The Normal Distribution

A random normal variable $X$ has mean $\mu$ and standard deviation $3$. Given that $P (X<5)=0.754$. Find $P (4<X<5)$. [Use, $\varphi (0.686)=0.24925,\varphi (0.353)=0.36317$]

A random normal variable $X$ has mean $\mu$ and standard deviation $3$. Given that $P (X<5)=0.754$. Find the value of $\mu$. [Use, $\varphi \left(0.6871\right)=0.754$]

A packing machine produces bags of rice whose weights are normally distributed with mean $50.1$ kg and standard deviation $0.4$ kg If a bag produced by this machine is selected at random, find the probability that its weight is more than $49$ kg given that it is less than $49.5$ kg. [Use, $\varphi (2.75)=0.00298,\varphi (1.5)=0.06681$]

A packing machine produces bags of rice whose weights are normally distributed with mean $50.1$ kg and standard deviation $0.4$ kg If a bag produced by this machine is selected at random, find the probability that its weight is between $49.5$ kg and $50.5$ kg.[Use, $\varphi (1.5)=0.9332,\varphi \left(1\right)=0.8413$]

A packing machine produces bags of rice whose weights are normally distributed with mean $50.1$ kg and standard deviation $0.4$ kg If a bag produced by this machine is selected at random, find the probability that its weight is less than $49.5$ kg.  [Use, $\varphi (1.5)=0.06681$]

A survey is conducted in a large factory. It is found that $27\%$ of the factory workers weigh less than $65$ kg and that $25\%$ of the factory workers weigh more than $96$ kg.

Assuming that the weights of the factory workers is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$.

Find the probability that a factory worker chosen at random weighs more than $100$ kg.

A survey is conducted in a large factory. It is found that $27\%$ of the factory workers weigh less than $65$ kg and that $25\%$ of the factory workers weigh more than $96$ kg.

Assuming that the weights of the factory workers is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$.

Determine two simultaneous linear equations satisfied by $\mu$ and $\sigma$.

Find the values of $\mu$ and $\sigma$.

An exam consists of $20$ true/false questions. John knows the correct answers to $8$ of the questions and decides to choose at random the answers to the remaining questions.

A student gains $2$ marks for each correct answer, has one mark subtracted for each incorrect answer, and gains no marks if the question is left unanswered.

Compare the expected number of marks when the student answers all the questions with the marks she will gain if she just answers the questions to which she knows the correct answer.

A survey is conducted in a large factory. It is found that $27\%$ of the factory workers weigh less than $65$ kg and that $25\%$ of the factory workers weigh more than $96$ kg.

Assuming that the weights of the factory workers is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$.

Determine two simultaneous linear equations satisfied by $\mu$ and $\sigma$.

An exam consists of $20$ true/false questions. John knows the correct answers to $8$ of the questions and decides to choose at random the answers to the remaining questions.

Find the probability that John answers all the $20$ questions correctly.

An exam consists of $20$ true/false questions. John knows the correct answers to $8$ of the questions and decides to choose at random the answers to the remaining questions.

Find the probability that John answers $10$ questions correctly.

The factory has $1000$ workers, $630$ of which are males. The weights of the males are normally distributed with mean $80.5$ and standard deviation $10.1$.

If a male factory worker is selected at random, find the probability that he weights between $75$ and $85$ kilograms. 

Find the expected number of male factory workers that weight more than $85$ kilograms.

The factory has $1000$ workers, $630$ of which are males. The weights of the males are normally distributed with mean $80.5$ and standard deviation $10.1$.

If a male factory worker is selected at random, find the probability that he weights between $75$ and $85$ kilograms. 

Z is the standardised normal random variable with mean $0$ and variance $1$. Find the value of $a$ such that $P\left(\right|Z|\le a)=0.85$.

The diagrams below each show a standard normal distribution. Find the values of $z$

The diagrams below each show a standard normal distribution. Find the values of $z$

Find $a$ such that $\mathrm{P}\left(\right|Z|>a)=0.1096$

Use the standard normal distribution $Z~\mathrm{N}(0,1)$

Find $a$ such that $\mathrm{P}(-a<Z<a)=0.3$

Find $a$ such that $\mathrm{P}(a<Z<-0.3)=0.182$

Use the standard normal distribution $Z~\mathrm{N}(0,1)$

Find $a$ such that $\mathrm{P}(a<Z<1.6)=0.787$

Use the standard normal distribution $Z~\mathrm{N}(0,1)$