A charitable organization wishes to estimate, to within half...

A charitable organization wishes to estimate, to within half a percentage point, the proportion of all telephone solicitations to its donors that result in a gift, with $90 \%$ confidence. Estimate the minimum sample size required, using the information that in the past the response rate has been about $30 \%$.

A charitable organization wishes to estimate, to within half a percentage point, the proportion of all
telephone solicitations to its donors that result in a gift, with $90 \%$ confidence. Estimate the minimum sample
size required, using the information that in the past the response rate has been about $30 \%$.

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Key Concepts

Confidence Level

This is the percentage that reflects how sure we are that the true population parameter lies within our confidence interval. In sample size estimation, a higher confidence level requires a larger sample size to ensure that the estimated interval captures the true proportion with the specified probability.

Margin of Error

The margin of error determines the maximum allowable difference between the sample estimate and the actual population parameter. It sets the precision of the estimate; a smaller margin of error demands a larger sample size to maintain the desired confidence level.

Proportion Estimate

The sample proportion (or prior estimate) provides an initial guess of the population parameter, used in sample size formulas to estimate variability. It is critical because the variability of a proportion, captured by p(1-p), is maximized when p is 0.5, and prior estimates can lead to a lower required sample size if they deviate from 0.5.

Sample Size Calculation for Proportions

This concept involves using a formula that incorporates the confidence level (through the critical value), the margin of error, and the variability of the proportion estimate. The formula n = (Z^2 * p(1-p)) / E^2 provides the minimum number of observations needed to achieve the desired precision and confidence in a proportion estimate.

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