The population proportion is assumed to be 0.13 . If the desired standard error is 0.0182 or less, what is the minimum sample size needed?

To determine the minimum sample size, the formula to calculate the sample size for a population proportion is given by: \[ n = \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \] where: - \( Z \) is the Z-value corresponding to the confidence level, - \( p \) is the population proportion (0.13 in this case), - \( E \) is the desired margin of error (0.0182). Assuming a 95% confidence level, the Z-value is 1.96. Plugging in the numbers, we calculate: \[ n = \frac{1.96^2 \cdot 0.13 \cdot (1 - 0.13)}{0.0182^2} \] \[ n = \frac{3.8416 \cdot 0.13 \cdot 0.87}{0.00033124} \] \[ n = \frac{0.43801024}{0.00033124} \] \[ n \approx 1322.69 \] Therefore, the minimum sample size needed is approximately 1323. The real world loves sample sizes! Imagine you’re conducting a survey to understand whether people prefer pineapple on pizza. With a sample size of 1323, you'd be equipped to confidently gauge opinions in your community. Maintaining a standard error of 0.0182 means your results will be super reliable. So whether you love or hate pineapple on pizza, you'll know the real consensus! When calculating sample sizes, a common mistake is not factoring in the population proportion correctly or misjudging the margin of error. Always double-check your values and ensure you’re using the right confidence level, as this can dramatically alter the required sample size. Remember, larger sample sizes give you more accurate results, but also increase costs and resources required. Balance is key!