Linear relationship μ = kσ

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.

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 \(\mu + \sigma\) ?

4 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).

1. Given that \(5 \sigma = 3 \mu\), find \(\mathrm { P } ( X < 2 \mu )\).
2. With a different relationship between \(\mu\) and \(\sigma\), it is given that \(\mathrm { P } \left( X < \frac { 1 } { 3 } \mu \right) = 0.8524\). Express \(\mu\) in terms of \(\sigma\).

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]

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.

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.

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 \(s\).

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.

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 .

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 .

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 \(\sigma\)

8

1. The variable \(X\) has the distribution \(\mathrm { N } ( 20,9 )\).
   (a) Find \(\mathrm { P } ( X > 25 )\).
   (b) Given that \(\mathrm { P } ( X > a ) = 0.2\), find \(a\).
   (c) Find \(b\) such that \(\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5\).
2. The variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)\). Find \(\mathrm { P } ( Y > 1.5 \mu )\).