Sampling distribution problem

Question

Students height is a normal distribution with $mean=167cm$ and $standard~deviation=3cm$.
If we choose 100 students independently, what is the probability that at least 55 of them have height less than 167 cm?
My try:
$$\sigma_\bar{X}^2=\frac{\sigma^2}{n}=\frac{9}{100}=0.09$$
So:$$P(\bar{X}<167)=P(\frac{\bar{X}-mean}{\frac{\sigma_\bar{X}}{\sqrt{n}}}<\frac{167-mean}{\frac{\sigma_\bar{X}}{\sqrt{n}}})=P(Z<0)=0.5$$
But I dont know how to relate 55 students to the solution. Any ideas?

Answer 1

Comment on attempt: You computed the probability that the sample average of the 100 students' heights is less than 167 cm, which is going to be 0.5 regardless of the sample size because the population was assumed normally distributed with mean 167. But this is not what was asked for.

Hint 1: You're asked for the probability that at least 55 of the 100 students have some property. Do you know a distribution for the number of things in an independent sample that have some property? There is one, and it has a connection to the normal distribution, but it isn't actually the normal distribution.

Hint 2: the height of any one student, call it $X$, is normal with mean 167 and standard deviation 3, so the probability that any one student has height less than 167 cm is $P(X<167)=P(Z<0)=0.5$.