Questions S1 - A-Level Maths

Stuart is investigating the relationship between Gross Domestic Product (GDP) and the size of the population for a particular country.
He takes a random sample of 9 years and records the size of the population, $t$ millions, and the GDP, $g$ billion dollars for each of these years.

The data are summarised as $$n = 9 \quad \sum t = 7.87 \quad \sum g = 144.84 \quad \sum g ^ { 2 } = 3624.41 \quad S _ { t t } = 1.29 \quad S _ { t g } = 40.25$$

Calculate the product moment correlation coefficient between $t$ and $g$
Give an interpretation of your product moment correlation coefficient.
Find the equation of the least squares regression line of $g$ on $t$ in the form $g = a + b t$
Give an interpretation of the value of $b$ in your regression line.
1. Use the regression line from part (c) to estimate the GDP, in billions of dollars, for a population of 7000000
2. Comment on the reliability of your answer in part (i). Give a reason, in context, for your answer. Using the regression line from part (c), Stuart estimates that for a population increase of $x$ million there will be an increase of 0.1 billion dollars in GDP.
Find the value of $x$

Edexcel S1 2022 June Q3

Moderate -0.3

Gill buys a bag of logs to use in her stove. The lengths, $l \mathrm {~cm}$, of the 88 logs in the bag are summarised in the table below.

Length $( \boldsymbol { l } )$	Frequency $( \boldsymbol { f } )$
$15 < l \leqslant 20$	19
$20 < l \leqslant 25$	35
$25 < l \leqslant 27$	16
$27 < l \leqslant 30$	15
$30 < l \leqslant 40$	3

A histogram is drawn to represent these data.
The bar representing logs with length $27 < l \leqslant 30$ has a width of 1.5 cm and a height of 4 cm .

Calculate the width and height of the bar representing log lengths of $20 < l \leqslant 25$
Use linear interpolation to estimate the median of $l$ The maximum length of log Gill can use in her stove is 26 cm .
Gill estimates, using linear interpolation, that $x$ logs from the bag will fit into her stove.
Show that $x = 62$ Gill randomly selects 4 logs from the bag.
Using $x = 62$, find the probability that all 4 logs will fit into her stove. The weights, $W$ grams, of the logs in the bag are coded using $y = 0.5 w - 255$ and summarised by $$n = 88 \quad \sum y = 924 \quad \sum y ^ { 2 } = 12862$$
Calculate
1. the mean of $W$
2. the variance of $W$

Edexcel S1 2022 June Q4

Moderate -0.3

The events $H$ and $W$ are such that

$$\mathrm { P } ( H ) = \frac { 3 } { 8 } \quad \mathrm { P } ( H \cup W ) = \frac { 3 } { 4 }$$ Given that $H$ and $W$ are independent,

show that $\mathrm { P } ( W ) = \frac { 3 } { 5 }$ The event $N$ is such that $$\mathrm { P } ( N ) = \frac { 1 } { 15 } \quad \mathrm { P } ( H \cap N ) = \mathrm { P } ( N )$$
Find $\mathrm { P } \left( N ^ { \prime } \mid H \right)$ Given that $W$ and $N$ are mutually exclusive,
draw a Venn diagram to represent the events $H , W$ and $N$ giving the exact probabilities of each region in the Venn diagram.

Edexcel S1 2022 June Q5

Moderate -0.8

A red spinner is designed so that the score $R$ is given by the following probability distribution.

$r$	2	3	4	5	6
$\mathrm { P } ( R = r )$	0.25	0.3	0.15	0.1	0.2

Show that $\mathrm { E } \left( R ^ { 2 } \right) = 15.8$ Given also that $\mathrm { E } ( R ) = 3.7$
find the standard deviation of $R$, giving your answer to 2 decimal places. A yellow spinner is designed so that the score $Y$ is given by the probability distribution in the table below. The cumulative distribution function $\mathrm { F } ( y )$ is also given.
$y$ 2 3 4 5 6
$\mathrm { P } ( Y = y )$ 0.1 0.2 0.1 $a$ $b$
$\mathrm {~F} ( y )$ 0.1 0.3 0.4 $c$ $d$
Write down the value of $d$ Given that $\mathrm { E } ( Y ) = 4.55$
find the value of $c$ Pabel and Jessie play a game with these two spinners.
Pabel uses the red spinner.
Jessie uses the yellow spinner.
They take turns to spin their spinner.
The winner is the first person whose spinner lands on the number 2 and the game ends. Jessie spins her spinner first.
Find the probability that Jessie wins on her second spin.
Calculate the probability that, in a game, the score on Pabel's first spin is the same as the score on Jessie’s first spin.

