L3 Probability Distributions
Question Bank
The weights of gold kiwifruit from an orchard near Te Puke can be modelled by a normal distribution with mean 118 g and standard deviation 14 g.
Find the probability that a randomly chosen kiwifruit from this orchard weighs less than 100 g.
On a full charge, the battery of a particular phone model lasts for a time that can be modelled by a normal distribution with mean 16.5 hours and standard deviation 1.8 hours.
(a) Find the probability that a full charge lasts more than 18 hours.
(b) A full charge "runs out early" if it lasts less than 14 hours. Find the probability that a full charge does not run out early.
In a provincial rugby competition, the masses of the forwards can be modelled by a normal distribution with mean 108 kg and standard deviation 9 kg.
(a) Find the probability that a randomly chosen forward has a mass between 100 kg and 115 kg.
(b) The teams in the competition have 180 forwards altogether. How many of them would you expect to have a mass between 100 kg and 115 kg?
The time taken to walk the Tongariro Alpine Crossing can be modelled by a normal distribution with mean 7.2 hours and standard deviation 0.9 hours.
The slowest 10% of walkers take longer than a certain time. Find this time, giving your answer in hours and minutes.
A Year 13 athlete's times for the 400 m can be modelled by a normal distribution with mean 58.2 seconds and standard deviation 0.9 seconds.
Answer these without calculating any probabilities: use features of the normal distribution.
(a) Estimate the percentage of her races in which she runs faster than 56.4 seconds.
(b) In her next race, which is more likely: a time under 57.0 seconds, or a time over 59.6 seconds? Explain your answer.
The annual rainfall at a weather station in Hawke's Bay can be modelled by a normal distribution with mean 820 mm and standard deviation 140 mm.
(a) A year is classed as "dry" if its rainfall is in the lowest 10% of years. Find the rainfall below which a year is classed as dry.
(b) Between what two values would you expect the middle 90% of annual rainfall totals to lie?
The shell lengths of green-lipped mussels harvested in the Marlborough Sounds can be modelled by a normal distribution with mean 95 mm and standard deviation 10 mm.
One mussel farm in the Sounds says that about 30% of the mussels it harvests are longer than 100 mm.
Is this normal model consistent with the farm's statement? Support your answer with a calculation.
A netball coach measures the vertical jump heights of 90 secondary school netball players. The heights have mean 41.6 cm and standard deviation 6.3 cm. The lowest jump is 24.5 cm, the highest is 59.0 cm, and a dot plot of the heights is roughly symmetric with a single peak.
(a) Suggest a normal model for the jump heights of players like these, giving its parameters. Use the information above to explain why your model is reasonable.
(b) Players who jump at least 50 cm are invited to a development squad. Use your model to estimate the probability that a randomly chosen player jumps at least 50 cm.
The weights of Hass avocados from a Northland orchard can be modelled by a normal distribution with mean 230 g and standard deviation 30 g. Avocados heavier than 250 g are sold as "large".
(a) Find the probability that a randomly chosen avocado is large.
(b) Given that an avocado is large, find the probability that it weighs more than 280 g.
The journey time for a school bus route is modelled by a normal distribution with mean 42 minutes. Records show that 12% of journeys take more than 50 minutes.
(a) Find the standard deviation of the journey times.
(b) Use your answer to (a) to estimate the probability that a journey takes less than 35 minutes.
At a wildlife sanctuary, the weights of adult female North Island brown kiwi are modelled by a normal distribution with standard deviation 0.35 kg. The sanctuary's records show that 6% of the adult females weigh more than 3.4 kg.
(a) Find the mean weight of the adult females.
(b) Any adult female that weighs less than 2.2 kg is checked by the sanctuary's vet. Estimate the percentage of adult females that are checked by the vet.
The weights of eggs from a free-range farm can be modelled by a normal distribution with mean 63 g and standard deviation 5.5 g. An egg is graded Size 7 if it weighs at least 62 g but less than 68 g.
Given that an egg is graded Size 7, find the probability that it weighs less than 64 g.
A water park's hydroslide has a minimum height of 120 cm: shorter children can't ride. The heights of 10-year-old children can be modelled by a normal distribution with mean 139 cm and standard deviation 6.5 cm.
