Central Limit Theorem help

Question

Suppose that a random sample of n independent measurements of the specific gravity of a certain body are to be taken by a physicist. It is assumed that these measurements follow a certain distribution with mean μ and standard deviation σ. Determine the smallest number of measurements n that must be taken in order to satisfy the following relation:

Pr(|X - μ| < $σ/6$) $\geq$ 0.99

Hint: Use the Central Limit Theorem

I have seen steps to solve this where one equation is:

Pr(|X - μ| < 2.58σ/$\sqrt{n}$)

What I don't understand is where 2.58 comes from.

Answer 1

It is related to the standard normal table.

Let $Z$ be a standard normal random variable. Then the value 2.58 corresponds to 0.99 in the following way: $\Pr(Z\leq 2.58) \approx 0.99$.

Answer 2

It comes from the tables of the cumulative probability of the standard Normal distribution - if you look up the z-score of 2.58, $P(Z<2.58)=0.99506$. Thus the probability $P(|Z|<2.58)=0.99$ when $Z$ comes from the standard Normal distribution. If you look up the Central Limit Theorem, you'll see how it is linked to the standard Normal distribution.