I am trying to understand this question and solution. I'm not sure why it was was able to simply put the 2 next to the square root of 75. I've read the section and I thought I understood the concepts well but maybe not. can someone please explain this conceptually and show each step mathematically?
edit: i understand the square root evaluation now, just not everything else.
The width of a confidence interval is $$ \frac{C}{\sqrt n}$$ where $C$ is some number that is proportional to the standard deviation and also has a factor (like $1.96$) that is determined by what type of confidence interval and what percentile. The only important thing here is how it depends on the sample size.
So the width of your first interval is $$ w_1 = \frac{C}{\sqrt{75}}$$ since the sample size is $75.$ If you make your second sample size $n_2$ rather than $75,$ the width will be $$ w_2 = \frac{C}{\sqrt{n_2}}.$$ We want the second to be half as wide, so $$ w_2=\frac{1}{2}w_1.$$
Plugging in for $w_1$ and $w_2,$ we want $$ \frac{C}{\sqrt{n_2}} = \frac{1}{2}\frac{C}{\sqrt{75}}.$$
Then we solve for $n_2$: $$ \sqrt{n_2} = 2\sqrt{75}\implies n_2 = 4\cdot 75 = 300.$$