Practice IB Mathematics Applications & Interpretation (AI) Topic SL 4.9—normal Distribution with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 4.9—normal Distribution and mirrors Paper 1, 2, 3 style where relevant.
Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.
A vehicle's consumption rate , in litres per km, is modelled by . Assume each vehicle maintains its sampled rate throughout a journey and that different vehicles have independent rates.
In an earlier experiment, vehicles each received the same unknown quantity litres. Exactly completed a km journey without refuelling.
…
A vehicle's fuel-consumption rate , in litres per km, is modelled by . The rate is assumed constant over the journey being considered. A journey of km therefore requires litres.
An earlier calibration specifies and . Treat these as exact model probabilities, determined independently of the current sample.
…
(g) the probability that an agent chosen at random took a break of at least minutes.
(h) the expected number of agents that took a break of at least minutes.
(i) the sampling technique used in this sampling method.
(e) Find the probability that this nurse attended or more patients.
(f) Given that the first nurse chosen at random attended or more patients, find the probability that both nurses attended exactly patients.
(g) the probability that a nurse chosen at random spent at least minutes with a patient.
(h) the expected number of nurses that spent at least minutes with a patient.
(i) the sampling technique used in this sampling method.
The table below shows the distribution of calls handled per shift by a group of call centre agents surveyed on a working day in Manila.
| Calls | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 |
|---|---|---|---|---|---|---|---|---|---|
| Frequency | 1 | 3 | 9 | 14 | 23 | 24 | 15 | 8 | 3 |
Two different agents are chosen at random from the group of agents, one after the other, without replacement.
The length of breaks taken by the agents was normally distributed with a mean of minutes and a standard deviation of minutes.
The agents were selected for the survey by dividing the full workforce into three shift groups and randomly choosing a fixed number from each shift group.
(a) Find the mean number of calls handled by the agents.
(b) Find the standard deviation of the number of calls handled.
(c) Find the median number of calls handled by the agents.
(d) Find the interquartile range.
(e) Find the probability that this agent handled or more calls.
(f) Given that the first agent selected handled or more calls, find the probability that both selected agents handled exactly calls.
The lifetimes , in hours, of rechargeable batteries follow a normal distribution . A battery is labeled as unreliable if it lasts less than 18 hours. To reduce the proportion of unreliable batteries, the manufacturer sets the probability of an unreliable battery to 0.005. They decide to increase the mean while keeping the standard deviation constant. The previous adjustment is found unsatisfactory, so the manufacturer sets the mean back to 22 and instead decreases to maintain the 0.005 probability threshold.
The table below shows the distribution of patients attended per shift by a group of nurses surveyed on a working day in Dublin.
| Patients | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
|---|---|---|---|---|---|---|---|---|---|
| Frequency | 2 | 4 | 9 | 12 | 25 | 22 | 16 | 8 | 2 |
A nurse is chosen at random from the group of nurses.
A second nurse is chosen at random from the remaining nurses.
For each of the nurses, the time spent with one randomly selected patient is modelled by a normal distribution with mean minutes and standard deviation minutes.
The nurses were selected for the survey by ensuring nurses from each of the hospital's departments were chosen at random.
(a) Find the mean number of patients attended by the nurses.
(b) Find the standard deviation of the number of patients attended.
(c) Find the median number of patients attended by the nurses.
(d) Find the interquartile range.
(a) Assuming and , the probability that a battery selected at random is unreliable.
(b) the new mean required if to result in an unreliable probability of 0.005. Give your result to two decimal places.
(c) the new value for the standard deviation given that achieves the 0.005 probability.
