Normal Distribution (Cambridge (CIE) A Level Maths: Probability & Statistics 1): Exam Questions

For the random variable, $X~\mathrm{N}(32,9)$,

(i) Write down the mean and standard deviation.

(ii) Draw a sketch of the graph and clearly label the mean.  

(i) With the help of your diagram, explain how you should know, without carrying out any calculations, that  $\mathrm{P}(X\le 34)>0.5.$

(ii) With the help of your diagram, explain how you should know, without carrying out any calculations, that $\mathrm{P}(X\ge 39)<0.5.$

Use the formula  $Z=\frac{X-\mu }{\sigma }$  to find appropriate -values, to three decimal places, and then use the table of $\mathrm{z}$-values to calculate, to three decimal places,

(i) $\mathrm{P}(X\le 34)$

(ii) $\mathrm{P}(X\ge 39)$

(iii) $P\left(34\le X\le 39\right).$

For the standard normal distribution $Z~\mathrm{N}(0,1^2)$, use the table of $\mathrm{z}$-values to find:

(i) $\mathrm{P}(Z<1.5)$

(ii) $\mathrm{P}(Z>-0.8)$

(iii) $\mathrm{P}(-2.1<Z<-0.3)$

The random variable $X~\mathrm{N}(2,0.1^2).$ 

Using the coding relationship between $X$ and $Z$, find the values of $a$, $b$, $c$ and $d$ such that:

(i) $\mathrm{P}(X<a)=\mathrm{P}(Z<1.5)$

(ii) $\mathrm{P}(X>b)=\mathrm{P}(Z>-0.8)$

(iii) $\mathrm{P}(c<X<d)=\mathrm{P}(-2.1<Z<-0.3)$

The random variable $X~\mathrm{N}(50,49)$

(i) Write down the values of $\mu$ and $\sigma$ .

(ii)Use the table of critical values to find the value of $\mathrm{z}$ such that $\mathrm{P}(Z\le z)=0.975$.

(iii)Use the formula $Z=\frac{X-\mu }{\sigma }$ to find the value of $a$ such that $\mathrm{P}(X\le a)=0.975$.

(i) Find the value of $\mathrm{z}$ such that $\mathrm{P}(Z\ge z)=0.001$.

(ii) Use the formula $Z=\frac{X-\mu }{\sigma }$ to find the value of $b$ such that $\mathrm{P}(X\ge b)=0.001.$

For the random variable $X~\mathrm{N}(\mu ,36)$ it is known that $\mathrm{P}(X<51)=0.8665$.

(i) Write down the value of $\sigma$.

(ii)Calculate the value of $\mathrm{z}$ such that $\mathrm{P}(Z<z)=0.8665.$

(iii)Use the formula $Z=\frac{X-\mu }{\sigma }$ to show that $51-\mu =6.66$.

(iv) Hence calculate the value of $\mu$.

For the random variable $Y~\mathrm{N}(93,{\sigma }^2)$ it is known that $\mathrm{P}(Y>101)=0.0735.$

(i) Calculate the value of $\mathrm{z}$ such that $\mathrm{P}(Z>z)=0.0735$.

(ii) Use the formula $Z=\frac{X-\mu }{\sigma }$ to show that $1.45\sigma =8$.

(iii) Hence calculate the value of $\sigma$.

For the random variable $X~\mathrm{N}(23,4^2)$ find the following probabilities:

(i) $\mathrm{P}(X<20)$

(ii) $\mathrm{P}(X\ge 29)$

(iii)  $\mathrm{P}(20\le X<29)$

For the random variable $Y~\mathrm{N}(100,225)$ find the following probabilities:

(i) $\mathrm{P}(Y\le 90)$

(ii) $\mathrm{P}(Y>140)$

(iii) $\mathrm{P}(85\le Y\le 115)$

The random variable $X~\mathrm{N}(10,9)$.

Find the value of $a$ and the value of $b$, each to 2 decimal places, such that:

(i) $\mathrm{P}(X<a)=0.4$

(ii) $\mathrm{P}(X>b)=0.25$

Use a sketch of the distribution of $X$ to explain why $\mathrm{P}(a\le X\le b)=0.35.$

The random variable $X~N(\mu ,{\sigma }^2)$. It is known that $\mathrm{P}(X>36.88)=0.025$ and $\mathrm{P}(X<27.16)=0.1$

Find the values of $a$ and $b$ for which $\mathrm{P}(Z>a)=0.025$ and $\mathrm{P}(Z<b)=0.1,$where $Z$ is the standard normal variable.  Give your answers correct to 3 decimal places.

