Hypothesis testing: normal vs. non-normal I have the following hypothesis testing problem:
$$H_0:X=Y,\quad\text{vs.}\quad H_1:X=Y+Z$$
where $Y\sim\mathcal{N}(0,\sigma^2)$ and $Z$ is a  random variable with  non-normal continuous distribution. I am not very familiar with statistics.
Is there a well-known way to solve this problem? 
 A: If data are normally distributed, then points in a a normal probability plot (normal Q-Q plot) tend to lie on a straight line. 
In your case, the random variable $X_H = Y$ is normally distributed and the random variable $X_A = Y + Z$ is not. To be specific suppose we
have $n = 100$ observations from $X_H \sim Norm(100, 15).$
Let's look at three relevant plots. The ECDF of a dataset puts probability $1/n$ at each of the $n$ datapoints of a sample. 
Starting from height 0 at the left, it moves to 1 at the right
through $n$ increments of $1/n$.
The EDCF imitates the population CDF, shown as a blue curve on
the left plot.
In the Q-Q plot at the right, the vertical scale is distorted
to make the normal CDF a straight line and points of the ECDF of a normal sample almost a straight line. (Simulated samples and
plots are from R statistical software.)
 x.h = rnorm(100, 100, 15)
 par(mfrow=c(1,2))  # 2 panels side by side
   plot.ecdf(x.h, pch=20)
      curve(pnorm(x, 100, 15), lwd=2, col="blue", add=T)
   qqnorm(x.h, datax=T)
 par(mfrow=c(1,1))


Now we show Q-Q plots of data from hypothetical (normal)
and alternative (non-normal) distributions. 
I have used $X_A = X_H + Z$ where $Z$ is exponential with mean 50.
 x.a = x.h + rexp(100, 1/50)
 par(mfrow=c(1,2))
    qqnorm(x.h, datax=T)
    qqnorm(x.a, datax=T)
par(mfrow=c(1,1))


The random variable $X_A$ is far from normal because of the added
exponential component. The nonnormality of $X_A$ results in
the markedly nonlinear Q-Q plot on the right.
The Shapiro-Wilk test is one of several tests of normality.
Roughly speaking, it measures the degree of nonlinearity in the
Q-Q plot. So you don't have to judge 'linearlity' just by eye.
Here are Shapiro-Wilk tests for $X_H$, with P-value
far above 5% (consistent with normality), and for $X_A$, with
P-value far below 5% (not consistent with normality).
 shapiro.test(x.h)

 ##        Shapiro-Wilk normality test

 ## data:  x.h 
 ## W = 0.9939, p-value = 0.935

 shapiro.test(x.a)

 ##        Shapiro-Wilk normality test

 ## data:  x.a 
 ## W = 0.9127, p-value = 5.913e-06

To make a good demonstration, I have used samples of moderate size and an alternative $X_A$
that is far from normal. For smaller samples or for alternatives
that are more nearly normal, you cannot expect such clear-cut 
results. 
This demonstration should get you started down the right path.
For more information you can look in a statistics text or online
for 'normal probability plot', 'quantile plot', 'Q-Q plot',
'tests of normality', and so on. 
