In which of the following situations can the sampling distribution of the sample proportion be safely approximated by a normal distribution?

• A. When both $np$ and $n-np$ are at least 5
• B. When both $np$ and $n-np$ are at least 10, that is, $np\ge 10$ and $n-np\ge 10$
• C. When both $np$ and $n-np$ are at least 10
• D. When the sample size $n$ is greater than 30, regardless of $p$

Answer: (B)

Verified step by step guidance

1. Understand that the sampling distribution of the sample proportion \( \hat{p} \) can be approximated by a normal distribution when certain conditions related to the sample size \( n \) and the population proportion \( p \) are met.
2. Recall the two key conditions for the normal approximation to be valid: both \( n p \) and \( n (1 - p) \) must be sufficiently large to ensure the distribution is not too skewed.
3. Identify the commonly accepted rule of thumb, which states that both \( n p \geq 10 \) and \( n (1 - p) \geq 10 \) are required for the normal approximation to hold well.
4. Note that this condition ensures there are enough expected successes (\( n p \)) and failures (\( n (1 - p) \)) in the sample to approximate the binomial distribution with a normal distribution.
5. Recognize that simply having \( n > 30 \) is not sufficient if \( p \) is very close to 0 or 1, because the distribution could still be skewed; the conditions on \( n p \) and \( n (1 - p) \) are more precise criteria.