Questions - Edexcel S1 - A-Level Maths

$\boldsymbol { x }$	2	4	5	7	8
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.25	0.3	0.2	0.1	0.15

Find $\mathrm { P } ( 2 X - 3 > 5 )$ Given that $\mathrm { E } ( X ) = 4.6$
show that $\operatorname { Var } ( X ) = 4.14$ The random variable $Y = a X - b$ where $a$ and $b$ are positive constants.
Given that $$\mathrm { E } ( Y ) = 13.4 \quad \text { and } \quad \operatorname { Var } ( Y ) = 66.24$$
find the value of $a$ and the value of $b$ In a game Sam and Alex each spin the spinner once, landing on $X _ { 1 }$ and $X _ { 2 }$ respectively.
Sam's score is given by the random variable $S = X _ { 1 }$
Alex's score is given by the random variable $R = 2 X _ { 2 } - 3$
The person with the higher score wins the game. If the scores are the same it is a draw.
Find the probability that Sam wins the game.

Edexcel S1 2024 June Q3

The lengths, $x \mathrm {~mm}$, of 50 pebbles are summarised in the table below.

Length	Frequency
$20 \leqslant x < 30$	2
$30 \leqslant x < 32$	16
$32 \leqslant x < 36$	20
$36 \leqslant x < 40$	8
$40 \leqslant x < 45$	3
$45 \leqslant x < 50$	1

A histogram is drawn to represent these data.
The bar representing the class $32 \leqslant x < 36$ is 2.5 cm wide and 7.5 cm tall.

Calculate the width and the height of the bar representing the class $30 \leqslant x < 32$
Using linear interpolation, estimate the median of $x$ The weight, $w$ grams, of each of the 50 pebbles is coded using $10 y = w - 20$ These coded data are summarised by $$\sum y = 104 \quad \sum y ^ { 2 } = 233.54$$
Show that the mean of $w$ is 40.8
Calculate the standard deviation of $w$ The weight of a pebble recorded as 40.8 grams is added to the sample.
Without carrying out any further calculations, state, giving a reason, what effect this would have on the value of
1. the mean of $w$
2. the standard deviation of $w$

Edexcel S1 2024 June Q4

A biologist is studying bears. The biologist records the length, $d \mathrm {~cm}$, and the girth, $g \mathrm {~cm}$, of 8 bears. The biologist summarises the data as follows

$$\begin{gathered} \sum d = 1456.8 \quad \sum g = 713.2 \quad \sum d g = 141978.84 \quad \sum g ^ { 2 } = 72675.98
S _ { d d } = 16769.78 \end{gathered}$$

Calculate the exact value of $S _ { d g }$ and the exact value of $S _ { g g }$
Calculate the value of the product moment correlation coefficient between $d$ and $g$
Show that the equation of the regression line of $g$ on $d$ can be written as $$g = - 42.3 + 0.722 d$$ where the values of the intercept and gradient are given to 3 significant figures.
Give an interpretation, in context, of the gradient of the regression line. Using the equation of the regression line given in part (c)
1. estimate the girth of a bear with a length of 2.5 metres,
2. explain why an estimate for the girth of a bear with a length of 0.5 metres is not reliable. Using the regression line from part (c), the biologist estimates that for each $x \mathrm {~cm}$ increase in the length of a bear there will be a 17.3 cm increase in the girth.
Find the value of $x$

Edexcel S1 2024 June Q5

A competition consists of two rounds.

The time, in minutes, taken by adults to complete round one is modelled by a normal distribution with mean 15 minutes and standard deviation 2 minutes.

