Use the standard normal distribution or the t-distribution to construct a $90 \%$ confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.
In a random sample of 49 people, the mean body mass index (BMI) was 26.6 and the standard deviation was 6.02.
Which distribution should be used to construct the confidence interval? Choose the correct answer below.
A. Use a normal distribution because the sample is random, the population is normal, and $\sigma$ is known.
B. Use a $t$-distribution because the sample is random, $n \geq 30$, and $\sigma$ is unknown.
C. Use a normal distribution because the sample is random, $n \geq 30$, and $\sigma$ is known.
D. Use a t-distribution because the sample is random, the population is normal, and $\sigma$ is unknown.
E. Neither a normal distribution nor a t-distribution can be used because either the sample is not random, or $\mathrm{n}< 30$, and the population is not known to be normal.
Answer
\(\boxed{\text{Final Answer: The correct answer is C. Use a normal distribution because the sample is random, } n \geq 30, \text{ and } \sigma \text{ is known.}}\)
Step 1 ：The question is asking us to determine which distribution should be used to construct a confidence interval for the population mean given a sample size, mean, and standard deviation.
Step 2 ：The standard deviation is given, which means we know the value of sigma. The sample size is greater than 30, which is generally the threshold for using the normal distribution. However, the question does not specify whether the population is normal or not.
Step 3 ：In general, if the population is normal and sigma is known, we use the normal distribution. If the population is normal and sigma is unknown, we use the t-distribution. If the sample size is large enough (n >= 30), we can use the normal distribution regardless of whether sigma is known or not.
Step 4 ：In this case, since the sample size is greater than 30 and sigma is known, we should use the normal distribution.
Step 5 ：\(\boxed{\text{Final Answer: The correct answer is C. Use a normal distribution because the sample is random, } n \geq 30, \text{ and } \sigma \text{ is known.}}\)