Edexcel S1 2022 June Q6

Standard +0.8

A manufacturer fills bottles with oil. The volume of oil in a bottle, $V \mathrm { ml }$, is normally distributed with $V \sim \mathrm {~N} \left( 100,2.5 ^ { 2 } \right)$
1. Find $\mathrm { P } ( V > 104.9 )$
2. In a pack of 150 bottles, find the expected number of bottles containing more than 104.9 ml
3. Find the value of $v$, to 2 decimal places, such that $\mathrm { P } ( V > v \mid V < 104.9 ) = 0.2801$

Edexcel S1 2023 June Q1

Moderate -0.8

The histogram shows the distances, in km , that 274 people travel to work.
\includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-02_1272_1582_296_175}

Given that 60 of these people travel between 10 km and 20 km to work, estimate

the number of people who travel between 22 km and 45 km to work,
the median distance travelled to work by these 274 people,
the mean distance travelled to work by these 274 people.

Edexcel S1 2023 June Q2

Moderate -0.3

Two students, Olive and Shan, collect data on the weight, $w$ grams, and the tail length, $t \mathrm {~cm}$, of 15 mice.

Olive summarised the data as follows $$\mathrm { S } _ { t t } = 5.3173 \quad \sum w ^ { 2 } = 6089.12 \quad \sum t w = 2304.53 \quad \sum w = 297.8 \quad \sum t = 114.8$$

Calculate the value of $\mathrm { S } _ { t w }$ and the value of $\mathrm { S } _ { w w }$
Calculate the value of the product moment correlation coefficient between $w$ and $t$
Show that the equation of the regression line of $w$ on $t$ can be written as $$w = - 16.7 + 4.77 t$$
Give an interpretation of the gradient of the regression line.
Explain why it would not be appropriate to use the regression line in part (c) to estimate the weight of a mouse with a tail length of 2 cm . Shan decided to code the data using $x = t - 6$ and $y = \frac { w } { 2 } - 5$
Write down the value of the product moment correlation coefficient between $x$ and $y$
Write down an equation of the regression line of $y$ on $x$ You do not need to simplify your equation.

Edexcel S1 2023 June Q3

Easy -1.2

Jim records the length, $l \mathrm {~mm}$, of 81 salmon. The data are coded using $x = l - 600$ and the following summary statistics are obtained.

$$n = 81 \quad \sum x = 3711 \quad \sum x ^ { 2 } = 475181$$

Find the mean length of these salmon.
Find the variance of the lengths of these salmon. The weight, $w$ grams, of each of the 81 salmon is recorded to the nearest gram. The recorded results for the 81 salmon are summarised in the box plot below.
\includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-10_362_1479_849_296}
Find the maximum number of salmon that have weights in the interval $$4600 < w \leqslant 7700$$ Raj says that the box plot is incorrect as Jim has not included outliers.
For these data an outlier is defined as a value that is more than
$1.5 \times$ IQR above the upper quartile or $1.5 \times$ IQR below the lower quartile
Show that there are no outliers.

Edexcel S1 2023 June Q4

Moderate -0.8

A bag contains a large number of coloured counters. Each counter is labelled A, B or C
$30 \%$ of the counters are labelled A
$45 \%$ of the counters are labelled B
The rest of the counters are labelled C
It is known that
2\% of the counters labelled A are red
4\% of the counters labelled B are red
6\% of the counters labelled C are red
One counter is selected at random from the bag.
1. Complete the tree diagram on the opposite page to illustrate this information.
2. Calculate the probability that the counter is labelled A and is not red.
3. Calculate the probability that the counter is red.
4. Given that the counter is red, find the probability that it is labelled C
\includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-15_1155_1000_285_456}

Edexcel S1 2023 June Q6

Moderate -0.8

6\% of the counters labelled C are red
One counter is selected at random from the bag.

Complete the tree diagram on the opposite page to illustrate this information.
Calculate the probability that the counter is labelled A and is not red.
Calculate the probability that the counter is red.
Given that the counter is red, find the probability that it is labelled C \end{enumerate} \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-15_1155_1000_285_456}
5. A discrete random variable $Y$ has probability function $$\mathrm { P } ( \mathrm { Y } = \mathrm { y } ) = \left\{ \begin{array} { c l } \mathrm { k } ( 3 - \mathrm { y } ) & y = 1,2 \\ \mathrm { k } \left( \mathrm { y } ^ { 2 } - 8 \right) & y = 3,4,5 \\ \mathrm { k } & y = 6 \\ 0 & \text { otherwise } \end{array} \right.$$ where $k$ is a constant.
Show that $k = \frac { 1 } { 30 }$ Find the exact value of
$\mathrm { P } ( 1 < Y \leqslant 4 )$
$\mathrm { E } ( Y )$ The random variable $X = 15 - 2 Y$
Calculate $\mathrm { P } ( Y \geqslant X )$
Calculate $\operatorname { Var } ( X )$
1. Three events $A , B$ and $C$ are such that
$$\mathrm { P } ( A ) = 0.1 \quad \mathrm { P } ( B \mid A ) = 0.3 \quad \mathrm { P } ( A \cup B ) = 0.25 \quad \mathrm { P } ( C ) = 0.5$$ Given that $A$ and $C$ are mutually exclusive
find $\mathrm { P } ( A \cup C )$
Show that $\mathrm { P } ( B ) = 0.18$ Given also that $B$ and $C$ are independent,
draw a Venn diagram to represent the events $A , B$ and $C$ and the probabilities associated with each region.

