Question

the idea of a sampling distribution and the effect of the sample size, n. how it is possible for the individual data to have just two discrete values—0 for the blue candidate or 1 for the red candidate—yet the sample mean is a continuous random variable.

Answer 1

Sample mean is a continuous random variable, because it can take continuous values.

For example, consider the case in your question.

If sample size is n and the probability of red is p, then the sample mean is n*p.

Now, since p lies in [0,1], n*p is any value between 0 and n, depending on the value of p.

Hence, the sample mean is a continuous random variable.