Final answer:
To have the standard error of a population proportion estimation not exceed one percent in the worst-case scenario (p \u2248 0.5), a simple random sample of at least 2500 participants is required.
Step-by-step explanation:
To determine the needed size of a simple random sample so that the standard error of the population proportion does not exceed one percent, we must consider the worst-case scenario, which is when the population proportion (p) is approximately 0.5. The formula for the standard error (SE) of a proportion is SE = \(\sqrt{\frac{p(1-p)}{n}}\), and we aim for this to be less than or equal to 0.01. Since p is at its maximum variability when p = 0.5, the formula simplifies to SE = \(\frac{0.5}{\sqrt{n}}\).
To find the minimum sample size n that satisfies SE \(\leq\) 0.01, we square both sides of the inequality to get \(\frac{0.25}{n} \leq 0.0001\), giving us n \(\geq\) \(\frac{0.25}{0.0001}\) which simplifies to n \(\geq\) 2500. Therefore, you need a sample of at least 2500 participants to ensure that the standard error does not exceed one percent for a 90 percent confidence interval.