How to perform this test for the fairness of a coin?

Suppose the task is to test whether a coin is fair (i.e. has equal probabilities of producing a head or a tail). If the coin is flipped 100 times and the results are recorded, the raw data can be represented as a sequence of 100 heads and tails. If there is interest in the marginal probability of obtaining a head, only the number T out of the 100 flips that produced a head needs to be recorded. But T can also be used as a test statistic in one of two ways:

• the exact sampling distribution of T under the null hypothesis is the binomial distribution with parameters 0.5 and 100.
• the value of T can be compared with its expected value under the null hypothesis of 50, and since the sample size is large, a normal distribution can be used as an approximation to the sampling distribution either for T or for the revised test statistic T−50.

My question is about the second bullet (I asked a different question for the first).

I've been reading about the central limit theorem and how the distribution of averages of samples approaches a normal distribution whose mean is equal to the population mean. I feel like these are connected concepts, but am not clear as to how to carry out the test described here.

If I had a sample of 100 flips, and I got some value for $$T$$ I should compare it to "the expected value under the null hypothesis". Is this expected value supposed to be the expected value of the distribution of sample means if the 100 flip experiments was repeated many times?

If so, how do I make that comparison? To be clear, I'm asking how I would do this if I was given pen and paper, and some data on a 100 flip trial and told to determine if the coin was fair.