The population proportion is assumed to be 0.45 . If the desired standard error is 0.0174 or less, what is the minimum sample size needed?

To find the minimum sample size needed to achieve a desired standard error of 0.0174 or less, we can use the formula for the standard error of a proportion: \[ SE = \sqrt{\frac{p(1-p)}{n}} \] where \( p \) is the population proportion and \( n \) is the sample size. Rearranging the formula to solve for \( n \), we get: \[ n = \frac{p(1-p)}{SE^2} \] Substituting \( p = 0.45 \) and \( SE = 0.0174 \): \[ n = \frac{0.45(1 - 0.45)}{(0.0174)^2} \] Calculating \( n \): \[ n = \frac{0.45 \times 0.55}{0.00030276} \approx \frac{0.2475}{0.00030276} \approx 816.55 \] Since the sample size must be a whole number, we round up to get \( \text{n} \approx 817 \). Thus, you need a minimum sample size of 817 to achieve the desired standard error.