Refer to the accompanying data display that results from a simple random sample of times (minutes) between eruptions of the Old Faithful geyser. The confidence level of $95 \%$ was used. Complete parts (a) through (d) below.
TInterval
$(85.74,91.76)$
$\bar{x}=88.75$
$\mathrm{Sx}=8.897431411$
$n=36$
a. What are the requirements for using the methods for the mean to construct a confidence interval estimate of a population mean?
A. The sample is a simple random sample, the population is normally distributed, and $n> 30$.
B. The sample is a simple random sample, the population is normally distributed, and $n \leq 30$.
C. The sample is a simple random sample and the population is normally distributed or $n \leq 30$.
D. The sample is a simple random sample and the population is normally distributed or $n> 30$.
Answer
Final Answer: \(\boxed{\text{A. The sample is a simple random sample, the population is normally distributed, and } n>30}\)
Step 1 ：The question is asking about the requirements for using the methods for the mean to construct a confidence interval estimate of a population mean. The requirements are: The sample is a simple random sample, the population is normally distributed, and the sample size is large enough (at least 30). Looking at the options given, option A states that the sample is a simple random sample, the population is normally distributed, and the sample size is greater than 30. This matches the requirements, so the answer is A.
Step 2 ：Final Answer: \(\boxed{\text{A. The sample is a simple random sample, the population is normally distributed, and } n>30}\)