Use your answers to part (a), along with the relationship between $Z$ and $X$, to show that the following simultaneous equations must be true:

                  $\mu +1.96\sigma =36.88$

                  $\mu -1.2816\sigma =27.16$

By solving the simultaneous equations in (b), determine the values of $\mu$ and $\sigma$. Give your answers correct to 2 decimal places.

The test scores, $X$, of a group of RAF recruits in an aptitude test are modelled as a normal distribution with $X~\mathrm{N}(210,27.8^2)$.

(i) Find the values of $a$ and $b$ such that  $\mathrm{P}(X<a)=0.25$ and $\mathrm{P}(X>b)=0.25$.

(ii) Hence find the interquartile range of the scores.

Those who score in the top 30% on the test move on to the next stage of training.

One of the recruits, Amelia, achieves a score of 231. Determine whether Amelia will move on to the next stage of training.

The random variable  $X~\mathrm{N}(13,15).$

Find the value of $a$, to 3 significant figures, such that:

(i) $\mathrm{P}(X>a)=0.2$

(ii) $\mathrm{P}(a\le X\le 14)=0.5$

(iii) $\mathrm{P}(13-a\le X\le 13+a)=0.9372$

Explain why there are no values of $a$ such that $\mathrm{P}(14\le X\le a)=0.5$.

The standard normal distribution is $Z~\mathrm{N}(0,1^2).$

(i) Find the value of $\mathrm{z}$ for which $\mathrm{P}(Z<z)=0.9$.

(ii) Use your answer to part (a)(i) along with the properties of the normal distribution to work out the values of  $a$ and $b$ for which $\mathrm{P}(Z>a)=0.1$ and $\mathrm{P}(Z>b)=0.9$.

The weights, $W$ $kg,$ of coconuts grown on the Coconutty As They Come coconut plantation are modelled as a normal distribution with mean 1.25 kg and standard deviation 0.38 kg.  The plantation only considers coconuts to be exportable if their weights are greater than 10% but less than 90%. 

Use your answer to part (a)(ii) to find the range of possible weights, to the nearest 0.01 $kg$, for an exportable coconut.

The random variable $X~\mathrm{N}(\mu ,{\sigma }^2)$. It is known that  $\mathrm{P}(X>34.451)=0.001$ and $\mathrm{P}(X<14.792)=0.1966.$

(i) Use the relationship between $X$ and the standard normal variable $Z$ to show that the following equation must be true:

            $\mu +3.090\sigma =34.451$

(ii) Write down a second equation in terms of $\mu$ and  $\sigma$.

By solving the simultaneous equations in (a), determine the values of $\mu$ and $\sigma$. 

A machine is used to fill bottles of a particular brand of soft drink.  The volume, $V$ ml, of soft drink in the bottles is normally distributed with mean 450 ml and standard deviation $\sigma$ ml.  Given that 5% of the cans contain less than 429 ml of soft drink, find: 

The value of $\sigma$,

 $\mathrm{P}(V\ge 485)$ correct to 3 decimal places,

The probability that the volume of a randomly selected bottle is no more than one standard deviation away from the mean.

Paul enjoys solving sudoku puzzles.  The lengths of time he spends on sudokus in a week are normally distributed with a mean of 2048 minutes and a standard deviation of 64 minutes. 

Find the probability that in a given week Paul spends less than 1945 minutes solving sudoku puzzles.

Estimate the number of weeks in a year that Paul spends between 2019 and 2091 minutes solving sudoku puzzles.

Assuming it takes Paul exactly 10 minutes to solve any sudoku puzzle, find the greatest number, $n$, of sudoku puzzle such that the probability of Paul solving less than $n$ puzzles in a week is less than 0.01.

A machine is used to fill bags of potatoes for a supermarket chain.  The weight, $W$ kg, of potatoes in the bags is normally distributed with mean 3 $kg$ and standard deviation $\sigma$ $kg$. 

