Is my data normally distributed? (QQ plot and histogram analysis) I am trying to create a regression model for prediction. I need to generate prediction/confidence intervals for my model.
I am trying to decide whether to use a quantile regression or linear regression model. If I use a linear regression model I need to ensure my prediction residuals are normally distributed to compute valid confidence intervals.
I'm not experienced in testing for data normality. My sample size is too high to generate a meaningful p-value using the Shapiro test, so I have plotted the residuals on a histogram and produced a QQ-plot.
I am not sure how to interpret these results and would appreciate any input. The shape looks broadly normal, but the peak is very short and high, and the tails are pretty long.
What do you think?
Is my data normally distributed?



This is an extended comment rather than an answer.
The answer to your actual question is: No, both the histogram and the QQ plot indicate the distribution (of the residuals? of the response? of some other variable?) is not Normal.
The histogram calls to mind the Laplace distribution, which is symmetric like the Normal but with heavier tails; it puts less probability near the center/mean, so the peak is sharp.

Source: Wikipedia
Is this observation relevant to you? It's hard to say without knowing any details. Keep in mind that quantile regression (QR) makes no assumptions about the error distribution while linear regression (LR) assumes that the errors are Normal. Since QR makes fewer assumptions than LR, it would seem that it's a "safer" choice. The flip side is that QR is not efficient at estimating the median as LR is at estimating the mean. As usual, if we make more assumptions and those assumptions are reasonably satisfied, we get better estimates of the model parameters.
Finally, comparing QQ plots may not be the best way to choose between quantile and linear regression.
What are the assumptions for applying a quantile regression model? 
When is quantile regression worse than OLS? 
What are the advantages of linear regression over quantile regression? 
Appendix
There is an interesting (but probably not relevant) connection between the Laplace distribution, mean absolute deviation from the median and quantile regression.
For the 0.5 quantile, QR effectively minimizes the mean absolute error $\sum_i|e_i| = \sum_i|y_i - x_i\beta|$ and the solution is the (conditional) median. The mean absolute deviation from the median is also the maximum likelihood estimate of the Laplace scale parameter.
