I would like to clarify the meaning of this question.
If Z is a standardized random variable, which of the following statements is correct?
A) Its distribution is always Normal.
B) We always have E(Z²) = 1.
C) Mean and variance cannot be determined without additional information.
D) It cannot be a continuous random variable.
In my understanding, standardization implies mean 0 and variance 1, but not necessarily normality. Is option B therefore the correct one?
The correct answer is whatever your textbook or course instructor defines as a “standardized” random variable, so you need to look up the working definition for your course.
It is quite common for $Z$ to denote a standard normal random variable, $N(0,1)$, which could explain why your classmates are selecting the first option, but that is not what I would mean by standardized, which is that the mean has been subtracted out (zero mean) and then divide through by the variance (variance of one). In that case, the correct answer would be $B$ by squaring the mean and adding the variance. I suspect this is the “full-credit” answer. However, standardization could involve dividing by the range of a data set, in which case, the second moment would not have to be $1$ under every circumstance.
Standardization typically means rescaling a random variable $X$ as $$Z=\dfrac {(X-\mu_X)} {\sigma_X}$$ This is to give the standardized distribution $Z$ a mean of 0, and a standard deviation of 1 ($\mu_Z=0, \sigma_Z=1$). This is done to give us a common "yardstick" to compare/describe different distributions. See e.g. here.
There may be other definitions (e.g. rescaling to $[0,1]$, or even to $[0,100]$ -percentages), but I would argue that these definitions are inadvisable, as using the term standardized is confusing, and instead the term rescaled should be used.
Note that one can standardize any distribution (as long as it has a finite mean and standard deviation). One can of course standardize the normal distribution, but also a uniform one, an exponentional, a log-normal, etc. But one can not rescale all distributions, say, to $[0,1]$, certainly not a normal one, the support of which is $[-\infty,+\infty]$, or an exponential, etc. So this is another reason why I would call different definitions of "standardized" inadvisable.
Now, the reason classmates picked A) is the use of $Z$ as the name of the variable, because the standard normal distribution is almost always called $Z$. But not all standardized distributions called $Z$ need to be standard normal. Below is a standardized exponential distribution.
So, the correct answer is B). And the "giveaway" is the word always; it implies that, regardless of the original distribution of $X$, we will have $${\sigma_Z}^2=E[(Z-\mu_Z)^2]=E[(Z-0)^2]=E[Z^2]=1$$ because standardization implies ${\sigma_Z}^2=\sigma_z=1$.
Standardisation usually refers to rescaling so that mean = 0 and variance = 1, but it does not necessarily require that. To be precise, that would be called z-standardisation. So I think the best most accurate answer is C: Mean and variance cannot be determined without additional information.
But as Dave indicates in his answer, this is one of those terms that gets defined inconsistently and people are likely to have different opinions based on their experience. So what is deemed correct will depend on the opinion of your instructor.
EDIT: Our standardization tag guidance agrees with my interpretation above. But I'm now persuaded (see other answers as well as comments by Nick Cox) that at this level of training, B would be the most sensible answer for you to give.
