Normal distribution parameters found then approximation applied

Questions where unknown normal distribution parameters (mean or SD) must first be determined before using the resulting probability in a binomial/normal approximation for a sample.

7 questions · Standard +0.3

2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
Sort by: Default | Easiest first | Hardest first
CAIE S1 2023 June Q5
12 marks Standard +0.3
5 The lengths of Western bluebirds are normally distributed with mean 16.5 cm and standard deviation 0.6 cm . A random sample of 150 of these birds is selected.
  1. How many of these 150 birds would you expect to have length between 15.4 cm and 16.8 cm ?
    The lengths of Eastern bluebirds are normally distributed with mean 18.4 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that \(72 \%\) of Eastern bluebirds have length greater than 17.1 cm .
  2. Find the value of \(\sigma\).
    A random sample of 120 Eastern bluebirds is chosen.
  3. Use an approximation to find the probability that fewer than 80 of these 120 bluebirds have length greater than 17.1 cm .
CAIE S1 2024 March Q4
12 marks Standard +0.3
4 A company sells small and large bags of rice. The masses of the small bags of rice are normally distributed with mean 1.20 kg and standard deviation 0.16 kg .
  1. In a random sample of 500 of these small bags of rice, how many would you expect to have a mass greater than 1.26 kg ?
    The masses of the large bags of rice are normally distributed with mean 2.50 kg and standard deviation \(\sigma \mathrm { kg } .20 \%\) of these large bags of rice have a mass less than 2.40 kg .
  2. Find the value of \(\sigma\).
    A random sample of 80 large bags of rice is chosen.
  3. Use a suitable approximation to find the probability that fewer than 22 of these large bags of rice have a mass less than 2.40 kg .
CAIE S1 2014 June Q7
11 marks Standard +0.3
7 The time Rafa spends on his homework each day in term-time has a normal distribution with mean 1.9 hours and standard deviation \(\sigma\) hours. On \(80 \%\) of these days he spends more than 1.35 hours on his homework.
  1. Find the value of \(\sigma\).
  2. Find the probability that, on a randomly chosen day in term-time, Rafa spends less than 2 hours on his homework.
  3. A random sample of 200 days in term-time is taken. Use an approximation to find the probability that the number of days on which Rafa spends more than 1.35 hours on his homework is between 163 and 173 inclusive.
CAIE S1 2018 November Q6
8 marks Standard +0.3
6 The lifetimes, in hours, of a particular type of light bulb are normally distributed with mean 2000 hours and standard deviation \(\sigma\) hours. The probability that a randomly chosen light bulb of this type has a lifetime of more than 1800 hours is 0.96 .
  1. Find the value of \(\sigma\).
    New technology has resulted in a new type of light bulb. It is found that on average one in five of these new light bulbs has a lifetime of more than 2500 hours.
  2. For a random selection of 300 of these new light bulbs, use a suitable approximate distribution to find the probability that fewer than 70 have a lifetime of more than 2500 hours.
  3. Justify the use of your approximate distribution in part (ii).
OCR S2 2010 June Q3
9 marks Standard +0.3
3 Tennis balls are dropped from a standard height, and the height of bounce, \(H \mathrm {~cm}\), is measured. \(H\) is a random variable with the distribution \(\mathrm { N } \left( 40 , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 32 ) = 0.2\).
  1. Find the value of \(\sigma\).
  2. 90 tennis balls are selected at random. Use an appropriate approximation to find the probability that more than 19 have \(H < 32\).
Edexcel S2 2017 October Q1
9 marks Standard +0.3
  1. A shop sells rods of nominal length 200 cm . The rods are bought from a manufacturer who uses a machine to cut rods of length \(L \mathrm {~cm}\), where \(L \sim \mathrm {~N} \left( \mu , 0.2 ^ { 2 } \right)\)
The value of \(\mu\) is such that there is only a \(5 \%\) chance that a rod, selected at random from those supplied to the shop, will have length less than 200 cm .
  1. Find the value of \(\mu\) to one decimal place. A customer buys a random sample of 8 of these rods.
  2. Find the probability that at least 3 of these rods will have length less than 200 cm . Another customer buys a random sample of 60 of these rods.
  3. Using a suitable approximation, find the probability that more than 5 of these rods will have length less than 200 cm .
Edexcel S2 Q5
13 marks Standard +0.3
A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length L where L ~ N(\(\mu\), 0.3²).
  1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]
A customer buys 10 of these canes from the garden centre.
  1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]
Another customer buys 500 canes.
  1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]