The question I wish to solve is as follows
According to records of 520 employees, the weekly overtime work of an employee follows a Normal distribution with mean 135 minutes and standard deviation 30 minutes.
Calculate the probability that a randomly selected employee has worked overtime more than 165 minutes in at least 4 of the previous 5 weeks.
My thinking for this is to use the following formula for a normal distribution with x = 165 and mu= 135 and sigma = 30.
$P(z >\frac{(x-\mu)}{\sigma})$
If I find the corresponding probability using z tables for this would I be right in assuming that this is the probability that a randomly selected employee works overtime more than 165 minutes in any given week?
how do I account for at least 4 of the previous 5 weeks?
Is this normal approximation to the binomial? I'm not quite sure about that part.
Best Answer
You'll want to find the probability that someone works overtime more than $165$ minutes in a given week. You're on the right track with this. The probability of it occurring in a given week is
$$1-\Phi\left(\frac{165-135}{30}\right)\approx0.1587$$
To find the probability of it happening in at least four of the five weeks, you'll want to make use of the binomial distribution:
$$P(X=k)={n \choose k}p^k(1-p)^{n-k}$$
We wish to find $P(X=4)+P(X=5)$ where $n=5$ and $p$ is the probability obtained above.