Answer these without calculating any probabilities.
(a) Explain why very few 10-year-olds will be turned away from the hydroslide.
(b) The park is thinking of raising the minimum height to 135 cm. A staff member says that this would turn away about half of all 10-year-olds. Explain why this is too high, and give a range for the percentage of 10-year-olds who would be turned away.
A company packs frozen peas into bags labelled 1 kg. The weight of peas the filling machine puts into a bag is normally distributed with standard deviation 9 g, and the machine's mean can be adjusted. The company wants no more than 2% of bags to contain less than 1 kg (1000 g).
(a) Find the lowest mean setting that meets the company's requirement.
(b) By mistake, the machine is set to a mean of 1010 g. What percentage of bags will contain less than 1 kg?
The weights of King salmon harvested from a farm in the Marlborough Sounds can be modelled by a normal distribution with mean 3.8 kg and standard deviation 0.6 kg. Salmon heavier than 3 kg are sold whole; the rest are filleted.
(a) Find the probability that a harvested salmon is sold whole.
(b) Of the salmon that are sold whole, the heaviest 15% are labelled "premium".
(i) What proportion of all the harvested salmon are premium?
(ii) Find the minimum weight of a premium salmon.
At a large secondary school, the amount of sleep that Year 13 students get on a school night is modelled by a normal distribution. 10% of these students sleep less than 6 hours on a school night, and 10% sleep more than 9 hours.
Find the mean and standard deviation of the amount of sleep these students get on a school night.
A teacher models the marks in a Year 12 test, marked out of 100, with a normal distribution with mean 78 and standard deviation 12.
(a) Use the model to find the probability that a randomly chosen student scores more than 90.
(b) Comment on the suitability of the model's parameters for these marks. Support your answer with a calculation.
A forester measures the trunk diameters (in cm, at chest height) of all 250 radiata pine trees in one block of a plantation. The histogram shows the results.
In this block, 20 of the 250 trees (8%) have a trunk diameter greater than 48 cm.
(a) Using the histogram, suggest a value for the mean of a normal model for the trunk diameters.
(b) Use the 8% and your answer to (a) to find the standard deviation of the model. Check that your value is consistent with the histogram.
A honey producer fills jars labelled 500 g. The amount of honey the filling machine puts in a jar is normally distributed with standard deviation 6 g, and the machine's mean can be adjusted. The producer's rule is that no more than 1% of jars may contain less than 500 g, so the mean is set to 514 g.
(a) Show that the setting of 514 g meets the producer's rule.
(b) A new filling head would reduce the standard deviation to 3 g, so the mean setting could be lowered while still meeting the rule. The new head costs 3500 dollars. The producer fills 30 000 jars a year, and the honey is worth 20 dollars per kg to the producer.
(i) Find the lowest mean setting the producer can use with the new head.
(ii) Should the producer buy the new filling head? Support your answer with calculations.
Across New Zealand, the weaning weights of lambs can be modelled by a normal distribution with mean 30 kg and standard deviation 4 kg.
A Canterbury sheep farmer's records over several seasons show that about 5% of her lambs weigh less than 26 kg at weaning, and about 5% weigh more than 34 kg.
Discuss whether the national model is appropriate for modelling the weaning weights of lambs on this farm. Support your answer with statistical calculations, and refer to features of the normal distribution.
A commuter can drive to work by either of two routes. Her travel time on each route can be modelled by a normal distribution:
Route A (city streets): mean 41 minutes, standard deviation 3 minutes.
Route B (motorway): mean 37 minutes, standard deviation 7.5 minutes.
(a) On most days she leaves home 45 minutes before she starts work. Which route gives her the better chance of arriving on time?
(b) On Mondays she leaves home only 40 minutes before she starts work. Which route should she take on Mondays? Support your answer with probabilities.
(c) Use features of the two models to explain why the better route is different in (a) and (b).
At a café, the temperature of a flat white when it is served is modelled by a normal distribution with mean 63 °C. The café has two rules:
Rule 1: no more than 3% of flat whites may be served colder than 57 °C.
Rule 2: no more than 0.2% of flat whites may be served hotter than 70 °C.
Find the largest standard deviation that meets both rules. Justify your answer.