When working with a normal approximation to a binomial distribution to find the margin of error for a sample proportion [tex]\(\hat{p}\)[/tex] and a sample size [tex]\(n\)[/tex], there are conditions that need to be satisfied to ensure the distribution of the sample proportion is approximately normal.
Specifically, both the number of successes [tex]\(n \hat{p}\)[/tex] and the number of failures [tex]\(n(1-\hat{p})\)[/tex] should typically be at least 10. This condition helps to ensure that the sample size is sufficiently large to invoke the Central Limit Theorem, leading to a normal distribution.
Therefore, the correct relationship must be:
[tex]\[ n(1-\hat{p}) \geq 10 \][/tex]
This ensures that the number of failures in the sample is large enough so that the sampling distribution of [tex]\(\hat{p}\)[/tex] can be approximated by a normal distribution.
Thus, the correct answer is:
D. [tex]\(n(1-\hat{p}) \geq 10\)[/tex]
