Questions S1 - A-Level Maths

Write down the time that is exceeded by $75 \%$ of Westyou members. The times for the Eastyou members are summarised by the stem and leaf diagram below.
Stem Leaf
2 0 2 3 4 $( 4 )$
2 5 6 8 8 8 9 9
3 0 0 0 0 0 1 1 1 2 2 2 2 3 4 $( 14 )$
3 5 5 5 7 9 $( 5 )$
Key: 2|0 means 20 minutes
Find the value of the median and interquartile range for the Eastyou members. An outlier is a value that falls either
On the grid on page 7, draw a box plot to represent the times of the Eastyou members.
State the skewness of each distribution. Give reasons for your answers. $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 } \\ & \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
\includegraphics[max width=\textwidth, alt={}]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-06_2255_50_314_1976}
\includegraphics[max width=\textwidth, alt={}, center]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-07_406_1390_2224_262} Turn over for a spare grid if you need to redraw your box plot. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Only use this grid if you need to redraw your box plot.} \includegraphics[alt={},max width=\textwidth]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-09_401_1399_2261_258}
\end{figure}

Edexcel S1 2018 June Q3

7 marks Moderate -0.3

A manufacturer of electric generators buys engines for its generators from three companies, $R , S$ and $T$.

Company $R$ supplies 40\% of the engines. Company $S$ supplies $25 \%$ of the engines. The rest of the engines are supplied by company $T$. It is known that $2 \%$ of the engines supplied by company $R$ are faulty, $1 \%$ of the engines supplied by company $S$ are faulty and $2 \%$ of the engines supplied by company $T$ are faulty. An engine is chosen at random.

Draw a tree diagram to show all the possible outcomes and the associated probabilities.
Calculate the probability that the engine is from company $R$ and is not faulty.
Calculate the probability that the engine is faulty. Given that the engine is faulty,
find the probability that the engine did not come from company $S$.

Edexcel S1 2018 June Q4

10 marks Moderate -0.3

4. A discrete random variable $X$ has probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } k ( 2 - x ) & x = 0,1 \\ k ( 3 - x ) & x = 2,3 \\ k ( x + 1 ) & x = 4 \\ 0 & \text { otherwise } \end{array} \right.$$ where $k$ is a constant.

Show that $k = \frac { 1 } { 9 }$ Find the exact value of
$\mathrm { P } ( 1 \leqslant X < 4 )$
$\mathrm { E } ( X )$
$\mathrm { E } \left( X ^ { 2 } \right)$
$\operatorname { Var } ( 3 X + 1 )$

Edexcel S1 2018 June Q5

13 marks Moderate -0.8

5. The weights, in grams, of a random sample of 48 broad beans are summarised in the table.

Weight in grams ( $\boldsymbol { x }$ )	Frequency (f)	Class midpoint (y)
$0.9 < x \leqslant 1.1$	9	1.0
$1.1 < x \leqslant 1.3$	12	1.2
$1.3 < x \leqslant 1.5$	11	1.4
$1.5 < x \leqslant 1.7$	8	1.6
$1.7 < x \leqslant 1.9$	3	1.8
$1.9 < x \leqslant 2.1$	3	2.0
$2.1 < x \leqslant 2.7$	2	2.4

(You may assume $\sum \mathrm { fy } { } ^ { 2 } = 101.56$ ) A histogram was drawn to represent these data. The $2.1 < x \leqslant 2.7$ class was represented by a bar of width 1.5 cm and height 1 cm .

Find the width and height of the $0.9 < x \leqslant 1.1$ class.
Give a reason to justify the use of a histogram to represent these data.
Estimate the mean and the standard deviation of the weights of these broad beans.
Use linear interpolation to estimate the median of the weights of these broad beans. One of these broad beans is selected at random.
Estimate the probability that its weight lies between 1.1 grams and 1.6 grams. One of these broad beans having a recorded weight of 0.95 grams was incorrectly weighed. The correct weight is 1.4 grams.
State, giving a reason, the effect this would have on your answers to part (c). Do not carry out any further calculations.