Use standardisation to find the proportion of adults that take less than 18 minutes to complete round one. Only the fastest $60 \%$ of adults from round one take part in round two.
Use standardisation to find the longest time that an adult can take to complete round one if they are to take part in round two. The time, $T$ minutes, taken by adults to complete round two is modelled by a normal distribution with mean $\mu$ Given that $\mathrm { P } ( \mu - 10 < T < \mu + 10 ) = 0.95$
find $\mathrm { P } ( T > \mu - 5 \mid T > \mu - 10 )$

Edexcel S1 2024 June Q6

The Venn diagram shows the probabilities related to teenagers playing 3 particular board games.
$C$ is the event that a teenager plays Chess
$S$ is the event that a teenager plays Scrabble
$G$ is the event that a teenager plays Go
where $p$ and $q$ are probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-22_684_935_598_566}
1. Find the probability that a randomly selected teenager plays Chess but does not play Go.
Given that the events $C$ and $S$ are independent,
find the value of $p$
Hence find the value of $q$
Find (i) $\mathrm { P } \left( ( C \cup S ) \cap G ^ { \prime } \right)$
(ii) $\mathrm { P } ( C \mid ( S \cap G ) )$ A youth club consists of a large number of teenagers.
In this youth club 76 teenagers play Chess and Go.
Use the information in the Venn diagram to estimate how many of the teenagers in the youth club do not play Scrabble.

Edexcel S1 2016 October Q1

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

Given that $\mathrm { P } ( X > \mu + a ) = 0.35$ where $a$ is a constant, find

$\mathrm { P } ( X > \mu - a )$
$\mathrm { P } ( \mu - a < X < \mu + a )$
$\mathrm { P } ( X < \mu + a \mid X > \mu - a )$

Edexcel S1 2016 October Q2

The discrete random variable $X$ has probability distribution

$x$	- 2	- 1	1	2	3
$\mathrm { P } ( X = x )$	$b$	$a$	$a$	$b$	$\frac { 1 } { 5 }$

where $a$ and $b$ are constants.

Write down an equation for $a$ and $b$.
Calculate $\mathrm { E } ( X )$. Given that $\mathrm { E } \left( X ^ { 2 } \right) = 3.5$
1. find a second equation in $a$ and $b$,
2. hence find the value of $a$ and the value of $b$.
Find $\operatorname { Var } ( X )$. The random variable $Y = 5 - 3 X$
Find $\mathrm { P } ( Y > 0 )$.
Find
1. $\mathrm { E } ( Y )$,
2. $\operatorname { Var } ( Y )$.

Edexcel S1 2016 October Q3

Hugo recorded the purchases of 80 customers in the ladies fashion department of a large store. His results were as follows

20 customers bought a coat
12 customers bought a coat and a scarf
23 customers bought a pair of gloves
13 customers bought a pair of gloves and a scarf no customer bought a coat and a pair of gloves 14 customers did not buy a coat nor a scarf nor a pair of gloves.

Draw a Venn diagram to represent all of this information.
One of the 80 customers is selected at random.
1. Find the probability that the customer bought a scarf.
2. Given that the customer bought a coat, find the probability that the customer also bought a scarf.
3. State, giving a reason, whether or not the event 'the customer bought a coat' and the event 'the customer bought a scarf' are statistically independent. Hugo had asked the member of staff selling coats and the member of staff selling gloves to encourage customers also to buy a scarf.
By considering suitable conditional probabilities, determine whether the member of staff selling coats or the member of staff selling gloves has the better performance at selling scarves to their customers. Give a reason for your answer.

Edexcel S1 2016 October Q4

A doctor is studying the scans of 30 -week old foetuses. She takes a random sample of 8 scans and measures the length, $f \mathrm {~mm}$, of the leg bone called the femur. She obtains the following results.

$$\begin{array} { l l l l l l l l } 52 & 53 & 56 & 57 & 57 & 59 & 60 & 62 \end{array}$$

Show that $\mathrm { S } _ { f f } = 80$ The doctor also measures the head circumference, $h \mathrm {~mm}$, of each foetus and her results are summarised as $$\sum h = 2209 \quad \sum h ^ { 2 } = 610463 \quad \mathrm {~S} _ { f h } = 182$$
Find $\mathrm { S } _ { h h }$
Calculate the product moment correlation coefficient between the length of the femur and the head circumference for these data. The doctor believes that there is a linear relationship between the length of the femur and the head circumference of 30-week old foetuses.
State, giving a reason, whether or not your calculation in part (c) supports the doctor's belief.
Find an equation of the regression line of $h$ on $f$. The doctor plans in future to measure the femur length, $f$, and then use the regression line to estimate the corresponding head circumference, $h$. A statistician points out that there will always be the chance of an error between the true head circumference and the estimated value of the head circumference. Given that the error, $E \mathrm {~mm}$, has the normal distribution $\mathrm { N } \left( 0,4 ^ { 2 } \right)$
find the probability that the estimate is within 3 mm of the true value.

