Questions - Edexcel S1 - A-Level Maths

Body weight (kg)	75-79	80-84	85-89	90-94	95-99	100-104	105-109
Number of Players (2010)	1	2	2	4	3	2	1

Find the estimated values in kg of the summary statistics $a$, $b$ and $c$ in the table below.
Estimate in 1990 Estimate in 2010
Mean 83.0 $a$
Median 82.0 $b$
Variance 44.0 $c$
Give your answers to 3 significant figures. The rugby coach claims that players’ body weight increased between 1990 and 2010.
Using the table in part (a), comment on the rugby coach's claim. \includegraphics[max width=\textwidth, alt={}, center]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-05_104_97_2613_1784}

Edexcel S1 2014 January Q3

3. Jean works for an insurance company. She randomly selects 8 people and records the price of their car insurance, $\pounds p$, and the time, $t$ years, since they passed their driving test. The data is shown in the table below.

$t$	10	13	17	18	22	24	25	27
$p$	720	650	430	490	500	390	280	300

$$\text { (You may use } \bar { t } = 19.5 , \bar { p } = 470 , S _ { t p } = - 6080 , S _ { t t } = 254 , S _ { p p } = 169200 \text { ) }$$

On the graph below draw a scatter diagram for these data.
Comment on the relationship between $p$ and $t$.
Find the equation of the regression line of $p$ on $t$.
Use your regression equation to estimate the price of car insurance for someone who passed their driving test 20 years ago. Jack passed his test 39 years ago and decides to use Jean's data to predict the price of his car insurance.
Comment on Jack's decision. Give a reason for your answer.
\includegraphics[max width=\textwidth, alt={}, center]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-06_951_1365_1603_294}

Edexcel S1 2014 January Q4

4. A discrete random variable $X$ has the probability distribution given in the table below, where $a$ and $b$ are constants.

$x$	- 1	0	1	2	3
$\mathrm { P } ( X = x )$	$a$	$\frac { 1 } { 10 }$	$\frac { 1 } { 5 }$	$\frac { 3 } { 10 }$	$b$

Given $\mathrm { E } ( X ) = \frac { 9 } { 5 }$

1. find two simultaneous equations for $a$ and $b$,
2. show that $a = \frac { 1 } { 20 }$ and find the value of $b$.
Specify the cumulative distribution function $\mathrm { F } ( x )$ for $x = - 1,0,1$, 2 and 3
Find $\mathrm { P } ( X < 2.5 )$.
Find $\operatorname { Var } ( 3 - 2 X )$.
\includegraphics[max width=\textwidth, alt={}, center]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-13_90_68_2613_1877}

Edexcel S1 2014 January Q5

5. A group of 100 students are asked if they like folk music, rock music or soul music. \begin{displayquote} All students who like folk music also like rock music No students like both rock music and soul music 75 students do not like soul music 12 students who like rock music do not like folk music 30 students like folk music

Draw a Venn diagram to illustrate this information.
State two of these types of music that are mutually exclusive. \end{displayquote} Find the probability that a randomly chosen student
does not like folk music, rock music or soul music,
likes rock music,
likes folk music or soul music. Given that a randomly chosen student likes rock music,
find the probability that he or she also likes folk music.

Edexcel S1 2014 January Q6

6. A manufacturer has a machine that fills bags with flour such that the weight of flour in a bag is normally distributed. A label states that each bag should contain 1 kg of flour.

The machine is set so that the weight of flour in a bag has mean 1.04 kg and standard deviation 0.17 kg . Find the proportion of bags that weigh less than the stated weight of 1 kg . The manufacturer wants to reduce the number of bags which contain less than the stated weight of 1 kg . At first she decides to adjust the mean but not the standard deviation so that only $5 \%$ of the bags filled are below the stated weight of 1 kg .
Find the adjusted mean. The manufacturer finds that a lot of the bags are overflowing with flour when the mean is adjusted, so decides to adjust the standard deviation instead to make the machine more accurate. The machine is set back to a mean of 1.04 kg . The manufacturer wants $1 \%$ of bags to be under 1 kg .
Find the adjusted standard deviation. Give your answer to 3 significant figures.

Edexcel S1 2014 January Q7

7. In a large college, $\frac { 3 } { 5 }$ of the students are male, $\frac { 3 } { 10 }$ of the students are left handed and $\frac { 1 } { 5 }$ of the male students are left handed. A student is chosen at random.

