suppose a sample is taken from a symmetric distribution whose tails decrease more slowly than those of a normal distribution I was wondering how to go about this question about Probability QQ Plots, the question is,
suppose a sample is taken from a symmetric distribution whose tails decrease more slowly than those of a normal distribution. what would be the qualitative shape of a normal probability plot of this sample?
 A: The shape of the plot will be influenced by factors other than light/heavy tails (e.g., skewness, etc.).  To focus on the tail influence, let's suppose the distribution sampled is symmetrical and continuous. 
Thus, given a random sample of size $n$ from some $X$-distribution with CDF $F$, consider a plot of the sample order statistics $X_{(i)}$ versus $\Phi^{-1}(p_i)$, where $p_i \approx \frac{i}{n+1}$.
Now, it can be shown that (when the expectation exists, and if $n$ is sufficiently large) 
$$E(X_{(i)}) \approx F^{-1}(p_i),$$
 so if the $X$-distribution has heavier-than-normal tails, it will be the case that sufficiently far into the tails 
$$E(X_{(i)}) \approx F^{-1}(p_i)
\begin{cases}
& > \Phi^{-1}(p_i)\quad \text{if } p_i \gg \frac{1}{2}\\
& < \Phi^{-1}(p_i)\quad \text{if } p_i \ll \frac{1}{2}
\end{cases}$$
whereas if the $X$-distribution has lighter-than-normal tails, it will be the case that sufficiently far into the tails
$$E(X_{(i)}) \approx F^{-1}(p_i)
\begin{cases}
& < \Phi^{-1}(p_i)\quad \text{if } p_i \gg \frac{1}{2}\\
& > \Phi^{-1}(p_i)\quad \text{if } p_i \ll \frac{1}{2}
\end{cases}$$ 
In other words, if the tails are heavier (resp. lighter) than normal, this shows up as the plot tailing off below (resp. above) the normal line to the left and above (resp. below) the normal line to the right. 
Here's a sketch, exaggerating the behavior:

Here are some online plot examples illustrating this:
lighter-than-normal tails
heavier-than-normal tails
NB: In plotting $X_{(i)}$ vs. $\Phi^{-1}(p_i)$ (instead of $\Phi^{-1}(p_i)$ vs. $X_{(i)}$), I'm following the conventions of NIST and the WP article on Q-Q plots. 
A: Maybe it helps to have an example. The Laplace distribution has 'fatter' tails than a normal. We can easily generate some data
from a Laplace distribution. The difference of two exponential
distributions with the same rate is Laplace. (See Wikipedia on
'Laplace distribution', third bullet under Related Distributions. if it's not in your text.) Specifically, if $Y_1 \sim Exp(1)$ and, independently, $Y_2 \sim Exp(1),$ then $Y_1 - Y_2$ has a
Laplace distribution with mean $\mu = 0$ and SD $\sigma = \sqrt{2}.$
Below, we simulate a sample of size $n = 1000$ from such a
Laplace distribution and a separate random sample of the same
size from $Norm(\mu = 0,\, \sigma=\sqrt{2})$.
Then we compare the normal Q-Q plots of the Laplace and Normal samples.
 y1 = rexp(1000);  y2 = rexp(1000);  x = y1 - y2  # Laplace sample
 z = rnorm(1000, 0, sqrt(2))
 par(mfrow=c(1,2)) # enables side-by-side plots
   qqnorm(x, datax=T, main="Normal QQ-Plot of Laplace Data")
   qqnorm(z, datax=T, main="Normal QQ-plot of Normal Data")
 par(mfrow=c(1,1)) # return to single-panel plots


As anticipated the points on the normal probability plot (normal
QQ-plot) of normal data on the right fall pretty much in a straight
line. However, the points from the Laplace sample follow an S-shaped
curve with straggling points far the the left near the bottom and far to the right near the to top.
In case it helps you to visualize the relatively heavy tails of the Laplace distribution, here are histograms of these two samples.
(The minimum and maximum of the Laplace sample are approximately
-7.6 and 7.4, respectively. For the Normal sample the extremes are
approximately -4.8 and 4.1.)

Addendum: In response to a comment, I have added graphs of 
PDFs of normal and Laplace distributions with 0 means and 
equal standard deviations. If you look closely you will see
that the normal PDF (black) is below the Laplace PDF (blue) in the far tails.