Edexcel S1 2016 October Q5

The label on a jar of Amy’s jam states that the jar contains about 400 grams of jam. For each jar that contains less than 388 grams of jam, Amy will be fined $\pounds 100$. If a jar contains more than 410 grams of jam then Amy makes a loss of $\pounds 0.30$ on that jar.

The weight of jam, $A$ grams, in a jar of Amy's jam has a normal distribution with mean $\mu$ grams and standard deviation $\sigma$ grams. Amy chooses $\mu$ and $\sigma$ so that $\mathrm { P } ( A < 388 ) = 0.001$ and $\mathrm { P } ( A > 410 ) = 0.0197$

Find the value of $\mu$ and the value of $\sigma$. Amy can sell jars of jam containing between 388 grams and 410 grams for a profit of $\pounds 0.25$
Calculate the expected amount, in £s, that Amy receives for each jar of jam.

Edexcel S1 2016 October Q6

The stem and leaf diagram gives the blood pressure, $x \mathrm { mmHg }$, for a random sample of 19 female patients.

10	1	2
11	2	7	7	8	8
12	0	2	2	3	4	4	5	5	7
13	1	2	9

Key: 10 | 1 means blood pressure of 101 mmHg

Find the median and the quartiles for these data.
Find the interquartile range ( $Q _ { 3 } - Q _ { 1 }$ ) An outlier is a value that is greater than $Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$ or less than $Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$
Showing your working clearly, identify any outliers for these data.
On the grid on page 21 draw a box and whisker plot to represent these data. Show any outliers clearly. The above data can be summarised by $$\sum x = 2299 \text { and } \sum x ^ { 2 } = 279709$$
Calculate the mean and the standard deviation for these data. For a random sample taken from a normal distribution, a rule for determining outliers is: an outlier is more than $2.7 \times$ standard deviation above or below the mean.
Find the limits to determine outliers using this rule.
State, giving a reason based on some of the above calculations, whether or not a normal distribution is a suitable model for these data. \includegraphics[max width=\textwidth, alt={}, center]{8ff7539e-fa44-4388-af8c-80656f081528-21_2281_73_308_15}
Turn over for a spare diagram if you need to redraw your plot.
\includegraphics[max width=\textwidth, alt={}]{8ff7539e-fa44-4388-af8c-80656f081528-24_2639_1830_121_121}

Edexcel S1 2018 October Q1

The heights above sea level ( $h$ hundred metres) and the temperatures ( $t ^ { \circ } \mathrm { C }$ ) at 12 randomly selected places in France, at 7 am on July 31st, were recorded.
The data are summarised as follows
1. Find the value of $S _ { t t }$
2. Calculate the product moment correlation coefficient for these data.
3. Interpret the relationship between $t$ and $h$.
4. Find an equation of the regression line of $t$ on $h$.
At 7 am on July 31st Yinka is on holiday in South Africa. He uses the regression equation to estimate the temperature when the height above sea level is 500 m .
Find the estimated temperature Yinka calculates.
Comment on the validity of your answer in part (e). $$\sum h = 112 \quad \sum t = 136 \quad \sum t ^ { 2 } = 1828 \quad S _ { h t } = - 236 \quad S _ { h h } = 297$$
Find the value of $S$ (2)

Edexcel S1 2018 October Q2

The weights, to the nearest kilogram, of a sample of 33 female spotted hyenas living in the Serengeti are summarised in the stem and leaf diagram below.