Edexcel S1 2018 June Q6

9 marks Standard +0.3

6. The waiting time, $L$ minutes, to see a doctor at a health centre is normally distributed with $L \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$. Given that $\mathrm { P } ( L < 15 ) = 0.9$ and $\mathrm { P } ( L < 5 ) = 0.05$

find the value of $\mu$ and the value of $\sigma$. There are 23 people waiting to see a doctor at the health centre.
Determine the expected number of these people who will have a waiting time of more than 12 minutes.

Edexcel S1 2018 June Q7

12 marks Standard +0.3

Events $A$ and $B$ are such that

$$\mathrm { P } ( A ) = 0.5 \quad \mathrm { P } ( A \mid B ) = \frac { 2 } { 3 } \quad \mathrm { P } \left( A ^ { \prime } \cup B ^ { \prime } \right) = 0.6$$

Find $\mathrm { P } ( B )$
Find $\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)$ The event $C$ has $\mathrm { P } ( C ) = 0.15$ The events $A$ and $C$ are mutually exclusive. The events $B$ and $C$ are independent.
Find $\mathrm { P } ( B \cap C )$
Draw a Venn diagram to illustrate the events $A , B$ and $C$ and the probabilities for each region.

Edexcel S1 2019 June Q1

9 marks Easy -1.2

The heights, $x$ metres, of 40 children were recorded by a teacher. The results are summarised as follows

$$\sum x = 58 \quad \sum x ^ { 2 } = 84.829$$

Find the mean and the variance of the heights of these 40 children. The teacher decided that these statistics would be more useful in centimetres.
Find
1. the mean of these heights in centimetres,
2. the standard deviation of these heights in centimetres. Two more children join the group. Their heights are 130 cm and 160 cm .
1. State, giving a reason, the mean height of the 42 children.
2. Without recalculating the standard deviation, state, giving a reason, whether the standard deviation of the heights of the 42 children will be greater than, less than or the same as the standard deviation of the heights of the group of 40 children.

Edexcel S1 2019 June Q2

13 marks Easy -1.2

2. Chi wanted to summarise the scores of the 39 competitors in a village quiz. He started to produce the following stem and leaf diagram Key: 2|5 is a score of 25 \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Score}

1	1	5	8	9
2	0	2	5	8	9
3	3	5	5	7	8	9	$\ldots$

\end{table} He did not complete the stem and leaf diagram but instead produced the following box plot.
\includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-04_357_1237_772_356} Chi defined an outlier as a value that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$ or
less than $Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$

Find
1. the interquartile range
2. the range.
Describe, giving a reason, the skewness of the distribution of scores. Albert and Beth asked for their scores to be checked.
Albert's score was changed from 25 to 37
Beth's score was changed from 54 to 60
On the grid on page 5, draw an updated box plot. Show clearly any calculations that you used. Some of the competitors complained that the questions were biased towards the younger generation. The product moment correlation coefficient between the age of the competitors and their score in the quiz is - 0.187
State, giving a reason, whether or not the complaint is supported by this statistic. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-05_360_1242_2238_351} Turn over for a spare grid if you need to redraw your box plot. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-07_367_1246_2261_351}

Edexcel S1 2019 June Q3

13 marks Challenging +1.2

3. A certain disease occurs in a population in 2 mutually exclusive types. It is difficult to diagnose people with type $A$ of the disease and there is an unknown proportion $p$ of the population with type $A$.
It is easier to diagnose people with type $B$ of the disease and it is known that $2 \%$ of the population have type $B$. A test has been developed to help diagnose whether or not a person has the disease. The event $T$ represents a positive result on the test. After a large-scale trial of the test, the following information was obtained. For a person with type $B$ of the disease the probability of a positive test result is 0.96 For a person who does not have the disease the probability of a positive test result is 0.05 For a person with type $A$ of the disease the probability of a positive test result is $q$

Complete the tree diagram.
\includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-08_776_965_1050_484} The probability of a randomly selected person having a positive test result is 0.169 For a person with a positive test result, the probability that they do not have the disease is $\frac { 41 } { 169 }$
Find the value of $p$ and the value of $q$. A doctor is about to see a person who she knows does not have type $B$ of the disease but does have a positive test result.
1. Find the probability that this person has type $A$ of the disease.
2. State, giving a reason, whether or not the doctor will find the test useful.

