How do you know the random variable x\"bar\" has a normal distribution? The aver

Question

How do you know the random variable x"bar" has a normal distribution?

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 of these residents is to be computed. We know the random variable x has approximately a Normal distribution because of the central limit theorem. the law of large numbers. the 68-95-99.7 rule. the population from which we're sampling has a Normal distribution.

Explanation / Answer

The centrl limit theorem

The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

In this case sample size is 100 which is large.

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