Given that 7% of the bags contain a weight of potatoes that is at least 50 $g$ more than  the mean, find:

         $\mathrm{P}(2.9\le W\le 3.1).$

Twelve of the bags of potatoes are chosen at random.

Find the probability that not more than one of the bags will contain less than 2.96 $\mathrm{kg}$ of potatoes.

The test scores, $X$, of a group of Royal Navy recruits in an aptitude test are modelled as a normal distribution with $X~\mathrm{N}(520,89.9^2)$.

Find the interquartile range of the scores.

Those who get a score over 670 are offered a role within the submarine service.  Only those who get a score over 750 are offered a leadership role within the submarine service. 

Given that Mervyn, one of the recruits, has been offered a role within the submarine service, find the probability that he is offered a leadership role.

The distribution of the test scores, $X$, of a group of British Army officer cadets on an aviation aptitude test is modelled as a normal distribution with  $X~N(120,26.5^2)$. 

Only cadets who score in the top 10% on the test are eligible to proceed directly to helicopter pilot training.  Cadets whose scores are between the 40th and 90th percentiles, however, are eligible to resit the test in an attempt to improve their scores. 

Given that it is only possible to receive an integer number of marks as a score on the test, determine the range of test scores for which cadets would be eligible to resit the test.

(i) Find $\mathrm{P}(X>180)$.

(ii) The maximum score it is possible to receive on the test is 180.  Use this fact, and your answer to part (b)(i), to criticise the model being used for the score distribution.

The weight, $W$ kg, of the feed in a sack of pheasant feed produced by a certain manufacturer is modelled as  $W~N\left(\begin{array}{cc}20,& \frac{1}{3600}\end{array}\right)$ .

Roger buys twelve sacks of the manufacturer’s pheasant feed to feed to the pheasants who have begun showing up at his backyard bird feeding station.

Find the probability that all twelve sacks contain feed with a weight that is within 35 $\mathrm{g}$ of 20 $\mathrm{kg}$.

The masses of turkeys can be modelled using a normal distribution with a mean of $4.7\mathrm{kg}$ and a variance of $1.9\mathrm{kg}²$. Nicholas, a farmer, classes a turkey as ‘holiday ready’ if it weighs more than $5.5\mathrm{kg}$. 

A turkey is selected at random. Given that it is ‘holiday ready’, find the probability that it weighs less than $6\mathrm{kg}$.

Nicholas takes a large sample of ‘holiday ready’ turkeys, estimate the median mass of the turkeys in his sample.

An archaeologist has devoted his life to studying ancient Greek vases produced by a particular Boeotian pottery workshop.  The vases were made to a standard pattern, and after measuring a very large number of them the archaeologist has found that 5% of the vases have a mass greater than $2.237\mathrm{kg}$, while only 1% of them have a mass less than $1.906\mathrm{kg}$. 

Given that the masses of the vases may be assumed to be distributed normally, find the mean and standard deviation of the distribution.

The archaeologist has found that vases made by the workshop with a mass less than $m\mathrm{kg}$ are particularly fragile and require special care. A museum has just purchased a collection of  vases produced by the workshop. The probability of all 10 vases requiring special care is $9.766\times {10}^{-24}$. 

Find the value of $m$.

The Monkey Puzzle Tree Marketing Board has proposed a new scheme that will measure the puzzlingness of monkey puzzle trees in terms of a unit called the ‘fuddle’.  It is found that the puzzlingness measurements, $X$, of monkey puzzle trees grown on the We ♥ Puzzling Monkeys monkey puzzle tree plantation can be modelled as a normal distribution with mean $\mu$ fuddles and standard deviation $\sigma$ fuddles.  Because the owners of the plantation are committed to making sure that the monkey puzzle trees they sell to gardeners are neither too puzzling nor not puzzling enough, the plantation only considers monkey puzzle trees to be saleable if their puzzlingness is between the $10\%\mathrm{and}97.5\%$percentiles of the plantation’s monkey puzzle tree puzzlingness measurements. 

Find the range of possible puzzlingness measurements for a saleable monkey puzzle tree. Your answer should be given in terms of $\mu$ and $\sigma$.

Given that $\mathrm{P}(X<\mu -8)=0.2,$ find $\mathrm{P}(X>\mu +8│X>\mu -8).$