The average age of residents in a large residential retirement community is 69 years with standard deviation 5.8 years. A simple random sample of 100 residents is to be selected, and the sample mean age ¯ x x¯ of these residents is to be computed. We know the random variable ¯ x x¯ has approximately a Normal distribution because____________.

a. of the central limit theorem.
b. of the 68‑95‑99.7 rule.
c. the population from which we’re sampling has a Normal distribution.
d. of the law of large numbers.

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size, larger than 30, can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex]

In this problem, the sample size is 100, so it is sufficiently large to use the Central Limit Theorem. The mean of the sample in 69 and the standard deviation of the sample is 0.58.

So the correct answer is:

a. of the central limit theorem.