Normal Distribution (Edexcel A Level Maths: Statistics): Exam Questions

Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, $x$°C, for Beijing in 2015

Show that, to 3 significant figures, the standard deviation is 5.19 °C.

Simon decides to model the air temperatures with the random variable

$T~\mathrm{N}(22.6,5.{19}^2)$

Using Simon’s model, calculate the 10th to 90th interpercentile range.

Simon wants to model another variable from the large data set for Beijing using a normal distribution.

State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer.

The weight, in grams, of a chocolate bar produced by a certain manufacturer is modelled as $\mathrm{N}(200,1.{75}^2)$.

Find the probability that a randomly selected chocolate bar weighs less than 195 grams.

Heledd buys a pack containing 12 of the chocolate bars.  It may be assumed that the 12 bars in the pack represent a random sample.

Find the probability that all of the bars in the pack have a weight of at least 195 g.

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

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

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

For the random variable $Y~\mathrm{N}(100,225)$ find $\mathrm{P}(85\le Y\le 115)$.

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

Using the model, 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 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 weight falls into the 20% to 80% interpercentile range.

Using the model, find the range of possible weights, to the nearest 0.01 kg, for an exportable coconut. Give your answer as an inequality.

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

Given that $\mathrm{P}(X>36.88)=0.025$ show that $\mu +1.96\sigma =36.88$

Given that $\mathrm{P}(X<27.16)=0.1$, find another equation in terms of $\mu$ and $\sigma$.

Hence, find the values of $\mu$ and $\sigma$. Give your answers correct to 2 decimal places.

The random variable $X~\mathrm{N}(330,{10}^2)$. 

 Find the value of $a$, to 2 decimal places, such that $\mathrm{P}(315\le X\le a)=0.5$

The random variable $X~\mathrm{N}(13,4^2)$. 

Find the value of $a$, to 3 decimal places, such that $\mathrm{P}(a\le X\le 14)=0.5$.

A machine is used to fill cans of a particular brand of soft drink.  The volume, $V$ ml, of soft drink in the cans is normally distributed with mean 330 ml and standard deviation $\sigma$ ml. 

Given that 15% of the cans contain more than 333.4 ml of soft drink, show that $\sigma =3.28$ to three significant figures.

Find $\mathrm{P}(320\le V\le 340).$

Six cans of the soft drink are chosen at random.

Find the probability that all of the cans contain less than 329 ml of soft drink.

A manufacturer uses a machine to make metal rods.

The length of a metal rod, $L$ cm, is normally distributed with

• a mean of 8cm

• a standard deviation of $x$ cm

Given that the proportion of metal rods less than 7.902 cm in length is 2.5%, show that $x=0.05$ to 2 decimal places.

Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length.

The cost of producing a single metal rod is 20p

A metal rod

• where $L<7.94$ is sold for scrap for 5p

• where $7.94\le L\le 8.09$ is sold for 50p

• where $L>8.09$ is shortened for an extra cost of 10p and then sold for 50p

Calculate the expected profit per 500 of the metal rods.

Give your answer to the nearest pound.

The heights of females from a country are normally distributed with

• a mean of 166.5cm

• a standard deviation of 6.1 cm

Given that 1% of females from this country are shorter than $k$ cm, find the value of $k$.

Find the proportion of females from this country with heights between 150cm and 175cm.

A female, from this country, is chosen at random from those with heights between 150 cm and 175cm.

Find the probability that her height is more than 160cm.

The time, $T$ minutes, taken by a courier to deliver a package is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$. It is known that 10% of deliveries take less than 22 minutes and 5% of deliveries take more than 45 minutes.

Find the value of $\mu$ and the value of $\sigma$, giving your answers to 2 decimal places.

A delivery is selected at random. Given that the delivery takes longer than 30 minutes, find the probability that it takes longer than 40 minutes.

The heights, in cm, of adult women in the UK follow a normal distribution. The distribution is shown in the graph below.

Using the graph, estimate the mean and standard deviation of the heights.

The weight, in kilograms, of the feed in a sack of partridge feed produced by a certain manufacturer is modelled as $\mathrm{N}(20,0.1^2)$.

Find the probability that a randomly selected sack of partridge feed weighs less than 19.9 kilograms.

Roger buys ten sacks of the manufacturer’s partridge feed.

Find the probability that at least one of the weights of the ten sacks differs from 20 kilograms by more than 100 grams.

