How do I calculate a point estimate of the largest 10%?

Question

Here is the provided data:

The question asked is:
Calculate a point estimate of the value that separates the largest 10% of all values in the thickness distribution from the remaining 90% and state which estimator you used. [Hint: Express what you are trying to estimate in terms of μ and σ.]

The answer: 1.78138

How is the above 1.78138? I got that the σ = .3385 and μ = 1.348125, how do these values help me reach the answer? What must I do to further understand this?

Answer 1

What you are doing is you are trying to fit a normal distribution on your data. Your assumption is that the numbers you provided come from a normally distributed random variable, let's call it $X$.

You correctly calculated $\sigma$ and $\mu$, i.e., your assumption now is that $X$ is distributed as $N(\mu,\sigma)$. Now, the question is:

What is the value $x_0$, such that $90\%$ of all values given by $X$ are below $x_0$?

Or, in other words,

What is the value of $x_0$ such that the probability of a random sample from $X$ being below $x_0$ is $90\%$?