Edexcel S1 2019 June Q4

13 marks Moderate -0.3

The weights of packages delivered to Susie are normally distributed with a mean of 510 grams and a standard deviation of 45 grams.
1. Find the probability that a randomly selected package delivered to Susie weighs less than 450 grams.
The heaviest 5\% of packages delivered to Susie are delivered by Rav in his van, the others are delivered by Taruni on foot.
Find the weight of the lightest package that Rav would deliver to Susie. Susie randomly selects a package from those delivered by Taruni.
Find the probability that this package weighs more than 450 grams. On Tuesday there are 5 packages delivered to Susie.
Find the probability that 4 are delivered by Taruni and 1 is delivered by Rav.

Edexcel S1 2019 June Q5

14 marks Standard +0.3

The discrete random variable $X$ represents the score when a biased spinner is spun. The probability distribution of $X$ is given by

$x$	- 2	- 1	0	2	3
$\mathrm { P } ( X = x )$	$p$	$p$	$q$	$\frac { 1 } { 4 }$	$p$

where $p$ and $q$ are probabilities.

Find $\mathrm { E } ( X )$. Given that $\operatorname { Var } ( X ) = 2.5$
find the value of $p$.
Hence find the value of $q$. Amar is invited to play a game with the spinner.
The spinner is spun once and $X _ { 1 }$ is the score on the spinner. If $X _ { 1 } > 0$ Amar wins the game.
If $X _ { 1 } = 0$ Amar loses the game.
If $X _ { 1 } < 0$ the spinner is spun again and $X _ { 2 }$ is the score on this second spin and if $X _ { 1 } + X _ { 2 } > 0$ Amar wins the game, otherwise Amar loses the game.
Find the probability that Amar wins the game. Amar does not want to lose the game.
He says that because $\mathrm { E } ( X ) > 0$ he will play the game.
State, giving a reason, whether or not you would agree with Amar.

Edexcel S1 2019 June Q6

13 marks Moderate -0.8

Ranpose hospital offers services to a large number of clinics that refer patients to a range of hospitals.
The manager at Ranpose hospital took a random sample of 16 clinics and recorded

the distance, $x \mathrm {~km}$, of the clinic from Ranpose hospital
the percentage, $y \%$, of the referrals from the clinic who attend Ranpose hospital.

The data are summarised as $$\bar { x } = 8.1 \quad \bar { y } = 20.5 \quad \sum y ^ { 2 } = 8266 \quad \mathrm {~S} _ { x x } = 368.16 \quad \mathrm {~S} _ { x y } = - 630.9$$

Find the product moment correlation coefficient for these data.
Give an interpretation of your correlation coefficient. The manager at Ranpose hospital believes that there may be a linear relationship between the distance of a clinic from the hospital and the percentage of the referrals who attend the hospital. She drew the following scatter diagram for these data.
\includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-20_1106_926_1133_511}
State, giving a reason, whether or not these data support the manager's belief.
(1)
\section*{[The summary data and the scatter diagram are repeated below.]} The data are summarised as $$\bar { x } = 8.1 \quad \bar { y } = 20.5 \quad \sum y ^ { 2 } = 8266 \quad \mathrm {~S} _ { x x } = 368.16 \quad \mathrm {~S} _ { x y } = - 630.9$$ \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-22_1118_936_612_504}
Find the equation of the regression line of $y$ on $x$, giving your answer in the form $$y = a + b x$$
Give an interpretation of the gradient of your regression line.
Draw your regression line on the scatter diagram. The manager believes that Ranpose hospital should be attracting an "above average" percentage of referrals from clinics that are less than 5 km from the hospital. She proposes to target one clinic with some extra publicity about the services Ranpose offers.
On the scatter diagram circle the point representing the clinic she should target.
VIIIV SIHI NI JIIYM ION OC NAMV SIHIL NI JAHAM ION OC VJ4V SIHII NI JIIYM ION OO

Edexcel S1 2020 June Q1

5 marks Moderate -0.8

The discrete random variable $X$ takes the values $- 1,2,3,4$ and 7 only.

