The breakthrough in generative artificial intelligence (AI) has unlocked new possibilities for higher education. There are many studies on educational chatbots in the fields of science, technology, engineering, and mathematics; however, studies on designing and leveraging chatbots in a multidisciplinary field like project management have been scarce. Although some studies have incorporated elements to facilitate personalized or adaptive learning experience, more research on the design and evaluation of personalized adaptive learning chatbots is needed to offer learners more engaging learning experiences and help them achieve better outcomes. This study applies the concept of personalized adaptive learning, investigating how to design a personalized chatbot for training project management skills in higher education. By leveraging chatbot technology and generative AI, we built PMTutor, with personalized adaptive learning features, integrating it into a project management course. The results suggest that this type of chatbot can complement learning and engage learners by providing customized feedback during an exercise. This study provides insights into the design and usability of personalized adaptive learning chatbots for project management training.