Suppose that annual expenses for a family of four are normally distributed with a mean expense equal to $3000$ and standard deviation of $500$.
What is the probability that the yearly expenses are within three standard deviations around the mode? I tried to find the probability using: $\operatorname{mean}+3SD$ and $\operatorname{mean}-3SD$.
Since it's within $3 SD$, I calculated the area between $z=3$ and $z=-3$. Using the $z$-score table this gives the required probability: $0.99865-0.00135= 0.9973$. Is this correct?
Assuming the distribution is Normal, famously, yes. Note that$$\Phi(-x)=1-\Phi(x)\implies \Phi(x)-\Phi(-x)=2\Phi(x)-1$$would have saved you consulting the table twice.