Identifying sleep problem severity from overnight polysomnography (PSG) recordings plays an important role in diagnosing and treating sleep disorders such as the Obstructive Sleep Apnea (OSA). This analysis traditionally is done by specialists manually through visual inspections, which can be tedious, time-consuming, and is prone to subjective errors. One of the solutions is to use Convolutional Neural Networks (CNN) where the convolutional and pooling layers behave as feature extractors and some fully-connected (FCN) layers are used for making final predictions for the OSA severity. In this paper, a CNN architecture with 1D convolutional and FCN layers for classification is presented. The PSG data for this project are from the Cleveland Children's Sleep and Health Study database and classification results confirm the effectiveness of the proposed CNN method. The proposed 1D CNN model achieves excellent classification results without manually preprocesssing PSG signals such as feature extraction and feature reduction.