Questions S2 - A-Level Maths

Find the value of $p$ The probability that the delivery company loses more than $k$ packages in a randomly selected month is at least $2 p$
Find the largest possible value of $k$ In a randomly selected month,
find the probability that exactly 4 packages were lost in each half of the month. In a randomly selected two-month period, 21 packages were lost.
Find the probability that at least 10 packages were lost in each of these two months.
Using a suitable approximation, find the probability that more than 27 packages are lost during a randomly selected 4-month period.

Edexcel S2 2018 January Q6

8 marks Standard +0.3

In a local council, $60 \%$ of households recycle at least half of their waste. A random sample of 80 households is taken.

The random variable $X$ represents the number of households in the sample that recycle at least half of their waste.

Using a suitable approximation, find the smallest number of households, $n$, such that $$\mathrm { P } ( X \geqslant n ) < 0.05$$ The number of bags recycled per family per week was known to follow a Poisson distribution with mean 1.5 Following a recycling campaign, the council believes the mean number of bags recycled per family per week has increased. To test this belief, 6 families are selected at random and the total number of bags they recycle the following week is recorded. The council wishes to test, at the 5\% level of significance, whether or not there is evidence that the mean number of bags recycled per family per week has increased.
Find the critical region for the total number of bags recycled by the 6 families.

Edexcel S2 2018 January Q7

15 marks Standard +0.3

The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by

$$f ( x ) = \begin{cases} \frac { 1 } { 16 } x ^ { 2 } & 1 \leqslant x < 3 \\ k ( 4 - x ) & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

Show that $k = \frac { 11 } { 12 }$
Sketch $\mathrm { f } ( x )$ for $1 \leqslant x \leqslant 4$
Write down the mode of $X$ Given that $\mathrm { E } ( X ) = \frac { 25 } { 9 }$
use algebraic integration to find $\operatorname { Var } ( X )$, giving your answer to 3 significant figures. The cumulative distribution function of $X$ is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 48 } \left( x ^ { 3 } + c \right) & 1 \leqslant x < 3 \\ \frac { 11 } { 12 } \left( 4 x - \frac { 1 } { 2 } x ^ { 2 } + d \right) & 3 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
1. Find the exact value of $C$
2. Find the exact value of $d$
Calculate, to 3 significant figures, the upper quartile of $X$

\includegraphics[max width=\textwidth, alt={}]{a814156d-6945-4601-9cae-d28d8ae0db1e-28_2632_1826_121_121}

Edexcel S2 2019 January Q1

7 marks Moderate -0.3

A bus company sells tickets for a journey from London to Oxford every Saturday. Past records show that $5 \%$ of people who buy a ticket do not turn up for the journey.

The bus has seats for 48 people.
Each week the bus company sells tickets to exactly 50 people for the journey.
The random variable $X$ represents the number of these people who do not turn up for the journey.

State one assumption required to model $X$ as a binomial distribution. For this week's journey find,
1. the probability that all 50 people turn up for the journey,
2. $\mathrm { P } ( X = 1 )$ The bus company receives $\pounds 20$ for each ticket sold and all 50 tickets are sold. It must pay out $\pounds 60$ to each person who buys a ticket and turns up for the journey but does not have a seat.
Find the bus company's expected total earnings per journey.

Edexcel S2 2019 January Q2

12 marks Standard +0.3

During morning hours, employees arrive randomly at an office drinks dispenser at a rate of 2 every 10 minutes.

The number of employees arriving at the drinks dispenser is assumed to follow a Poisson distribution.

Find the probability that fewer than 5 employees arrive at the drinks dispenser during a 10-minute period one morning. During a 30 -minute period one morning, the probability that $n$ employees arrive at the drinks dispenser is the same as the probability that $n + 1$ employees arrive at the drinks dispenser.
Find the value of $n$ During a 45-minute period one morning, the probability that between $c$ and 12, inclusive, employees arrive at the drinks dispenser is 0.8546
Find the value of $C$
Find the probability that exactly 2 employees arrive at the drinks dispenser in exactly 4 of the 6 non-overlapping 10-minute intervals between 10 am and 11am one morning.

