A simple random sample of size $\mathrm{n}$ is drawn from a population that is normally distributed. The sample mean, $\bar{x}$, is found to be 114 , and the sample standard deviation, $\mathrm{s}$, is found to be 10 . Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? A. Yes, the population does not need to be normally distributed. B. No, the population does not need to be normally distributed. C. Yes, the population needs to be normally distributed. D. No, the population needs to be normally distributed.

Step 1 ：The Central Limit Theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). If the population is normal, then the theorem holds true even for samples smaller than 30.

Step 2 ：Since the question does not provide the sample size, we cannot definitively answer the question. However, we can say that if the sample size was large enough, we could have computed the confidence intervals even if the population had not been normally distributed.

Step 3 ：So, the answer depends on the sample size. If the sample size is large enough, the answer is A. Yes, the population does not need to be normally distributed. If the sample size is not large enough, the answer is D. No, the population needs to be normally distributed.

Step 4 ：Since we don't have information about the sample size, we can't provide a definitive answer.

Step 5 ：\(\boxed{\text{The answer depends on the sample size. If the sample size is large enough, the answer is A. Yes, the population does not need to be normally distributed. If the sample size is not large enough, the answer is D. No, the population needs to be normally distributed.}}\)