\begin{table}[h]

\captionsetup{labelformat=empty} \caption{Weight (kg)}

3	2	3	7
4	1	3	3	4	5	5	6	9
5	1	2	2	3	4	4	5	5	5	7	8	8	9	9	9
6	2	3	3
7	1	4	7
8	4

\end{table} Totals

Find the median and quartiles for the weights of the female spotted hyenas. An outlier is defined as any value greater than $c$ or any value less than $d$ where $$\begin{aligned} & c = Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right)
& d = Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \end{aligned}$$
Showing your working clearly, identify any outliers for these data.
(3) The weights, to the nearest kilogram, of a sample of male spotted hyenas living in the Serengeti are summarised below.
\includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-06_755_1568_1537_185}
In the space provided in the grid above, draw a box and whisker plot to represent the weights of female spotted hyenas living in the Serengeti. Indicate clearly any outliers. (A copy of this grid is on page 9 if you need to redraw your box and whisker plot.)
Compare the weights of male and female spotted hyenas living in the Serengeti. Key: 3|2 means 32
\includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-09_2658_101_107_9}

Edexcel S1 2018 October Q3

3. The parking times, $t$ hours, for cars in a car park are summarised below.

Time (t hours)	Frequency (f)	Time midpoint (m)
$0 \leqslant t < 1$	10	0.5
$1 \leqslant t < 2$	18	1.5
$2 \leqslant t < 4$	15	3
$4 \leqslant t < 6$	12	5
$6 \leqslant t < 12$	5	9

$$\text { (You may use } \sum \mathrm { fm } = 182 \text { and } \sum \mathrm { fm } ^ { 2 } = 883 \text { ) }$$ A histogram is drawn to represent these data.
The bar representing the time $1 \leqslant t < 2$ has a width of 1.5 cm and a height of 6 cm .

Calculate the width and the height of the bar representing the time $4 \leqslant t < 6$
Use linear interpolation to estimate the median parking time for the cars in the car park.
Estimate the mean and the standard deviation of the parking time for the cars in the car park.
Describe, giving a reason, the skewness of the data. One of these cars is selected at random.
Estimate the probability that this car is parked for more than 75 minutes.

Edexcel S1 2018 October Q4

4. Pieces of wood cladding are produced by a timber merchant. There are three types of fault, $A , B$ and $C$, that can appear in each piece of wood cladding. The Venn diagram shows the probabilities of a piece of wood cladding having the various types of fault.
\includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-14_602_1120_497_413} A piece of wood cladding is chosen at random.

Find the probability that the piece of wood cladding has more than one type of fault. Fault types $A$ and $C$ occur independently.
Find the probability that the piece of wood cladding has no faults. Given that the piece of wood cladding has fault $A$,
find the probability that it also has fault $B$ but not fault $C$. Two pieces of the wood cladding are selected at random.
Find the probability that both have exactly 2 types of fault.

Edexcel S1 2018 October Q5

The discrete random variable $X$ is defined by the cumulative distribution function

$x$	1	2	3	4	5
$\mathrm {~F} ( x )$	$\frac { 3 k } { 2 }$	$4 k$	$\frac { 15 k } { 2 }$	$12 k$	$\frac { 35 k } { 2 }$

where $k$ is a constant.

Find the probability distribution of $X$.
Find $\mathrm { P } ( 1.5 < X \leqslant 3.5 )$ The random variable $Y = 12 - 7 X$
Calculate Var(Y)
Calculate $\mathrm { P } ( 4 X \leqslant | Y | )$

Edexcel S1 2018 October Q6

A machine makes bolts such that the length, $L \mathrm {~cm}$, of a bolt has distribution $L \sim \mathrm {~N} \left( 4.1,0.125 ^ { 2 } \right)$

A bolt is selected at random.