Edexcel S2 2019 January Q3

8 marks Moderate -0.3

3. Figure 1 shows an accurate graph of the cumulative distribution function, $\mathrm { F } ( x )$, for the continuous random variable $X$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{17296edc-9ab4-4f81-ae68-c76190986fd1-08_535_1152_354_342} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

Find $\mathrm { P } ( 3 < X < 7 )$ The probability density function of $X$ is given by $$\mathrm { f } ( x ) = \begin{cases} a & 2 \leqslant x < 4 \\ b & 4 \leqslant x < 6 \\ c & 6 \leqslant x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$ where $a$, $b$ and $c$ are constants.
Find the value of $a$, the value of $b$ and the value of $c$
Find $\mathrm { E } ( X )$

Edexcel S2 2019 January Q4

9 marks Standard +0.3

At a shop, past figures show that $35 \%$ of customers pay by credit card. Following the shop’s decision to no longer charge a fee for using a credit card, a random sample of 20 customers is taken and 11 are found to have paid by credit card.

Hadi believes that the proportion of customers paying by credit card is now greater than 35\%

Test Hadi's belief at the $5 \%$ level of significance. State your hypotheses clearly. For a random sample of 20 customers,
show that 11 lies less than 2 standard deviations above the mean number of customers paying by credit card.
You may assume that $35 \%$ is the true proportion of customers who pay by credit card.

Edexcel S2 2019 January Q5

13 marks Standard +0.3

The continuous random variable $X$ is uniformly distributed over the interval $[ a , b ]$ where $0 < a < b$

Given that $\mathrm { P } ( X < b - 2 a ) = \frac { 1 } { 3 }$

1. show that $\mathrm { E } ( X ) = \frac { 5 a } { 2 }$
2. find $\mathrm { P } ( X > b - 4 a )$ The continuous random variable $Y$ is uniformly distributed over the interval [3, c] where $c > 3$ Given that $\operatorname { Var } ( Y ) = 3 c - 9$, find
1. the value of $c$
2. $\mathrm { P } ( 2 Y - 7 < 20 - Y )$
3. $\mathrm { E } \left( Y ^ { 2 } \right)$

Edexcel S2 2019 January Q6

12 marks Moderate -0.3

(i) (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

A factory produces tyres for bicycles and $0.25 \%$ of the tyres produced are defective. A company orders 3000 tyres from the factory.
(b) Find, using a Poisson approximation, the probability that there are more than 7 defective tyres in the company’s order.
(ii) At the company $40 \%$ of employees are known to cycle to work. A random sample of 150 employees is taken. The random variable $C$ represents the number of employees in the sample who cycle to work.
(a) Describe a suitable sampling frame that can be used to take this sample.
(b) Explain what you understand by the sampling distribution of $C$ Louis uses a normal approximation to calculate the probability that at most $\alpha$ employees in the sample cycle to work. He forgets to use a continuity correction and obtains the incorrect probability 0.0668 Find, showing all stages of your working,
(c) the value of $\alpha$
(d) the correct probability.

Edexcel S2 2019 January Q7

14 marks Standard +0.3

The continuous random variable $X$ has probability density function

$$f ( x ) = \begin{cases} c ( x + 3 ) & - 3 \leqslant x < 0 \\ \frac { 5 } { 36 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where $c$ is a positive constant.

Show that $c = \frac { 1 } { 12 }$
1. Sketch the probability density function.
2. Explain why the mode of $X = 0$
Find the cumulative distribution function of $X$, for all values of $x$
Find, to 3 significant figures, the value of $d$ such that $\mathrm { P } ( X > d \mid X > 0 ) = \frac { 2 } { 5 }$
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Q7

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Edexcel S2 2021 January Q1

14 marks Standard +0.3

Jim farms oysters in a particular lake. He knows from past experience that $5 \%$ of young oysters do not survive to be harvested.