Given that the student is left handed, find the probability that the student is male.
Given that the student is female, find the probability that she is left handed.
Find the probability that the randomly chosen student is male and right handed. Two students are chosen at random.
Find the probability that one student is left handed and one is right handed.

Edexcel S1 2014 January Q8

8. A manager records the number of hours of overtime claimed by 40 staff in a month. The histogram in Figure 1 represents the results. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-26_1107_1513_406_210} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

Calculate the number of staff who have claimed less than 10 hours of overtime in the month.
Estimate the median number of hours of overtime claimed by these 40 staff in the month.
Estimate the mean number of hours of overtime claimed by these 40 staff in the month. The manager wants to compare these data with overtime data he collected earlier to find out if the overtime claimed by staff has decreased.
State, giving a reason, whether the manager should use the median or the mean to compare the overtime claimed by staff.
(2)

Edexcel S1 2015 January Q1

The discrete random variable $X$ has probability function $\mathrm { p } ( x )$ and cumulative distribution function $\mathrm { F } ( x )$ given in the table below.

$x$	1	2	3	4	5
$\mathrm { p } ( x )$	0.10	$a$	0.28	$c$	0.24
$\mathrm {~F} ( x )$	0.10	0.26	$b$	0.76	$d$

Write down the value of $d$
Find the values of $a$, $b$ and $c$
Write down the value of $\mathrm { P } ( X > 4 )$ Two independent observations, $X _ { 1 }$ and $X _ { 2 }$, are taken from the distribution of $X$.
Find the probability that $X _ { 1 }$ and $X _ { 2 }$ are both odd. Given that $X _ { 1 }$ and $X _ { 2 }$ are both odd,
find the probability that the sum of $X _ { 1 }$ and $X _ { 2 }$ is 6 Give your answer to 3 significant figures.

Edexcel S1 2015 January Q2

A sports teacher recorded the number of press-ups done by his students in two minutes. He recorded this information for a Year 7 class and for a Year 11 class.

The back-to-back stem and leaf diagram shows this information.

Totals	Year 7 class		Year 11 class	Totals
(6)	876554	1
(10)	9776544422	2	0569	(4)
(7)	8754330	3	34588	(5)
(5)	99722	4	05679	(5)
(3)	840	5	03556677799	(11)
		6	0333348	(7)

Key: $2 | 4 | 0$ means 42 press-ups for a Year 7 student and 40 press-ups for a Year 11 student

Find the median number of press-ups for each class. For the Year 11 class, the lower quartile is 38 and the upper quartile is 59
Find the lower quartile and the upper quartile for the Year 7 class.
Use the medians and quartiles to describe the skewness of each of the two distributions.
Give two reasons why the normal distribution should not be used to model the number of press-ups done by the Year 11 class.

Edexcel S1 2015 January Q3

The table shows the price of a bottle of milk, $m$ pence, and the price of a loaf of bread, $b$ pence, for 8 different years.

$m$	29	29	35	39	41	43	44	46
$b$	75	83	91	121	120	126	119	126

(You may use $\mathrm { S } _ { b b } = 3083.875$ and $\mathrm { S } _ { m m } = 305.5$ )

Find the exact value of $\sum b m$
Find $\mathrm { S } _ { b m }$
Calculate the product moment correlation coefficient between $b$ and $m$
Interpret the value of the correlation coefficient. A ninth year is added to the data set. In this year the price of the bottle of milk is 46 pence and the price of a loaf of bread is 175 pence.
Without further calculation, state whether the value of the product moment correlation coefficient will increase, decrease or stay the same when all nine years are used. Give a reason for your answer.

Edexcel S1 2015 January Q4

4. Events $A$ and $B$ are shown in the Venn diagram below
where $x , y , 0.10$ and 0.32 are probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{c58f3e88-2dbc-40d6-a966-a5765a7c67ba-08_467_798_408_575}

Find an expression in terms of $x$ for
1. $\mathrm { P } ( A )$
2. $\mathrm { P } ( B \mid A )$
Find an expression in terms of $x$ and $y$ for $\mathrm { P } ( A \cup B )$ Given that $\mathrm { P } ( A ) = 2 \mathrm { P } ( B )$
find the value of $x$ and the value of $y$

Edexcel S1 2015 January Q5

The resting heart rate, $h$ beats per minute (bpm), and average length of daily exercise, $t$ minutes, of a random sample of 8 teachers are shown in the table below.