Find the probability that the length of this bolt is more than 4.3 cm .
Show that $\mathrm { P } ( 3.9 < L < 4.3 )$ is 0.890 correct to 3 decimal places. The machine makes 500 bolts.
The cost to make each bolt is 5 pence.
Only bolts with length between 3.9 cm and 4.3 cm can be used. These are sold for 9 pence each. All the bolts that cannot be used are recycled with a scrap value of 1 pence each.
Calculate an estimate for the profit made on these 500 bolts. Following adjustments to the machine, the length of a bolt, $B \mathrm {~cm}$, made by the machine is such that $B \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$ Given that $\mathrm { P } ( B > 4.198 ) = 0.025$ and $\mathrm { P } ( B < 4.065 ) = 0.242$
find the value of $\mu$ and the value of $\sigma$
State, giving a reason, whether the adjustments to the machine will result in a decrease or an increase in the profit made on 500 bolts.

Edexcel S1 2022 October Q1

The stem lengths of a sample of 120 tulips are recorded in the grouped frequency table below.

Stem length (cm)	Frequency
$40 \leqslant x < 42$	12
$42 \leqslant x < 45$	18
$45 \leqslant x < 50$	23
$50 \leqslant x < 55$	35
$55 \leqslant x < 58$	24
$58 \leqslant x < 60$	8

A histogram is drawn to represent these data.
The area of the bar representing the $40 \leqslant x < 42$ class is $16.5 \mathrm {~cm} ^ { 2 }$

Calculate the exact area of the bar representing the $42 \leqslant x < 45$ class. The height of the tallest bar in the histogram is 10 cm .
Find the exact height of the second tallest bar.
$Q _ { 1 }$ for these data is 45 cm .
Use linear interpolation to find an estimate for
1. $Q _ { 2 }$
2. the interquartile range. One measure of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
By calculating this measure, describe the skewness of these data.

Edexcel S1 2022 October Q2

The production cost, $\pounds c$ million, of a film and the total ticket sales, $\pounds t$ million, earned by the film are recorded for a sample of 40 films.

Some summary statistics are given below. $$\sum c = 1634 \quad \sum t = 1361 \quad \sum t ^ { 2 } = 82873 \quad \sum c t = 83634 \quad \mathrm {~S} _ { c c } = 28732.1$$

Find the exact value of $\mathrm { S } _ { t t }$ and the exact value of $\mathrm { S } _ { c t }$
Calculate the value of the product moment correlation coefficient for these data.
Give an interpretation of your answer to part (b)
Show that the equation of the linear regression line of $t$ on $c$ can be written as $$t = - 5.84 + 0.976 c$$ where the values of the intercept and gradient are given to 3 significant figures.
Find the expected total ticket sales for a film with a production cost of $\pounds 90$ million. Using the regression line in part (d)
find the range of values of the production cost of a film for which the total ticket sales are less than $80 \%$ of its production cost.

Edexcel S1 2022 October Q3

Morgan is investigating the body length, $b$ centimetres, of squirrels.

A random sample of 8 squirrels is taken and the data for each squirrel is coded using $$x = \frac { b - 21 } { 2 }$$ The results for the coded data are summarised below $$\sum x = - 1.2 \quad \sum x ^ { 2 } = 5.1$$

Find the mean of $b$
Find the standard deviation of $b$ A 9th squirrel is added to the sample. Given that for all 9 squirrels $\sum x = 0$
find
1. the body length of the 9th squirrel,
2. the standard deviation of $x$ for all 9 squirrels.

Edexcel S1 2022 October Q4

The cumulative distribution function of the discrete random variable $W$, which takes only the values 6,7 and 8 , is given by

$$F ( W ) = \frac { ( w + 3 ) ( w - 1 ) } { 77 } \text { for } w = 6,7,8$$ Find $\mathrm { E } ( W )$

VIAV SIHI NI III IM IONOOC

VIIIV SIHI NI III IM I I N O O

VI4V SIHI NI III IM I ON OC

Edexcel S1 2022 October Q5

The weights, $W$ grams, of kiwi fruit grown on a farm are normally distributed with mean 80 grams and standard deviation 8 grams.

The table shows the classifications of the kiwi fruit by their weight, where $k$ is a positive constant.

