This is what the AI just spit out.
To determine the sample size needed to confirm at least 50% of the heifers are pregnant with a 75% confidence level, you can use a formula for calculating sample size for proportions in a finite population.
The formula is:
n = [Z^2 * P(1-P) * N] / [(N-1) * MOE^2 + Z^2 * P(1-P)]
where:
n = sample size
Z = Z-score corresponding to the desired confidence level (for 75% confidence level, Z ≈ 1.15)
P = estimated proportion (0.5 for 50%)
N = population size (25 heifers)
MOE = margin of error
Assuming a margin of error of 0.05 (5%), you can plug in the values:
n = [(1.15)^2 * 0.5 (1-0.5) * 25] / [(25-1) * 0.05^2 + (1.15)^2 * 0.5 (1-0.5)]
n = 2.56 / (24 * 0.0025 + 1.56)
n = 2.56 / (0.06 + 1.56)
n ≈ 2.56 / 1.62
n ≈ 1.58
Therefore, you would need to initially test at least 2 heifers to confirm with 75% confidence that at least 50% of the 25 heifers are pregnant.