$t$	20	35	40	25	45	70	75	90
$h$	88	85	77	75	71	66	60	54

State, with a reason, which variable is the response variable. The equation of the least squares regression line of $h$ on $t$ is $$h = 93.5 - 0.43 t$$
Give an interpretation of the gradient of this regression line.
Find the value of $\bar { t }$ and the value of $\bar { h }$
Show that the point $( \bar { t } , \bar { h } )$ lies on the regression line.
Estimate the resting heart rate of a teacher with an average length of daily exercise of 1 hour.
Comment, giving a reason, on the reliability of the estimate in part (e). The resting heart rate of teachers is assumed to be normally distributed with mean 73 bpm and standard deviation 8 bpm . The middle $95 \%$ of resting heart rates of teachers lies between $a$ and $b$
Find the value of $a$ and the value of $b$.

Edexcel S1 2015 January Q6

The random variable $X$ has probability function

$$\mathrm { P } ( X = x ) = \frac { x ^ { 2 } } { k } \quad x = 1,2,3,4$$

Show that $k = 30$
Find $\mathrm { P } ( X \neq 4 )$
Find the exact value of $\mathrm { E } ( X )$
Find the exact value of $\operatorname { Var } ( X )$ Given that $Y = 3 X - 1$
find $\mathrm { E } \left( Y ^ { 2 } \right)$

Edexcel S1 2015 January Q7

The birth weights, $W$ grams, of a particular breed of kitten are assumed to be normally distributed with mean 99 g and standard deviation 3.6 g
1. Find $\mathrm { P } ( W > 92 )$
2. Find, to one decimal place, the value of $k$ such that $\mathrm { P } ( W < k ) = 3 \mathrm { P } ( W > k )$
3. Write down the name given to the value of $k$.
For a different breed of kitten, the birth weights are assumed to be normally distributed with mean 120 g Given that the 20th percentile for this breed of kitten is 116 g
find the standard deviation of the birth weight of this breed of kitten.

Edexcel S1 2016 January Q1

The discrete random variable $X$ has the probability distribution given in the table below.

$x$	- 2	1	3	4	6
$\mathrm { P } ( X = x )$	$\frac { 1 } { 4 }$	$\frac { 1 } { 6 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 12 }$	$\frac { 1 } { 6 }$

Write down the value of $\mathrm { F } ( 5 )$
Find $\mathrm { E } ( X )$
Find $\operatorname { Var } ( X )$ The random variable $Y = 7 - 2 X$
Find
1. $\mathrm { E } ( Y )$
2. $\operatorname { Var } ( Y )$
3. $\mathrm { P } ( Y > X )$ \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-03_2261_47_313_37}

Edexcel S1 2016 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{70137e9a-0a6b-48b5-8dd4-c436cb063351-04_284_1244_260_388} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows part of a box and whisker plot for the marks in an examination with a large number of candidates. Part of the lower whisker has been torn off.

Given that $75 \%$ of the candidates passed the examination, state the lowest mark for the award of a pass.
Given that the top $25 \%$ of the candidates achieved a merit grade, state the lowest mark for the award of a merit grade. An outlier is defined as any value greater than $c$ or any value less than $d$ where $$\begin{aligned} & c = Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right)
& d = Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \end{aligned}$$
Find the value of $c$ and the value of $d$.
Write down the 3 highest marks scored in the examination. The 3 lowest marks in the examination were 5, 10 and 15
On the diagram on page 7, complete the box and whisker plot. Three candidates are selected at random from those who took this examination.
Find the probability that all 3 of these candidates passed the examination but only 2 achieved a merit grade.
\includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-05_285_1628_2343_166} Turn over for a spare diagram if you need to redraw your plot.

