Weakly supervised learning (WSL) is a machine learning approach used when labeled data is scarce or expensive to obtain. In such scenarios, models are trained using weaker supervision sources instead of human-annotated data. However, these sources are often noisy and may introduce unquantified biases during training. This issue is particularly pronounced in automated scoring (AS) of second language (L2) learner output, where high variability and limited generalizability pose significant challenges.In this paper, we investigate analytical scoring of L2 learner responses under weak and semi-supervised learning conditions, leveraging Prediction-Powered Inference (PPI) to provide statistical guarantees on score validity. We compare two approaches: (1) synthetic scoring using large language models (LLMs), and (2) a semi-supervised setting in which a machine learning model, trained on a small gold-standard set, generates predictions for a larger unlabeled corpus. In both cases, PPI is applied to construct valid confidence intervals for assessing the reliability of the predicted scores.Our analysis, based on a dataset of L2 learner conversations with an AI agent, shows that PPI is highly informative for evaluating the quality of weakly annotated data. Moreover, we demonstrate that PPI can increase the effective sample size by over 150% relative to the original human-scored subset, enabling more robust inference in educational assessment settings where labeled data is scarce.