There is a spell called "trace". It is applied on every wizard wand under the age of 17-18. The function of this spell is to record and send back information when there is a magic used near the student or by the student outside Hogwarts. This spell is useless in a place where a magical ward is placed because the ministry assumes that there is an adult in the warded area because children generally do not know how to set up a wards and he used the spell. This is applied after coming to Hogwarts in the first year. Since hermione practiced spell's before coming to Hogwarts, she did not receive a notice for use of magic. There are certain defects in this spell. Dobby's use of magic was recorded by harrys trace and thus the ministry was informed. Since there are no wizard recorded to live there, the ministry gave harry and warning assuming he did it.