A company is about to launch an innovative electronic device. It opens pre-registration for those interested in buying this device on launch day. A figure of $352$ people registered. From past experience, it is known that the probability that a registered person will actually decide to buy a device on the day of launch is $0.6$. Assume that the decisions of different people are independent. According to the CLT, what is the minimum number of devices needed on the launch day to ensure that with a probability of $0.96$ there will be enough devices for all registrants who have decided to buy the device?
What I did: Let $n$ be the number of devices. Let $X_{i}$ be an indicator that is $1$ if person $i$ bought device, otherwise $0$. Let $X$ be the number of people who bought devices. So we get $X=\sum_{i=1}^{352}X_{i}$. We know what $X\sim Bin\left(352,0.6\right)$. We want to find $n$ so $P\left(X\leq n\right)=0.96$. So we get: $$ P\left(X\leq n\right)=P\left(\frac{X-352\cdot\frac{1056}{6}}{\sqrt{352\cdot\frac{2112}{25}}}\leq\frac{n-352\cdot\frac{1056}{6}}{\sqrt{352\cdot\frac{2112}{25}}}\right)\leq\Phi\left(\frac{n-352\cdot\frac{1056}{6}}{\sqrt{352\cdot\frac{2112}{25}}}\right) $$
But it gives strange results. What is wrong?