Multiple probability calculations only

2 The volume of Everwhite toothpaste in a pump-action dispenser may be modelled by a normal distribution with a mean of 106 ml and a standard deviation of 2.5 ml . Determine the probability that the volume of Everwhite in a randomly selected dispenser is:

1. less than 110 ml ;
2. more than 100 ml ;
3. between 104 ml and 108 ml ;
4. not exactly 106 ml .

5. The time taken in minutes, \(T\), for a mechanic to service a bicycle follows a normal distribution with a mean of 25 minutes and a variance of 16 minutes \(^ { 2 }\). Find

1. \(\mathrm { P } ( T < 28 )\),
2. \(\quad \mathrm { P } ( | T - 25 | < 5 )\). One afternoon the mechanic has 3 bicycles to service.
3. Find the probability that he will take less than 23 minutes on each of the three bicycles.
   (4 marks)

2 The heights of sunflowers may be assumed to be normally distributed with a mean of 185 cm and a standard deviation of 10 cm .

1. Determine the probability that the height of a randomly selected sunflower:
   1. is less than 200 cm ;
   2. is more than 175 cm ;
   3. is between 175 cm and 200 cm .
2. Determine the probability that the mean height of a random sample of 4 sunflowers is more than 190 cm .

1 The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes

1. more than 27 minutes,
2. between 20 and 25 minutes.

The random variable \(X\) has the normal distribution \(N(2, 1.7^2)\).

1. State the standard deviation of \(X\). [1 mark]
2. Find \(P(X < 0)\). [2 marks]
3. Find \(P(0.6 < X < 3.4)\). [4 marks]

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.

1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. [1]
2. 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. [4]
3. 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 \(T \sim N(5, 3.5^2)\) Using this model,
   1. find the probability that a routine appointment with the dentist takes less than 2 minutes [1]
   2. find \(P(T < 2 | T > 0)\) [3]
   3. hence explain why this normal distribution may not be a good model for \(T\). [1]
4. 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. [5]

The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes

1. more than 27 minutes, [3]
2. between 20 and 25 minutes. [3]

The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes

1. more than 27 minutes, [3]
2. between 20 and 25 minutes. [3]