Final answer:
To estimate the proportion of children under 12 with chicken pox within 1% with 95% confidence and using a pilot study estimate of 8.6%, the minimum sample size is approximately 3037 children.
Step-by-step explanation:
To estimate the true proportion of children under 12 who have had chicken pox to within 1% with 95% confidence, you should use the formula for sample size for estimating a population proportion:
n = (Z^2 * p * (1 - p)) / E^2
Where n is the sample size, Z is the z-score corresponding to the confidence level, p is the estimated proportion (from the pilot study), and E is the desired margin of error. From the standard normal distribution, a 95% confidence level corresponds to a z-score of approximately 1.96. The pilot study estimated that 8.6% of children under 12 have had chicken pox, so p = 0.086. The desired margin of error is 1%, or E = 0.01.
Plugging the numbers into the formula:
n = (1.96^2 * 0.086 * (1 - 0.086)) / 0.01^2
Calculating this gives:
n = (3.8416 * 0.086 * 0.914) / 0.0001
n = 303.606656 / 0.0001
n ≈ 3037
Therefore, the minimum sample size that must be taken is approximately 3037 children (rounding up to the next whole number, as you cannot survey a fraction of a child).