Edexcel S1 2016 January Q3

3. A publisher collects information about the amount spent on advertising, $\pounds x$, and the sales, $y$ books, for some of her publications. She collects information for a random sample of 8 textbooks and codes the data using $v = \frac { x + 50 } { 200 }$ and $s = \frac { y } { 1000 }$ to give

$v$	0.60	8.10	4.30	0.40	1.60	6.40	2.50	5.10
$s$	1.84	6.73	5.95	1.30	2.45	7.46	4.82	6.25

[You may use: $\sum v = 29 \sum s = 36.8 \sum s ^ { 2 } = 209.72 \sum v s = 177.311 \quad \mathrm {~S} _ { v v } = 55.275$ ]

Find $\mathrm { S } _ { v s }$ and $\mathrm { S } _ { s s }$
Calculate the product moment correlation coefficient for these data. The publisher believes that a linear regression model may be appropriate to describe these data.
State, giving a reason, whether or not your answer to part (b) supports the publisher's belief.
Find the equation of the regression line of $s$ on $v$, giving your answer in the form $s = a + b v$
Hence find the equation of the regression line of $y$ on $x$ for the sample of textbooks, giving your answer in the form $y = c + d x$ The publisher calculated the regression line for a sample of novels and obtained the equation $$y = 3100 + 1.2 x$$ She wants to increase the sales of books by spending more money on advertising.
State, giving your reasons, whether the publisher should spend more money on advertising textbooks or novels.

Edexcel S1 2016 January Q4

4. A training agency awards a certificate to each student who passes a test while completing a course.
Students failing the test will attempt the test again up to 3 more times, and, if they pass the test, will be awarded a certificate.
The probability of passing the test at the first attempt is 0.7 , but the probability of passing reduces by 0.2 at each attempt.

Complete the tree diagram below to show this information.
\includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-08_545_1244_639_340} A student who completed the course is selected at random.
Find the probability that the student was awarded a certificate.
Given that the student was awarded a certificate, find the probability that the student passed on the first or second attempt. The training agency decides to alter the test taken by the students while completing the course, but will not allow more than 2 attempts. The agency requires the probability of passing the test at the first attempt to be $p$, and the probability of passing the test at the second attempt to be ( $p - 0.2$ ). The percentage of students who complete the course and are awarded a certificate is to be $95 \%$
Show that $p$ satisfies the equation $$p ^ { 2 } - 2.2 p + 1.15 = 0$$
Hence find the value of $p$, giving your answer to 3 decimal places.
\includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-09_2261_47_313_37}

Edexcel S1 2016 January Q5

5. Rosie keeps bees. The amount of honey, in kg, produced by a hive of Rosie's bees in a season, is modelled by a normal distribution with a mean of 22 kg and a standard deviation of 10 kg .

Find the probability that a hive of Rosie's bees produces less than 18 kg of honey in a season. The local bee keepers’ club awards a certificate to every hive that produces more than 39 kg of honey in a season, and a medal to every hive that produces more than 50 kg in a season. Given that one of Rosie's bee hives is awarded a certificate
find the probability that this hive is also awarded a medal.
(5) Sam also keeps bees. The amount of honey, in kg, produced by a hive of Sam's bees in a season, is modelled by a normal distribution with mean $\mu \mathrm { kg }$ and standard deviation $\sigma \mathrm { kg }$. The probability that a hive of Sam’s bees produces less than 28 kg of honey in a season is 0.8413 Only 20\% of Sam's bee hives produce less than 18 kg of honey in a season.
Find the value of $\mu$ and the value of $\sigma$. Give your answers to 2 decimal places.
(6)

\includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-11_2261_47_313_37}

Edexcel S1 2016 January Q6

6. Yujie is investigating the weights of 10 young rabbits. She records the weight, $x$ grams, of each rabbit and the results are summarised below. $$\sum x = 8360 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 63840$$

Calculate the mean and the standard deviation of the weights of these rabbits. Given that the median weight of these rabbits is 815 grams,
describe, giving a reason, the skewness of these data. Two more rabbits weighing 776 grams and 896 grams are added to make a group of 12 rabbits.
State, giving a reason, how the inclusion of these two rabbits would affect the mean.
By considering the change in $\sum ( x - \bar { x } ) ^ { 2 }$, state what effect the inclusion of these two rabbits would have on the standard deviation.
END

Edexcel S1 2017 January Q1

Ralph records the weights, in grams, of 100 tomatoes. This information is displayed in the histogram below.
\includegraphics[max width=\textwidth, alt={}, center]{1130517e-33d0-41b1-9303-2d981379954d-02_981_1268_338_274}

Given that 5 of the tomatoes have a weight between 2 and 3 grams,

find the number of tomatoes with a weight between 0 and 2 grams. One of the tomatoes is selected at random.
Find the probability that it weighs more than 3 grams.
Estimate the proportion of the tomatoes with a weight greater than 6.25 grams.
Using your answer to part (c), explain whether or not the median is greater than 6.25 grams. Given that the mean weight of these tomatoes is 6.25 grams and using your answer to part (d),
describe the skewness of the distribution of the weights of these tomatoes. Give a reason for your answer. Two of these 100 tomatoes are selected at random.
Estimate the probability that both tomatoes weigh within 0.75 grams of the mean.

