Section 39I. Any person using water supplied by a city, town or district, measured by a meter, shall be entitled to an examination and test of such meter to determine the accuracy of the same in any quarter or period, upon written application therefor, which shall be made before the expiration of the time when the rate for such quarter or period is required to be paid, to the board, commissioner or officer in charge of the water works of the city, town or district. Such examination and test shall be made by a competent person employed by the city, town or district. A written report of the result of the examination and test shall be furnished to the person making the application, and if it appears that the meter has registered with substantial accuracy, the expense of the examination and test shall be paid by the person applying therefor, and in no case shall the expense so required to be paid exceed three dollars for each examination and test, but if it appears that the meter has not registered within two per cent of accuracy and that the person has been charged with, or has paid for, more water than he should have been charged with or should have paid for, the amount of such excess shall forthwith be credited to such person or remitted to him if he has paid the same, and the expense of the examination and test shall be borne by the city, town or district supplying the water; if, however, it appears that the person has been charged with, or has paid for, less water than he should have been charged with or should have paid for, he shall forthwith be charged with the proper additional amount and shall pay the same, together with the expense of the examination and test, to the city, town or district supplying the water.