Given that $$\mathrm { P } ( X = x ) = \frac { 8 - x } { k } \text { for } x = - 1,2,3,4 \text { and } 7$$ find the value of $\mathrm { E } ( X )$

Edexcel S1 2020 June Q2

13 marks Moderate -0.8

In a school canteen, students can choose from a main course of meat ( $M$ ), fish ( $F$ ) or vegetarian ( $V$ ). They can then choose a drink of either water ( $W$ ) or juice ( $J$ ).

The partially completed tree diagram, where $p$ and $q$ are probabilities, shows the probabilities of these choices for a randomly selected student. \section*{Drink} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Main course} \includegraphics[alt={},max width=\textwidth]{81d5e460-9559-4d25-aa08-6440559aec83-04_783_1013_593_463}

\end{figure}

Complete the tree diagram, giving your answers in terms of $p$ and $q$ where appropriate.
Find an expression, in terms of $p$ and $q$, for the probability that a randomly selected student chooses water to drink. The events "choosing a vegetarian main course" and "choosing water to drink" are independent.
Find a linear equation in terms of $p$ and $q$. A student who has chosen juice to drink is selected at random. The probability that they chose fish for their main course is $\frac { 7 } { 30 }$
Find the value of $p$ and the value of $q$. The canteen manager claims that students who choose water to drink are most likely to choose a fish main course.
State, showing your working clearly, whether or not the manager's claim is correct.

Edexcel S1 2020 June Q3

13 marks Standard +0.3

3. The distance achieved in a long jump competition by students at a school is normally Students who achieve a distance greater than 4.3 metres receive a medal.

Find the proportion of students who receive medals. The school wishes to give a certificate of achievement or a medal to the $80 \%$ of students who achieve a distance of at least $d$ metres.
Find the value of $d$. Of those who received medals, the $\frac { 1 } { 3 }$ who jump the furthest will receive gold medals.
Find the shortest distance, $g$ metres, that must be achieved to receive a gold medal. A journalist from the local newspaper interviews a randomly selected group of 3 medal winners.
Find the exact probability that there is at least one gold medal winner in the group.
\includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-08_79_1153_233_251} Students who achieve a distance greater than 4.3 metres receive a medal.
Find the proportion of students who receive medals.
VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel S1 2020 June Q4

14 marks Moderate -0.3

4. A group of students took some tests. A teacher is analysing the average mark for each student. Each student obtained a different average mark. For these average marks, the lower quartile is 24 , the median is 30 and the interquartile range (IQR) is 10
The three lowest average marks are 8, 10 and 15.5 and the three highest average marks are 45, 52.5 and 56 The teacher defines an outlier to be a value that is either
more than $1.5 \times$ IQR below the lower quartile or
more than $1.5 \times$ IQR above the upper quartile

Determine any outliers in these data.
On the grid below draw a box plot for these data, indicating clearly any outliers.
\includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-12_350_1223_1128_370}
Use the quartiles to describe the skewness of these data. Give a reason for your answer. Two more students also took the tests. Their average marks, which were both less than 45, are added to the data and the box plot redrawn. The median and the upper quartile are the same but the lower quartile is now 26
Redraw the box plot on the grid below.
(3)
\includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-12_350_1221_2106_367}
Give ranges of values within which each of these students' average marks must lie. Turn over for spare grids if you need to redraw your answer for part (b) or part (d).
VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Copy of grid for part (b)} \includegraphics[alt={},max width=\textwidth]{81d5e460-9559-4d25-aa08-6440559aec83-15_356_1226_1726_367}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Copy of grid for part (d)} \includegraphics[alt={},max width=\textwidth]{81d5e460-9559-4d25-aa08-6440559aec83-15_353_1226_2240_367}
\end{figure}

