This is a question from a past stat exam that I am studying because my final is in two days (lol). It'd be great if someone could guide me through how do both parts of the problem. I know gamma function was on the exam but I'm not sure if it was either this question or this other one...
Suppose that, from past experience, Klarifye Research, LLC, knows that a population of witbugs has a normal distribution with a mean weight of 250g and a standard deviation of 15.
a. What is the probability that a witbug will have a weight greater than 270g?
b. What weight would be the 85th percentile? Give a numeric answer rounded to 2 decimal places.
Assuming you have a standard normal distribution table handy...
(a)
You can use the table once you have converted your random variable (let's call it $X$) to the standard normal distribution $Z$. The conversion is $$Z = \frac{X - \mu}{\sigma}$$
where in this case $\mu = 250$ and $\sigma = 15$. So $$P(X > 270) = P\left(Z = \frac{X - \mu}{\sigma} > \frac{270 - \mu}{\sigma}\right) = P\left(Z > \frac{4}{3}\right)$$
The standard normal distribution table gives $\phi(z) = P(Z < z)$ so look up $\phi\left(\frac{4}{3}\right)$ and the answer is $1 - \phi\left(\frac{4}{3}\right)$.
(b)
$85$th percentile means that you are looking for the value $x$ for which $P(X < x) = 0.85$. From the table find the value of $z$ such that $\phi(z) = P(Z < z) = 0.85$ (this is an inverse look up). Then $x = \sigma z + \mu$ (the inverse of the conversion from $X$ to $Z$).