In a random sample of 30 young oysters, the random variable $X$ represents the number that do not survive to be harvested.

Write down a suitable model for the distribution of $X$.
State an assumption that has been made for the model in part (a).
Find the probability that
1. exactly 24 young oysters do survive to be harvested,
2. at least 3 young oysters do not survive to be harvested. A second random sample, of 200 young oysters, is taken. The probability that at least $n$ of these young oysters do not survive to be harvested is more than 0.8
Using a suitable approximation, find the maximum value of $n$. Jim believes that the level of salt in the lake water has changed and it has altered the survival rate of his oysters. He takes a random sample of 25 young oysters and places them in the lake.
When Jim harvests the oysters, he finds that 21 do survive to be harvested.
Use a suitable test, at the $5 \%$ level of significance, to assess whether or not there is evidence that the proportion of oysters not surviving to be harvested is more than $5 \%$. State your hypotheses clearly.

Edexcel S2 2021 January Q2

10 marks Standard +0.3

2. The distance, in metres, a novice tightrope artist, walking on a wire, walks before falling is modelled by the random variable $W$ with cumulative distribution function $$\mathrm { F } ( w ) = \left\{ \begin{array} { c c } 0 & w < 0 \\ \frac { 1 } { 3 } \left( w - \frac { w ^ { 4 } } { 256 } \right) & 0 \leqslant w \leqslant 4 \\ 1 & w > 4 \end{array} \right.$$

Find the probability that a novice tightrope artist, walking on the wire, walks at least 3.5 metres before falling. A random sample of 30 novice tightrope artists is taken.
Find the probability that more than 1 of these novice tightrope artists, walking on the wire, walks at least 3.5 metres before falling. Given $\mathrm { E } ( W ) = 1.6$
use algebraic integration to find $\operatorname { Var } ( W )$
DO NOT WRITEIN THIS AREA

Edexcel S2 2021 January Q3

17 marks Standard +0.3

3. The number of water fleas, in 100 ml of pond water, has a Poisson distribution with mean 7

Find the probability that a sample of 100 ml of the pond water does not contain exactly 4 water fleas. Aja collects 5 separate samples, each of 100 ml , of the pond water.
Find the probability that exactly 1 of these samples contains exactly 4 water fleas. Using a normal approximation, the probability that more than 3 water fleas will be found in a random sample of $n \mathrm { ml }$ of the pond water is 0.9394 correct to 4 significant figures.
1. Show that $n - 1.55 \sqrt { \frac { n } { 0.07 } } - 50 = 0$
2. Hence find the value of $n$ After the pond has been cleaned, the number of water fleas in a 100 ml random sample of the pond water is 15
Using a suitable test, at the $1 \%$ level of significance, assess whether or not there is evidence that the number of water fleas per 100 ml of the pond water has increased. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-11_2255_50_314_34}

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel S2 2021 January Q4

10 marks Standard +0.3

4. A continuous random variable $X$ has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( a - x ) ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where $k$ and $a$ are constants.

Show that $k a ^ { 3 } = 3$ Given that $\mathrm { E } ( X ) = 1.5$
use algebraic integration to show that $a = 6$
Verify that the median of $X$ is 1.2 to one decimal place.
\includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-15_2255_50_314_34}

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel S2 2021 January Q5

14 marks Standard +0.3

5. A piece of wood $A B$ is 3 metres long. The wood is cut at random at a point $C$ and the random variable $W$ represents the length of the piece of wood $A C$.

