Finding unknown boundaries

Questions that require finding unknown boundary values (e.g., 'greatest possible weight of small') given percentages or proportions, using inverse normal distribution.

4 questions

CAIE S1 2013 June Q3
3 Buildings in a certain city centre are classified by height as tall, medium or short. The heights can be modelled by a normal distribution with mean 50 metres and standard deviation 16 metres. Buildings with a height of more than 70 metres are classified as tall.
  1. Find the probability that a building chosen at random is classified as tall.
  2. The rest of the buildings are classified as medium and short in such a way that there are twice as many medium buildings as there are short ones. Find the height below which buildings are classified as short.
OCR MEI S2 2005 June Q2
2 The fuel economy of a car varies from day to day according to weather and driving conditions. Fuel economy is measured in miles per gallon (mpg). The fuel economy of a particular petrol-fuelled type of car is known to be Normally distributed with mean 38.5 mpg and standard deviation 4.0 mpg .
  1. Find the probability that on a randomly selected day the fuel economy of a car of this type will be above 45.0 mpg .
  2. The manufacturer wishes to quote a fuel economy figure which will be exceeded on \(90 \%\) of days. What figure should be quoted? The daily fuel economy of a similar type of car which is diesel-fuelled is known to be Normally distributed with mean 51.2 mpg and unknown standard deviation \(\sigma \mathrm { mpg }\).
  3. Given that on 75\% of days the fuel economy of this type of car is below 55.0 mpg , show that \(\sigma = 5.63\).
  4. Draw a sketch to illustrate both distributions on a single diagram.
  5. Find the probability that the fuel economy of either the petrol or the diesel model (or both) will be above 45.0 mpg on a randomly selected day. You may assume that the fuel economies of the two models are independent.
Edexcel S1 2017 June Q3
  1. At a school athletics day, the distances, in metres, achieved by students in the long jump are modelled by the normal distribution with mean 3.3 m and standard deviation 0.6 m
    1. Find an estimate for the proportion of students who jump less than 2.5 m
    The long jump competition consists of 2 jumps. All the students can take part in the first jump and the \(40 \%\) who jump the greatest distance in their first jump qualify for the second jump.
  2. Find an estimate for the minimum distance achieved in the first jump in order to qualify for the second jump.
    Give your answer correct to 4 significant figures.
  3. Find an estimate for the median distance achieved in the first jump by those who qualify for the second jump. The distance of the second jump is independent of the distance of the first jump and is modelled with the same normal distribution. Students who jump a distance greater than 4.1 m in their second jump receive a certificate. At the start of the long jump competition, a student is selected at random.
  4. Find the probability that this student will receive a certificate.
Edexcel S1 2021 October Q6
  1. Xiang is designing shelves for a bookshop. The height, \(H \mathrm {~cm}\), of books is modelled by the normal distribution with mean 25.1 cm and standard deviation 5.5 cm
    1. Show that \(\mathrm { P } ( H > 30.8 ) = 0.15\)
    Xiang decided that the smallest \(5 \%\) of books and books taller than 30.8 cm would not be placed on the shelves. All the other books will be placed on the shelves.
  2. Find the range of heights of books that will be placed on the shelves.
    (3) The books that will be placed on the shelves have heights classified as small, medium or large.
    The numbers of small, medium and large books are in the ratios \(2 : 3 : 3\)
  3. The medium books have heights \(x \mathrm {~cm}\) where \(m < x < d\)
    1. Show that \(d = 25.8\) to 1 decimal place.
    2. Find the value of \(m\) Xiang wants 2 shelves for small books, 3 shelves for medium books and 3 shelves for large books.
      These shelves will be placed one above another and made of wood that is 1 cm thick.
  4. Work out the minimum total height needed.