For the random variable $X~\mathrm{N}(\mu ,{\sigma }^2)$, let $\mathrm{P}\left(X<a\right)=p$ and $\mathrm{P}\left(X>b\right)=q$, where $a<\mu <b$.

Write the following in terms of $p$ and $q$.

(i) $\mathrm{P}\left(a<X<b\right)$

(ii) $\mathrm{P}\left(X<a|X<b\right)$

(iii) $\mathrm{P}\left(X<2\mu -a\right)$

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 score in the top 1% on the test are eligible to join the submarine service.

One of the recruits, Mervyn, is a keen would-be submariner. He achieves a score of 750 on the test.  Determine whether Mervyn will be eligible to join the submarine service.

The random variable $X~\mathrm{N}(\mu ,{\sigma }^2).$ It is known that $\mathrm{P}(X>34.451)=0.001$ and $\mathrm{P}(X<14.792)=0.2$

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

The lifetime, $L$ hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.

Alice’s calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.

Find the probability that a randomly selected battery will last for longer than 16 hours.

At the start of her exams Alice put 4 new batteries in her calculator.

She has used her calculator for 16 hours, but has another 4 hours of exams to sit.

Find the probability that her calculator will not stop working for Alice’s remaining exams.

Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.

Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures.

A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.

Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes.

The health centre also claims that the time a dentist spends with a patient during a routine appointment, $T$ minutes, can be modelled by the normal distribution where $T~\mathrm{N}\left(5,3.5^2\right)$

Using this model,

(i) find the probability that a routine appointment with the dentist takes less than 2 minutes

(ii) find $\mathrm{P}\left(T<2|T>0\right)$

(iii) hence explain why this normal distribution may not be a good model for $T$.

The dentist believes that she cannot complete a routine appointment in less than 2 minutes.

She suggests that the health centre should use a refined model only including values of $T>2$.

Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place.

A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, $D$ ml, follows a normal distribution with mean 25 ml.

Given that 15% of bottles contain less than 24.63 ml, find, to 2 decimal places, the value of $k$ such that $\mathrm{P}(24.63<D<k)=0.45$

The times taken to complete a puzzle follow a normal distribution with standard deviation 18 seconds. Given that

$\mathrm{median}:\mathrm{interquartile}\mathrm{range}=5:2$

find the mean time taken to complete the puzzle.

A machine is used to fill bags of potatoes for a supermarket chain.  The masses of the bags are normally distributed with mean 3 kilograms and standard deviation $\sigma$ kilograms.  It is known that 7% of the bags weigh at least 50 grams more than the mean.

Find the probability that the mass of a randomly selected bag does not differ from the mean by more than 100 grams.

Twelve of the bags of potatoes are chosen at random.

Find the probability that no more than one of the bags weigh less than 2.96 kilograms.

The random variable $X~\mathrm{N}(2.35,0.3^2)$. 

Given that $\mathrm{P}(X>a|X<2.5)=0.4$, find the value of $a$ to 2 decimal places.

For the standard normal variable  $Z~\mathrm{N}(0,1^2),$  the function $\mathrm{\Phi }$ is defined by

 $\mathrm{\Phi }\left(a\right)=\mathrm{P}(Z<a),a\in \mathrm{ℝ}$

The constants $m$ and $n$ are positive real numbers.

Find an expression for each of the following probabilities in terms of $\mathrm{\Phi }\left(m\right)$ and $\mathrm{\Phi }\left(n\right)$.

(i) $\mathrm{P}(Z>m)$

(ii) $\mathrm{P}(-m<Z<n)$

The random variable $X~\mathrm{N}(30,4^2)$.

(i) Given that $\mathrm{\Phi }\left(\sqrt{2}\right)=\mathrm{P}\left(X>a\right)$, find the exact value of $a$.

(ii) Given that $\mathrm{\Phi }\left(b\right)=1-\mathrm{P}\left(X<{\pi }^3\right)$, find the exact value of $b$.

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 kg, while only 1% of them have a mass less than 1.906 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 1.93 kg are particularly fragile and require special care.

 A museum has just purchased a collection of $k$ vases produced by the workshop.  The $k$ vases may be assumed to be a random sample.

Given that there is a less than 15% chance that the collection contains at least one vase that is particularly fragile and require special care, find the greatest possible value of $k$. Your answer should be supported by clear algebraic working.