Normal distribution probability then binomial/normal approximation on sample

Questions where a probability is first found from a normal distribution, then that probability is used in a binomial or normal approximation for a sample (e.g. find P(X > a) from N(μ,σ²), then approximate P for n items).

9 questions · Standard +0.2

2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
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CAIE S1 2023 November Q3
11 marks Standard +0.3
3 A farmer sells eggs. The weights, in grams, of the eggs can be modelled by a normal distribution with mean 80.5 and standard deviation 6.6. Eggs are classified as small, medium or large according to their weight. A small egg weighs less than 76 grams and \(40 \%\) of the eggs are classified as medium.
  1. Find the percentage of eggs that are classified as small.
  2. Find the least possible weight of an egg classified as large.
    150 of the eggs for sale last week were weighed.
  3. Use an approximation to find the probability that more than 68 of these eggs were classified as medium.
CAIE S1 2024 November Q5
9 marks Moderate -0.3
5 The weights of the green apples sold by a shop are normally distributed with mean 90 grams and standard deviation 8 grams.
  1. Find the probability that a randomly chosen green apple weighs between 83 grams and 95 grams. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-09_2717_29_105_22}
  2. The shop also sells red apples. \(60 \%\) of the red apples sold by the shop weigh more than 80 grams. 160 red apples are chosen at random from the shop. Use a suitable approximation to find the probability that fewer than 105 of the chosen red apples weigh more than 80 grams.
CAIE S1 2015 June Q5
9 marks Standard +0.3
5 The heights of books in a library, in cm, have a normal distribution with mean 21.7 and standard deviation 6.5. A book with a height of more than 29 cm is classified as 'large'.
  1. Find the probability that, of 8 books chosen at random, fewer than 2 books are classified as large.
  2. \(n\) books are chosen at random. The probability of there being at least 1 large book is more than 0.98 . Find the least possible value of \(n\).
CAIE S1 2016 June Q5
9 marks Standard +0.3
5 Plastic drinking straws are manufactured to fit into drinks cartons which have a hole in the top. A straw fits into the hole if the diameter of the straw is less than 3 mm . The diameters of the straws have a normal distribution with mean 2.6 mm and standard deviation 0.25 mm .
  1. A straw is chosen at random. Find the probability that it fits into the hole in a drinks carton.
  2. 500 straws are chosen at random. Use a suitable approximation to find the probability that at least 480 straws fit into the holes in drinks cartons.
  3. Justify the use of your approximation.
CAIE S1 2018 June Q6
8 marks Standard +0.3
6 The diameters of apples in an orchard have a normal distribution with mean 5.7 cm and standard deviation 0.8 cm . Apples with diameters between 4.1 cm and 5 cm can be used as toffee apples.
  1. Find the probability that an apple selected at random can be used as a toffee apple.
  2. 250 apples are chosen at random. Use a suitable approximation to find the probability that fewer than 50 can be used as toffee apples.
CAIE S1 2005 November Q7
10 marks Standard +0.3
7 In tests on a new type of light bulb it was found that the time they lasted followed a normal distribution with standard deviation 40.6 hours. 10\% lasted longer than 5130 hours.
  1. Find the mean lifetime, giving your answer to the nearest hour.
  2. Find the probability that a light bulb fails to last for 5000 hours.
  3. A hospital buys 600 of these light bulbs. Using a suitable approximation, find the probability that fewer than 65 light bulbs will last longer than 5130 hours.
OCR MEI S2 2012 January Q3
19 marks Standard +0.3
3 The lifetime of a particular type of light bulb is \(X\) hours, where \(X\) is Normally distributed with mean 1100 and variance 2000.
  1. Find \(\mathrm { P } ( 1100 < X < 1200 )\).
  2. Use a suitable approximating distribution to find the probability that, in a random sample of 100 of these light bulbs, no more than 40 have a lifetime between 1100 and 1200 hours.
  3. A factory has a large number of these light bulbs installed. As soon as \(1 \%\) of the bulbs have come to the end of their lifetimes, it is company policy to replace all of the bulbs. After how many hours should the bulbs need to be replaced?
  4. The bulbs are to be replaced by low-energy bulbs. The lifetime of these bulbs is Normally distributed and the mean is claimed by the manufacturer to be 7000 hours. The standard deviation is known to be 100 hours. A random sample of 25 low-energy bulbs is selected. Their mean lifetime is found to be 6972 hours. Carry out a 2 -tail test at the \(10 \%\) level to investigate the claim.
    [0pt] [Question 4 is printed overleaf.]
OCR MEI S2 2011 June Q3
20 marks Standard +0.3
3 The weights of Braeburn apples on display in a supermarket, measured in grams, are Normally distributed with mean 210.5 and standard deviation 15.2.
  1. Find the probability that a randomly selected apple weighs at least 220 grams.
  2. These apples are sold in packs of 3. You may assume that the weights of apples in each pack are independent. Find the probability that all 3 of the apples in a randomly selected pack weigh at least 220 grams.
  3. 100 packs are selected at random.
    (A) State the exact distribution of the number of these 100 packs in which all 3 apples weigh at least 220 grams.
    (B) Use a suitable approximating distribution to find the probability that in at most one of these packs all 3 apples weigh at least 220 grams.
    (C) Explain why this approximating distribution is suitable.
  4. The supermarket also sells Cox's Orange Pippin apples. The weights of these apples, measured in grams, are Normally distributed with mean 185 and standard deviation \(\sigma\).
    (A) Given that \(10 \%\) of randomly selected Cox's Orange Pippin apples weigh less than 170 grams, calculate the value of \(\sigma\).
    (B) Sketch the distributions of the weights of both types of apple on a single diagram.
Edexcel S2 2014 June Q4
12 marks Standard +0.3
4. Pieces of ribbon are cut to length \(L \mathrm {~cm}\) where \(L \sim \mathrm {~N} \left( \mu , 0.5 ^ { 2 } \right)\)
  1. Given that \(30 \%\) of the pieces of ribbon have length more than 100 cm , find the value of \(\mu\) to the nearest 0.1 cm . John selects 12 pieces of ribbon at random.
  2. Find the probability that fewer than 3 of these pieces of ribbon have length more than 100 cm . Aditi selects 400 pieces of ribbon at random.
  3. Using a suitable approximation, find the probability that more than 127 of these pieces of ribbon will have length more than 100 cm .