Final answer:
The conditions that should be checked to determine if the sample mean will be approximately normally distributed are that the sample size should be more than 30 (B), or that the sample comes from a normally distributed population (D). These conditions are part of the Central Limit Theorem and statistical sampling theory.
Step-by-step explanation:
To determine if the sample mean will be approximately normally distributed, certain conditions must be checked. While options A and C mentioned are not directly related to the conditions necessary for a sample mean to be normally distributed, options B and D relate to important concepts that are part of the Central Limit Theorem and sampling theory.
Option B mentions that the sample size should be more than 30. This is a general rule of thumb in statistics, suggesting that for larger sample sizes (typically n > 30), the distribution of the sample means will tend to be approximately normal regardless of the shape of the population distribution. This rule is a part of the Central Limit Theorem, which states that the distribution of sample means will approach a normal distribution as the sample size increases.
Option D highlights the condition where the sample comes from a normally distributed population. This condition ensures that the sample mean will also be normally distributed for any sample size, a direct result of the population's distribution.
Moreover, option E refers to the success-failure condition for binomial distributions, where 'np' and 'n(1-p)' (or 'nq') must both be greater than 5, which is a rule to approximate binomial distributions using a normal model when performing a hypothesis test for proportions.