Small		Large
Tiny	Petite	Extra	Jumbo	Mega
$w < 66$	$66 \leqslant w < 70$	$70 \leqslant w < 80$	$80 \leqslant w < k$	$w \geqslant k$

One kiwi fruit is selected at random from those grown on the farm.

Find the probability that this kiwi fruit is Large. 35\% of the kiwi fruit are Jumbo.
Find the value of $k$ to one decimal place. 75\% of Tiny kiwi fruit weigh more than $y$ grams.
Find the value of $y$ giving your answer to one decimal place.

Edexcel S1 2022 October Q6

The Venn diagram shows the events $A , B , C$ and $D$, where $p , q , r$ and $s$ are probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{1fda59cb-059e-4850-810f-cc3e69bc058e-20_504_826_296_621}
1. Write down the value of
  1. $\mathrm { P } ( A )$
  2. $\mathrm { P } ( A \mid B )$
  3. $\mathrm { P } ( A \mid C )$
Given that $\mathrm { P } \left( B ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }$ and $\mathrm { P } ( C \mid D ) = \frac { 3 } { 5 }$
find the exact value of $q$ and the exact value of $r$ Given also that $\mathrm { P } \left( B \cup C ^ { \prime } \right) = \frac { 5 } { 8 }$
find the exact value of $s$

Edexcel S1 2022 October Q7

Adana selects one number at random from the distribution of $X$ which has the following probability distribution.

$x$	0	5	10
$\mathrm { P } ( X = x )$	0.1	0.2	0.7

Given that the number selected by Adana is not 5 , write down the probability it is 0
Show that $\mathrm { E } \left( X ^ { 2 } \right) = 75$
Find $\operatorname { Var } ( X )$
Find $\operatorname { Var } ( 4 - 3 X )$ Bruno and Charlie each independently select one number at random from the distribution of $X$
Find the probability that the number Bruno selects is greater than the number Charlie selects. Devika multiplies Bruno's number by Charlie's number to obtain a product, $D$
Determine the probability distribution of $D$

Edexcel S1 2023 October Q1

Sally plays a game in which she can either win or lose.

A turn consists of up to 3 games. On each turn Sally plays the game up to 3 times. If she wins the first 2 games or loses the first 2 games, then she will not play the 3rd game.

The probability that Sally wins the first game in a turn is 0.7
If Sally wins a game the probability that she wins the next game is 0.6
If Sally loses a game the probability that she wins the next game is 0.2
1. Use this information to complete the tree diagram on page 3
2. Find the probability that Sally wins the first 2 games in a turn.
3. Find the probability that Sally wins exactly 2 games in a turn.

Given that Sally wins 2 games in a turn,

find the probability that she won the first 2 games. Given that Sally won the first game in a turn,

find the probability that she won 2 games. 1st game 2nd game win

Length	Frequency
\(20 \leqslant x < 30\)	2
\(30 \leqslant x < 32\)	16
\(32 \leqslant x < 36\)	20
\(36 \leqslant x < 40\)	8
\(40 \leqslant x < 45\)	3
\(45 \leqslant x < 50\)	1

\(x\)	- 2	- 1	1	2	3
\(\mathrm { P } ( X = x )\)	\(b\)	\(a\)	\(a\)	\(b\)	\(\frac { 1 } { 5 }\)

\(x\)	1	2	3	4	5
\(\mathrm {~F} ( x )\)	\(\frac { 3 k } { 2 }\)	\(4 k\)	\(\frac { 15 k } { 2 }\)	\(12 k\)	\(\frac { 35 k } { 2 }\)

\(\boldsymbol { x }\)	2	4	5	7	8
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.25	0.3	0.2	0.1	0.15

Stem length (cm)	Frequency
\(40 \leqslant x < 42\)	12
\(42 \leqslant x < 45\)	18
\(45 \leqslant x < 50\)	23
\(50 \leqslant x < 55\)	35
\(55 \leqslant x < 58\)	24
\(58 \leqslant x < 60\)	8

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