Find the probability that the length of the piece of wood $A C$ is more than 1.8 metres. The two pieces of wood $A C$ and $C B$ form the two shortest sides of a right-angled triangle. The random variable $X$ represents the length of the longest side of the right-angled triangle.
Show that $X ^ { 2 } = 2 W ^ { 2 } - 6 W + 9$
[0pt] [You may assume for random variables $S , T$ and $U$ and for constants $a$ and $b$ that if $S = a T + b U$ then $\mathrm { E } ( S ) = a \mathrm { E } ( T ) + b \mathrm { E } ( U ) ]$
Find $\mathrm { E } \left( X ^ { 2 } \right)$
Find $\mathrm { P } \left( X ^ { 2 } > 5 \right)$

Edexcel S2 2021 January Q6

10 marks Moderate -0.8

6. The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.

Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.
Suggest a suitable sampling frame.
Identify the sampling units. The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70
When a ball is selected at random, the random variable $X$ represents the number on the ball where $$\mathrm { P } ( X = 20 ) = p \quad \mathrm { P } ( X = 50 ) = q \quad \mathrm { P } ( X = 70 ) = r$$ A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag. The random variable $M$ is the mean of the 2 numbers recorded. Given that $$\mathrm { P } ( M = 20 ) = \frac { 25 } { 64 } \quad \mathrm { P } ( M = 60 ) = \frac { 1 } { 16 } \quad \text { and } \quad q > r$$
show that $\mathrm { P } ( M = 50 ) = \frac { 1 } { 16 }$
VIHV SIHII NI I IIIM I ON OC VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO
\includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-24_111_65_2525_1880}
\includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-24_140_233_2625_1733}

Edexcel S2 2022 January Q1

11 marks Standard +0.3

1 A local pottery makes cups. The number of faulty cups made by the pottery in a week follows a Poisson distribution with a mean of 6 In a randomly chosen week, the probability that there will be at least $x$ faulty cups made is 0.1528

Find the value of $x$
Use a normal approximation to find the probability that in 6 randomly chosen weeks the total number of faulty cups made is fewer than 32 A week is called a "poor week" if at least $x$ faulty cups are made, where $x$ is the value found in part (a).
Find the probability that in 50 randomly chosen weeks, more than 1 is a "poor week".

Edexcel S2 2022 January Q2

8 marks Moderate -0.8

2 The continuous random variable $X$ has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < - k \\ \frac { x + k } { 4 k } & - k \leqslant x \leqslant 3 k \\ 1 & x > 3 k \end{array} \right.$$ where $k$ is a positive constant.

Specify fully, in terms of $k$, the probability density function of $X$
Write down, in terms of $k$, the value of $\mathrm { E } ( X )$
Show that $\operatorname { Var } ( X ) = \frac { 4 } { 3 } k ^ { 2 }$
Find, in terms of $k$, the value of $\mathrm { E } \left( 3 X ^ { 2 } \right)$

Edexcel S2 2022 January Q3

9 marks Standard +0.3

3 A photocopier in a school is known to break down at random at a mean rate of 8 times per week.

Give a reason why a Poisson distribution could be used to model the number of breakdowns. The headteacher of the school replaces the photocopier with a refurbished one and wants to find out if the rate of breakdowns has increased or decreased.
Write down suitable null and alternative hypotheses that the headteacher should use. The refurbished photocopier was monitored for the first week after it was installed.
Using a $5 \%$ level of significance, find the critical region to test whether the rate of breakdowns has now changed.
Find the actual significance level of a test based on the critical region from part (c). During the first week after it was installed there were 4 breakdowns.
Comment on this finding in the light of the critical region found in part (c).

Edexcel S2 2022 January Q4

15 marks Standard +0.3

4 The continuous random variable $X$ has a probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 2 } k ( x - 1 ) & 1 \leqslant x \leqslant 3 \\ k & 3 < x \leqslant 6 \\ \frac { 1 } { 4 } k ( 10 - x ) & 6 < x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$ where $k$ is a positive constant.

Sketch $\mathrm { f } ( x )$ for all values of $x$
Show that $k = \frac { 1 } { 6 }$
Specify fully the cumulative distribution function $\mathrm { F } ( x )$ of $X$ Given that $\mathrm { E } ( X ) = \frac { 61 } { 12 }$
find $\mathrm { P } ( X > \mathrm { E } ( X ) )$
Describe the skewness of the distribution, giving a reason for your answer.