Edexcel S1 2020 June Q5

15 marks Moderate -0.3

A large company rents shops in different parts of the country. A random sample of 10 shops was taken and the floor area, $x$ in $10 \mathrm {~m} ^ { 2 }$, and the annual rent, $y$ in thousands of dollars, were recorded.
The data are summarised by the following statistics

$$\sum x = 900 \quad \sum x ^ { 2 } = 84818 \quad \sum y = 183 \quad \sum y ^ { 2 } = 3434$$ and the regression line of $y$ on $x$ has equation $y = 6.066 + 0.136 x$

Use the regression line to estimate the annual rent in dollars for a shop with a floor area of $800 \mathrm {~m} ^ { 2 }$
Find $\mathrm { S } _ { y y }$ and $\mathrm { S } _ { x x }$
Find the product moment correlation coefficient between $y$ and $x$. An 11th shop is added to the sample. The floor area is $900 \mathrm {~m} ^ { 2 }$ and the annual rent is 15000 dollars.
Use the formula $\mathrm { S } _ { x y } = \sum ( x - \bar { x } ) ( y - \bar { y } )$ to show that the value of $\mathrm { S } _ { x y }$ for the 11 shops will be the same as it was for the original 10 shops.
Find the new equation of the regression line of $y$ on $x$ for the 11 shops. The company is considering renting a larger shop with area of $3000 \mathrm {~m} ^ { 2 }$
Comment on the suitability of using the new regression line to estimate the annual rent. Give a reason for your answer.

Edexcel S1 2020 June Q6

15 marks

6. The random variable $A$ represents the score when a spinner is spun. The probability distribution for $A$ is given in the following table.

$a$	1	4	5	7
$\mathrm { P } ( A = a )$	0.40	0.20	0.25	0.15

Show that $\mathrm { E } ( A ) = 3.5$
Find $\operatorname { Var } ( A )$ The random variable $B$ represents the score on a 4 -sided die. The probability distribution for $B$ is given in the following table where $k$ is a positive integer.
$b$ 1 3 4 $k$
$\mathrm { P } ( B = b )$ 0.25 0.25 0.25 0.25
Write down the name of the probability distribution of $B$.
Given that $\mathrm { E } ( B ) = \mathrm { E } ( A )$ state, giving a reason, the value of $k$. The random variable $X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$ Sam and Tim are playing a game with the spinner and the die. They each spin the spinner once to obtain their value of $A$ and each roll the die once to obtain their value of $B$.
Their value of $A$ is taken as their value of $\mu$ and their value of $B$ is taken as their value of $\sigma$. The person with the larger value of $\mathrm { P } ( X > 3.5 )$ is the winner.
Given that Sam obtained values of $a = 4$ and $b = 3$ and Tim obtained $b = 4$ find, giving a reason, the probability that Tim wins.
Find the largest value of $\mathrm { P } ( X > 3.5 )$ achievable in this game.
Find the probability of achieving this value. \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-21_2255_50_314_34}

Edexcel S1 2021 June Q1

5 marks Easy -1.2

There are 7 red counters, 3 blue counters and 2 yellow counters in a bag. Gina selects a counter at random from the bag and keeps it. If the counter is yellow she does not select any more counters. If the counter is not yellow she randomly selects a second counter from the bag.
1. Complete the tree diagram.
First Counter
Second Counter
\includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-02_1147_1081_603_397} Given that Gina has selected a yellow counter,
find the probability that she has 2 counters.

Edexcel S1 2021 June Q2

12 marks Challenging +1.2

2. In the Venn diagram below, $A , B$ and $C$ are events and $p , q , r$ and $s$ are probabilities. The events $A$ and $C$ are independent and $\mathrm { P } ( A ) = 0.65$
\includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-04_373_815_397_568}

State which two of the events $A$, $B$ and $C$ are mutually exclusive.
Find the value of $r$ and the value of $s$. The events ( $A \cap C ^ { \prime }$ ) and ( $B \cup C$ ) are also independent.
Find the exact value of $p$ and the exact value of $q$. Give your answers as fractions.