Edexcel S1 2023 June Q7

Standard +0.3

A machine squeezes apples to extract their juice. The volume of juice, $J \mathrm { ml }$, extracted from 1 kg of apples is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$

Given that $\mu = 500$ and $\sigma = 25$ use standardisation to

1. show that $\mathrm { P } ( J > 510 ) = 0.3446$
2. calculate the value of $d$ such that $\mathrm { P } ( J > d ) = 0.9192$ Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.
Calculate the probability that each of the 5 bags of apples produce less than 510 ml of juice. Following adjustments to the machine, the volume of juice, $R \mathrm { ml }$, extracted from 1 kg of apples is such that $\mu = 520$ and $\sigma = k$ Given that $\mathrm { P } ( R < r ) = 0.15$ and $\mathrm { P } ( R > 3 r - 800 ) = 0.005$
find the value of $r$ and the value of $k$

Edexcel S1 2024 June Q1

Easy -1.2

A researcher is investigating the growth of two types of tree, Birch and Maple. The height, to the nearest cm, a seedling grows in one year is recorded for 35 Birch trees and 32 Maple trees. The results are summarised in the back-to-back stem and leaf diagram below.

Totals	Birch		Maple	Totals
(2)	98	2	57789	(5)
(8)	99965311	3	0266899	(7)
(9)	988763111	4	$111 \boldsymbol { k } 78$	(6)
(9)	777543210	5	0123444	(7)
(3)	765	6	346	(3)
(3)	654	7	07	(2)
(1)	5	8	00	(2)

Key: 5 | 6 | 3 means 65 cm for a Birch tree and 63 cm for a Maple tree
The median height that these Maple trees grow in one year is 45 cm .

Find the value of $\boldsymbol { k }$, used in the stem and leaf diagram.
Find the lower quartile and the upper quartile of the height grown in one year for these Birch trees. The researcher defines an outlier as an observation that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { or less than } Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
Show that there is only one outlier amongst the Birch trees. The grid on page 3 shows a box plot for the heights that the Maple trees grow in one year.
On the same grid draw a box plot for the heights that the Birch trees grow in one year.
Comment on any difference in the distributions of the growth of these Birch trees and the growth of these Maple trees.
State the values of any statistics you have used to support your comment. The researcher realises he has missed out 4 pieces of data for the Maple trees. The heights each seedling grows in one year, to the nearest cm, in ascending order, for these 4 Maple trees are $27 \mathrm {~cm} , a \mathrm {~cm} , 48 \mathrm {~cm} , 2 a \mathrm {~cm}$. Given that there is no change to the box plot for the Maple trees given on page 3
find the range of possible values for $a$ Show your working clearly.
\includegraphics[max width=\textwidth, alt={}]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-03_1243_1659_1464_210}
Only use this grid if you need to redraw your answer for part (d)
\includegraphics[max width=\textwidth, alt={}, center]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-05_1154_1643_1503_217}
(Total for Question 1 is 13 marks)

Edexcel S1 2024 June Q2

Moderate -0.8

2. A spinner can land on the numbers $2,4,5,7$ or 8 only. The random variable $X$ represents the number that this spinner lands on when it is spun once. The probability distribution of $X$ is given in the table below.

$\boldsymbol { x }$	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

Moderate -0.8

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

Moderate -0.3

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

Standard +0.3

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

Standard +0.3

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

Moderate -0.8

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

Moderate -0.3

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

Standard +0.3

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

Moderate -0.3

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

Challenging +1.2

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

Easy -1.2

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

Moderate -0.8

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

Easy -1.3

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}

Length \(( \boldsymbol { l } )\)	Frequency \(( \boldsymbol { f } )\)
\(15 < l \leqslant 20\)	19
\(20 < l \leqslant 25\)	35
\(25 < l \leqslant 27\)	16
\(27 < l \leqslant 30\)	15
\(30 < l \leqslant 40\)	3

\(y\)	2	3	4	5	6
\(\mathrm { P } ( Y = y )\)	0.1	0.2	0.1	\(a\)	\(b\)
\(\mathrm {~F} ( y )\)	0.1	0.3	0.4	\(c\)	\(d\)

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

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

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