Edexcel S2 2022 January Q5

14 marks Moderate -0.3

5 Applicants for a pilot training programme with a passenger airline are screened for colour blindness. Past records show that the proportion of applicants identified as colour blind is 0.045

Write down a suitable model for the distribution of the number of applicants identified as colour blind from a total of $n$ applicants.
State one assumption necessary for this distribution to be a suitable model of this situation.
Using a suitable approximation, find the probability that exactly 5 out of 120 applicants are identified as colour blind.
Explain why the approximation that you used in part (c) is appropriate. Jaymini claims that 75\% of all applicants for this training programme go on to become pilots. From a random sample of 96 applicants for this training programme 67 go on to become pilots.
Using a suitable approximation, test Jaymini's claim at the $5 \%$ level of significance. State your hypotheses clearly.

Edexcel S2 2022 January Q6

10 marks Standard +0.3

Explain what you understand by the sampling distribution of a statistic. At Sam's cafe a standard breakfast consists of 6 breakfast items. Customers can then choose to upgrade to a medium breakfast by adding 1 extra breakfast item or they can upgrade to a large breakfast by adding 2 extra breakfast items. Standard, medium and large breakfasts are sold in the ratio $6 : 3 : 2$ respectively. A random sample of 2 customers is taken from customers who have bought a breakfast from Sam's cafe on a particular day.
Find the sampling distribution for the total number, $T$, of breakfast items bought by these 2 customers. Show your working clearly.
Find $\mathrm { E } ( T )$

Edexcel S2 2022 January Q7

8 marks Standard +0.8

7 The sides of a square are each of length $L \mathrm {~cm}$ and its area is $A \mathrm {~cm} ^ { 2 }$ Given that $A$ is uniformly distributed on the interval [10,30]

find $\mathrm { P } ( L \geqslant 4.5 )$
find $\operatorname { Var } ( L )$

\includegraphics[max width=\textwidth, alt={}]{a009b02e-4cd3-497b-a141-4630c653e20b-28_2655_1947_114_116}

Edexcel S2 2023 January Q1

11 marks Moderate -0.3

A shop sells shoes at a mean rate of 4 pairs of shoes per hour on a weekday.
1. Suggest a suitable distribution for modelling the number of sales of pairs of shoes made per hour on a weekday.
2. State one assumption necessary for this distribution to be a suitable model of this situation.
3. Find the probability that on a weekday the shop sells
  1. more than 4 pairs of shoes in a one-hour period,
  2. more than 4 pairs of shoes in each of 3 consecutive one-hour periods.
The area manager visits the shop on a weekday, the day after an advert for the shop appears in a local paper. In a one-hour period during the manager's visit, the shop sells 7 pairs of shoes. This leads the manager to believe that the advert has increased the shop's sales of pairs of shoes.
Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is evidence of an increase in sales of pairs of shoes following the appearance of the advert.

Edexcel S2 2023 January Q2

11 marks Moderate -0.8

A bag contains a large number of coins. It only contains 20 p and 50 p coins. A random sample of 3 coins is taken from the bag.
1. List all the possible combinations of 3 coins that might be taken.
Let $\bar { X }$ represent the mean value of the 3 coins taken.
Part of the sampling distribution of $\bar { X }$ is given below.
$\bar { x }$ 20 $a$ $b$ 50
$\mathrm { P } ( \bar { X } = \bar { x } )$ $\frac { 4913 } { 8000 }$ $c$ $d$ $\frac { 27 } { 8000 }$
Write down the value of $a$ and the value of $b$ The probability of taking a 20p coin at random from the bag is $p$ The probability of taking a 50p coin at random from the bag is $q$
Find the value of $p$ and the value of $q$
Hence, find the value of $c$ and the value of $d$ Let $M$ represent the mode of the 3 coins taken at random from the bag.
Find the sampling distribution of $M$

\(\bar { x }\)	20	\(a\)	\(b\)	50
\(\mathrm { P } ( \bar { X } = \bar { x } )\)	\(\frac { 4913 } { 8000 }\)	\(c\)	\(d\)	\(\frac { 27 } { 8000 }\)

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