Edexcel S1 2021 June Q3

14 marks Moderate -0.8

A random sample of 100 carrots is taken from a farm and their lengths, $L \mathrm {~cm}$, recorded. The data are summarised in the following table.

Length, $L$ cm	Frequency, f	Class mid point, $\boldsymbol { x } \mathbf { c m }$
$5 \leqslant L < 8$	5	6.5
$8 \leqslant L < 10$	13	9
$10 \leqslant L < 12$	16	11
$12 \leqslant L < 15$	25	13.5
$15 \leqslant L < 20$	30	17.5
$20 \leqslant L < 28$	11	24

A histogram is drawn to represent these data.
The bar representing the class $5 \leqslant L < 8$ is 1.5 cm wide and 1 cm high.

Find the width and height of the bar representing the class $15 \leqslant L < 20$
Use linear interpolation to estimate the median length of these carrots.
Estimate
1. the mean length of these carrots,
2. the standard deviation of the lengths of these carrots. A supermarket will only buy carrots with length between 9 cm and 22 cm .
Estimate the proportion of carrots from the farm that the supermarket will buy. Any carrots that the supermarket does not buy are sold as animal feed. The farm makes a profit of 2.2 pence on each carrot sold to the supermarket, a profit of 0.8 pence on each carrot longer than 22 cm and a loss of 1.2 pence on each carrot shorter than 9 cm .
Find an estimate of the mean profit per carrot made by the farm.

Edexcel S1 2021 June Q4

13 marks Standard +0.3

Kris works in the mailroom of a large company and is responsible for all the letters sent by the company. The weights of letters sent by the company, $W$ grams, have a normal distribution with mean 165 g and standard deviation 35 g .
1. Estimate the proportion of letters sent by the company that weigh less than 120 g .
Kris splits the letters to be sent into 3 categories: heavy, medium and light, with $\frac { 1 } { 3 }$ of the letters in each category.
Find the weight limits that determine medium letters. A heavy letter is chosen at random.
Find the probability that this letter weighs less than 200 g . Kris chooses a random sample of 3 letters from those in the mailroom one day.
Find the probability that there is one letter in each of the 3 categories.

Edexcel S1 2021 June Q5

15 marks Standard +0.3

The discrete random variable $X$ has the following probability distribution

$x$	- 2	- 1	0	1	4
$\mathrm { P } ( X = x )$	$a$	$b$	$c$	$b$	$a$

Given that $\mathrm { E } ( X ) = 0.5$

find the value of $a$. Given also that $\operatorname { Var } ( X ) = 5.01$
find the value of $b$ and the value of $c$. The random variable $Y = 5 - 8 X$
Find
1. $\mathrm { E } ( Y )$
2. $\operatorname { Var } ( Y )$
Find $\mathrm { P } \left( 4 X ^ { 2 } > Y \right)$

Edexcel S1 2021 June Q6

16 marks Standard +0.3

Two economics students, Andi and Behrouz, are studying some data relating to unemployment, $x \%$, and increase in wages, $y \%$, for a European country. The least squares regression line of $y$ on $x$ has equation

$$y = 3.684 - 0.3242 x$$ and $$\sum y = 23.7 \quad \sum y ^ { 2 } = 42.63 \quad \sum x ^ { 2 } = 756.81 \quad n = 16$$

Show that $\mathrm { S } _ { y y } = 7.524375$
Find $\mathrm { S } _ { x x }$
Find the product moment correlation coefficient between $x$ and $y$. Behrouz claims that, assuming the model is valid, the data show that when unemployment is 2\% wages increase at over 3\%
Explain how Behrouz could have come to this conclusion. Andi uses the formula $$\text { range } = \text { mean } \pm 3 \times \text { standard deviation }$$ to estimate the range of values for $x$.
Find estimates of the minimum value and the maximum value of $x$ in these data using Andi's formula.
Comment, giving a reason, on the reliability of Behrouz's claim. Andi suggests using the regression line with equation $y = 3.684 - 0.3242 x$ to estimate unemployment when wages are increasing at $2 \%$
Comment, giving a reason, on Andi's suggestion.

