Thinking about the CLT

#1
Indicate whether the following statement is true, false, or uncertain, and provide a short explanation of why you think this is the case. “If height in centimeters in some given population is distributed with mean 175 and standard deviation 10, the central limit theorem implies that the probability of finding a person taller than 177 centimeters in any random sample of size 100 is approximately 2.5%.”


I have two theories:

1.
False.
CLT finds SAMPLE MEAN probability. This is 1 child, so not finding the mean of
the sample and therefore– not the Central Limit Theorum.

or

2.
Mu = 175
Sigma = 10

Sample Mean = Mu = 175
Sample Standard Deviation = Standard Error = 10/(100)^(1/2) = 1

Probability (1 - P(SM < 177) = 1 - P(z < (177 - 175)/1) = 1 - P(z < 2) = 1 - 0.9772 = 0.0228