degree of certainty that the population parameter lies within a specified range of the sample estimate. A 95% confidence level means that if we were to take many samples, approximately 95% of the calculated confidence intervals would contain the true population proportion, providing a strong assurance of accuracy.

Margin of Error

The margin of error indicates the range within which the true population proportion is expected to fall, based on the sample estimate. In this scenario, a margin of error of two percentage points means that the estimate from the sample can vary by plus or minus 2% from the actual population proportion, which is crucial for understanding the precision of the estimate.