According to a​ study, 47​% of all males between the ages of 18 and 24 live at home. ​ (Unmarried college students living in a dorm are counted as living at​ home.) Suppose that a survey is administered and 115 of 202 respondents indicated that they live at home.​ (a) Use the normal approximation to the binomial to approximate the probability that at least 115 respondents live at home.​ (b) Do the results from part​ (a) contradict the​ study?