Examinations for deputy or special inspectors shall be in writing and shall be held by the chief and a member of the board, or by at least two national board commissioned inspectors. Such examinations shall be confined to questions the answers to which will aid in determining the fitness and competency of the applicant for the intended service. In case an applicant for an inspector's appointment or commission fails to pass the examination, he or she may appeal to the board for another examination which shall be given by the chief within ninety days. The record of an applicant's examination shall be accessible to said applicant and his or her employer.