Linear relationship μ = kσ

The constraint is a linear relationship between mean and standard deviation (μ = kσ or σ = kμ), requiring standardization and using Z-tables to find the parameter.

15 questions · Standard +0.3

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 Q5
8 marks Standard +0.3
5
  1. The heights of the members of a club are normally distributed with mean 166 cm and standard deviation 10 cm .
    1. Find the probability that a randomly chosen member of the club has height less than 170 cm .
    2. Given that \(40 \%\) of the members have heights greater than \(h \mathrm {~cm}\), find the value of \(h\) correct to 2 decimal places.
  2. The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a randomly chosen value of \(X\) is positive.
CAIE S1 2007 June Q3
7 marks Standard +0.3
3
  1. The random variable \(X\) is normally distributed. The mean is twice the standard deviation. It is given that \(\mathrm { P } ( X > 5.2 ) = 0.9\). Find the standard deviation.
  2. A normal distribution has mean \(\mu\) and standard deviation \(\sigma\). If 800 observations are taken from this distribution, how many would you expect to be between \(\mu - \sigma\) and \(\mu + \sigma\) ?
CAIE S1 2013 June Q4
7 marks Standard +0.3
4
  1. The random variable \(Y\) is normally distributed with positive mean \(\mu\) and standard deviation \(\frac { 1 } { 2 } \mu\). Find the probability that a randomly chosen value of \(Y\) is negative.
  2. The weights of bags of rice are normally distributed with mean 2.04 kg and standard deviation \(\sigma \mathrm { kg }\). In a random sample of 8000 such bags, 253 weighed over 2.1 kg . Find the value of \(\sigma\). [4]
CAIE S1 2013 June Q1
3 marks Standard +0.3
1 The random variable \(Y\) is normally distributed with mean equal to five times the standard deviation. It is given that \(\mathrm { P } ( Y > 20 ) = 0.0732\). Find the mean.
CAIE S1 2017 June Q6
11 marks Standard +0.3
6
  1. The random variable \(X\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). You are given that \(\sigma = 0.25 \mu\) and \(\mathrm { P } ( X < 6.8 ) = 0.75\).
    1. Find the value of \(\mu\).
    2. Find \(\mathrm { P } ( X < 4.7 )\).
  2. The lengths of metal rods have a normal distribution with mean 16 cm and standard deviation 0.2 cm . Rods which are shorter than 15.75 cm or longer than 16.25 cm are not usable. Find the expected number of usable rods in a batch of 1000 rods.
CAIE S1 2017 June Q4
6 marks Standard +0.3
4
  1. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(\mu = 1.5 \sigma\). A random value of \(X\) is chosen. Find the probability that this value of \(X\) is greater than 0 .
  2. The life of a particular type of torch battery is normally distributed with mean 120 hours and standard deviation \(s\) hours. It is known that \(87.5 \%\) of these batteries last longer than 70 hours. Find the value of \(s\).
CAIE S1 2016 March Q7
11 marks Standard +0.3
7 The times taken by a garage to fit a tow bar onto a car have a normal distribution with mean \(m\) hours and standard deviation 0.35 hours. It is found that \(95 \%\) of times taken are longer than 0.9 hours.
  1. Find the value of \(m\).
  2. On one day 4 cars have a tow bar fitted. Find the probability that none of them takes more than 2 hours to fit. The times in hours taken by another garage to fit a tow bar onto a car have the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) where \(\mu = 3 \sigma\).
  3. Find the probability that it takes more than \(0.6 \mu\) hours to fit a tow bar onto a randomly chosen car at this garage.
CAIE S1 2012 November Q4
9 marks Standard +0.8
4 The mean of a certain normally distributed variable is four times the standard deviation. The probability that a randomly chosen value is greater than 5 is 0.15 .
  1. Find the mean and standard deviation.
  2. 200 values of the variable are chosen at random. Find the probability that at least 160 of these values are less than 5 .
CAIE S1 2018 November Q7
12 marks Standard +0.3
7
  1. The time, \(X\) hours, for which students use a games machine in any given day has a normal distribution with mean 3.24 hours and standard deviation 0.96 hours.
    1. On how many days of the year ( 365 days) would you expect a randomly chosen student to use a games machine for less than 4 hours?
    2. Find the value of \(k\) such that \(\mathrm { P } ( X > k ) = 0.2\).
    3. Find the probability that the number of hours for which a randomly chosen student uses a games machine in a day is within 1.5 standard deviations of the mean.
  2. The variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), where \(4 \sigma = 3 \mu\) and \(\mu \neq 0\). Find the probability that a randomly chosen value of \(Y\) is positive.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2011 November Q1
6 marks Standard +0.3
1 The random variable \(X\) is normally distributed and is such that the mean \(\mu\) is three times the standard deviation \(\sigma\). It is given that \(\mathrm { P } ( X < 25 ) = 0.648\).
  1. Find the values of \(\mu\) and \(\sigma\).
  2. Find the probability that, from 6 random values of \(X\), exactly 4 are greater than 25 .
Edexcel S1 2024 January Q8
9 marks Standard +0.8
  1. The random variable \(X\) is normally distributed with mean \(\mu\) and variance 36
Given that $$\mathrm { P } ( \mu - 2 k < X < \mu + 2 k ) = 0.6$$
  1. find the value of \(k\) The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\) Given that $$2 \mu = 3 \sigma ^ { 2 } \quad \text { and } \quad \mathrm { P } \left( \mathrm { Y } > \frac { 3 } { 2 } \mu \right) = 0.0668$$
  2. find the value of \(\mu\) and the value of \(\sigma\)
OCR H240/02 2018 June Q8
8 marks Standard +0.3
8
  1. The variable \(X\) has the distribution \(\mathrm { N } ( 20,9 )\).
    1. Find \(\mathrm { P } ( X > 25 )\).
    2. Given that \(\mathrm { P } ( X > a ) = 0.2\), find \(a\).
    3. Find \(b\) such that \(\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5\).
    4. The variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)\). Find \(\mathrm { P } ( Y > 1.5 \mu )\).
CAIE S1 2010 June Q4
6 marks Standard +0.3
The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).
  1. Given that \(5\sigma = 3\mu\), find \(\mathrm{P}(X < 2\mu)\). [3]
  2. With a different relationship between \(\mu\) and \(\sigma\), it is given that \(\mathrm{P}(X < \frac{4\mu}{3}) = 0.8524\). Express \(\mu\) in terms of \(\sigma\). [3]
CAIE S1 2014 November Q5
8 marks Moderate -0.8
  1. The time, \(X\) hours, for which people sleep in one night has a normal distribution with mean 7.15 hours and standard deviation 0.88 hours.
    1. Find the probability that a randomly chosen person sleeps for less than 8 hours in a night. [2]
    2. Find the value of \(q\) such that P\((X < q) = 0.75\). [3]
  2. The random variable \(Y\) has the distribution N\((\mu, \sigma^2)\), where \(2\sigma = 3\mu\) and \(\mu \neq 0\). Find P\((Y > 4\mu)\). [3]
Edexcel S1 Q4
13 marks Standard +0.3
The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma^2\).
  1. If \(2\mu = 3\sigma\), find P\((X < 2\mu)\). [5 marks]
  2. If, instead, P\((X < 3\mu) = 0.86\),
    1. find \(\mu\) in terms of \(\sigma\), [4 marks]
    2. calculate P\((X > 0)\). [4 marks]