Central Limit Theorem Practice Problem Confusion

Question

In the problem shown below, I was wondering why do we not multiply by root(n) as y/n is the sample mean or did I interpret that incorrectly?

https://www.slader.com/discussion/question/let-y-be-bn-055-find-the-smallest-value-of-n-such-that-approximately-p-left-y-n-1-2-right-geq-095/

Answer 1

$$\Pr\left(\frac Y n>\frac 1 2\right)\ge.95$$

The central limit theorem will have $\frac Y n\sim \text{Normal}\left(\frac 1 nE(Y),\frac 1 {n^2}Var(Y)\right)\equiv\text{Normal}\left(p, \frac{p(1-p)}{n}\right)$ because $Y$ is a sum of independent bernoulli trials.

Since $p=.55$ here, $N(.55, .55(.45)/n)$.

We want the $\Pr\left(Z>\frac{1/2-.55}{\sqrt{.55(.45)}}\sqrt n\right)\ge.95$ (so notice there is a $\sqrt n$ as you observed correctly that this is the sample mean in some sense).

$\Pr(Z>x)\ge.95$ iff $1-\Phi(x)\ge.95$ or $x\le \Phi^{-1}(.05)=-1.644854$.

So $\frac{1/2-.55}{\sqrt{.55(.45)}}\sqrt n\le-1.644854$ or $n\ge 267.8489$. So $n=268$ works.