So I have a general idea of how to solve this, I'm pretty sure I'm supposed to use the basic confidence interval equation and solve for n, but I'm not sure what to set the equation = to originally. I thought of making it 659.7 = 50+- 1.96(311.7/sqrt(n)) and solving for n, but the answers wrong. then I thought of putting 50 where the n is and solving, but I'm not sure I understand that approach conceptually. Could someone explain the process and show me the solving in detail? I'm a little lost on this conceptually.
Edit: I'm starting to think I'm supposed to take = 659.7 +- 1.96(311.7/sqrt(50)) to get the upper and lower confidence bound, then do the same equation with my new boundaries but with n instead of 50 and solve for n. is that correct? Actually that would be wrong, it comes out to 56.3 and the answer is 150.
We don't need to know the mean. We just want $$\mu \pm 1.96\cdot\frac{311.7}{\sqrt{n}}$$
where $$1.96\cdot\frac{311.7}{\sqrt{n}}=50$$
So this comes out to be $n=149.29$ but we must round up to $n=150$.
Checking that the bound on our estimation is within $50$,
$$1.96\cdot\frac{311.7}{\sqrt{150}}\approx 49.88 \lt 50$$
$n$ is the sample size we must solve for so I'm not sure why you put the desired bound of $50$ as $n$.