16 5. A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes. (a) Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. (1) Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes. (b) Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients’ complaint. (4) The health centre also claims that the time a dentist spends with a patient during a routine appointment, T minutes, can be modelled by the normal distribution where T ~ N(5, 3.52) (c) Using this model, (i) find the probability that a routine appointment with the dentist takes less than 2 minutes (1) (ii) find P (T < 2 | T > 0) (3) (iii) hence explain why this normal distribution may not be a good model for T. (1) The dentist believes that she cannot complete a routine appointment in less than 2 minutes. She suggests that the health centre should use a refined model only including values of  T > 2 (d) Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________