Question

The average age of cars owned by residents of a small city is 6 years with a standard deviation of 2.2 years. A simple random sample of 400 cars is to be selected, and the sample mean age of these cars is to be computed. We know the random variable has approximately a normal distribution because of

Answer 1

The random variable
`\bar x` has approximately a normal distribution because of the central limit theorem.

Explanation:

According to the Central Limit Theorem if we have an unknown population with mean μ and standard deviation σ and appropriately huge random samples (n ≥ 30) are selected from the population with replacement, then the sampling distribution of the sample mean will be approximately normally distributed.

Then, the mean of the sample means is given by,

`\mu_(\bar x)=\mu`

And the standard deviation of the sample means is given by,

`\sigma_(\bar x)=(\sigma)/(√(n))`

Let the random variable X be defined as the age of cars owned by residents of a small city.

It is provided that:

μ = 6 years

σ = 2.2 years

n = 400

As the sample selected is too large, i.e. n = 400 > 30, according to the central limit theorem the sampling distribution of the sample mean (
`\bar x`) will be approximately normally distributed.