\includegraphics[max width=\textwidth, alt={}]{a439724e-b570-434d-bf75-de2b50915042-20_2647_1835_118_116}

Edexcel S1 2022 June Q1

11 marks Easy -1.2

The company Seafield requires contractors to record the number of hours they work each week. A random sample of 38 weeks is taken and the number of hours worked per week by contractor Kiana is summarised in the stem and leaf diagram below.

Stem	Leaf
1	4	4	4	5	5	5	6	6	9	9	9	(11)
2	1	2	2	3	3	4	4	4	$w$	9		(10)
3	2	3	4	4	5	6	7	7	7	9		(10)
4	1	1	2	3								(4)
5	1	9										(2)
6	4											(1)

Key : 3|2 means 32 The quartiles for this distribution are summarised in the table below.

$Q _ { 1 }$	$Q _ { 2 }$	$Q _ { 3 }$
$x$	26	$y$

Find the values of $w , x$ and $y$ Kiana is looking for outliers in the data. She decides to classify as outliers any observations greater than $$Q _ { 3 } + 1.0 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
Showing your working clearly, identify any outliers that Kiana finds.
Draw a box plot for these data in the space provided on the grid opposite.
Use the formula $$\text { skewness } = \frac { \left( Q _ { 3 } - Q _ { 2 } \right) - \left( Q _ { 2 } - Q _ { 1 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }$$ to find the skewness of these data. Give your answer to 2 significant figures. Kiana's new employer, Landacre, wishes to know the average number of hours per week she worked during her employment at Seafield to help calculate the cost of employing her.
Explain why Landacre might prefer to know Kiana's mean, rather than median, number of hours worked per week. Turn over for a spare grid if you need to redraw your box plot.

Stem	Leaf
2	0	2	3	4											\(( 4 )\)
2	5	6	8	8	8	9	9
3	0	0	0	0	0	1	1	1	2	2	2	2	3	4	\(( 14 )\)
3	5	5	5	7	9										\(( 5 )\)

\(x\)	- 2	- 1	0	2	3
\(\mathrm { P } ( X = x )\)	\(p\)	\(p\)	\(q\)	\(\frac { 1 } { 4 }\)	\(p\)

\(x\)	- 2	- 1	0	1	4
\(\mathrm { P } ( X = x )\)	\(a\)	\(b\)	\(c\)	\(b\)	\(a\)

\(Q _ { 1 }\)	\(Q _ { 2 }\)	\(Q _ { 3 }\)
\(x\)	26	\(y\)

Weight in grams ( \(\boldsymbol { x }\) )	Frequency (f)	Class midpoint (y)
\(0.9 < x \leqslant 1.1\)	9	1.0
\(1.1 < x \leqslant 1.3\)	12	1.2
\(1.3 < x \leqslant 1.5\)	11	1.4
\(1.5 < x \leqslant 1.7\)	8	1.6
\(1.7 < x \leqslant 1.9\)	3	1.8
\(1.9 < x \leqslant 2.1\)	3	2.0
\(2.1 < x \leqslant 2.7\)	2	2.4

Length, \(L\) cm	Frequency, f	Class mid point, \(\boldsymbol { x } \mathbf { c m }\)
\(5 \leqslant L < 8\)	5	6.5
\(8 \leqslant L < 10\)	13	9
\(10 \leqslant L < 12\)	16	11
\(12 \leqslant L < 15\)	25	13.5
\(15 \leqslant L < 20\)	30	17.5
\(20 \leqslant L < 28\)	11	24

1	1	5	8	9
2	0	2	5	8	9
3	3	5	5	7	8	9	\(\ldots\)

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