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.
Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes.
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.
Stating your hypotheses clearly and using a \(5 \%\) significance level, test whether or not there is evidence to support the patients' complaint.
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 \(T \sim \mathrm {~N} \left( 5,3.5 ^ { 2 } \right)\)
Using this model,
find the probability that a routine appointment with the dentist takes less than 2 minutes
find \(\mathrm { P } ( T < 2 \mid T > 0 )\)
hence explain why this normal distribution may not be a good model for \(T\).
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 \(T > 2\)
Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place.