Edexcel S1 2017 January Q2

An integer is selected at random from the integers 1 to 50 inclusive.
$A$ is the event that the integer selected is prime.
$B$ is the event that the integer selected ends in a 3
$C$ is the event that the integer selected is greater than 20
The Venn diagram shows the number of integers in each region for the events $A , B$ and $C$
\includegraphics[max width=\textwidth, alt={}, center]{1130517e-33d0-41b1-9303-2d981379954d-04_607_1125_593_413}
1. Describe in words the event $( A \cap B )$
2. Write down the probability that the integer selected is prime.
3. Find $\mathrm { P } \left( [ A \cup B \cup C ] ^ { \prime } \right)$
Given that the integer selected is greater than 20
find the probability that it is prime. Using your answers to (b) and (d),
state, with a reason, whether or not the events $A$ and $C$ are statistically independent. Given that the integer selected is greater than 20 and prime,
find the probability that it ends in a 3

Edexcel S1 2017 January Q3

A scientist measured the salinity of water, $x \mathrm {~g} / \mathrm { kg }$, and recorded the temperature at which the water froze, $y ^ { \circ } \mathrm { C }$, for 12 different water samples. The summary statistics are listed below.

$$\begin{gathered} \sum x = 504 \quad \sum y = - 27 \quad \sum x ^ { 2 } = 22842 \quad \sum y ^ { 2 } = 62.98
\sum x y = - 1190.7 \quad \mathrm {~S} _ { x x } = 1674 \quad \mathrm {~S} _ { y y } = 2.23 \end{gathered}$$

Find the mean and variance of the recorded temperatures.
(3) Priya believes that the higher the salinity of water, the higher the temperature at which the water freezes.
1. Calculate the product moment correlation coefficient between $x$ and $y$
2. State, with a reason, whether or not this value supports Priya's belief.
Find the least squares regression line of $y$ on $x$ in the form $y = a + b x$ Give the value of $a$ and the value of $b$ to 3 significant figures.
Estimate the temperature at which water freezes when the salinity is $32 \mathrm {~g} / \mathrm { kg }$ The coding $w = 1.8 y + 32$ is used to convert the recorded temperatures from ${ } ^ { \circ } \mathrm { C }$ to ${ } ^ { \circ } \mathrm { F }$
Find an equation of the least squares regression line of $w$ on $x$ in the form $w = c + d x$
Find
1. the variance of the recorded temperatures when converted to ${ } ^ { \circ } \mathrm { F }$
2. the product moment correlation coefficient between $w$ and $x$
  \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

Edexcel S1 2017 January Q4

In a game, the number of points scored by a player in the first round is given by the random variable $X$ with probability distribution

$x$	5	6	7	8
$\mathrm { P } ( X = x )$	0.13	0.21	0.29	0.37

Find

$\mathrm { E } ( X )$
$\operatorname { Var } ( X )$
$\operatorname { Var } ( 3 - 2 X )$ The number of points scored by a player in the second round is given by the random variable $Y$ and is independent of the number of points scored in the first round. The random variable $Y$ has probability function $$\mathrm { P } ( Y = y ) = \frac { 1 } { 4 } \quad \text { for } y = 5,6,7,8$$
Write down the value of $\mathrm { E } ( Y )$
Find $\mathrm { P } ( X = Y )$
Find the probability that the number of points scored by a player in the first round is greater than the number of points scored by the player in the second round.

Edexcel S1 2017 January Q5

In a survey, people were asked if they use a computer every day.

Of those people under 50 years old, $80 \%$ said they use computer every day. Of those people aged 50 or more, $55 \%$ said they use computer every day. The proportion of people in the survey under 50 years old is $p$

Draw a tree diagram to represent this information. In the survey, 70\% of all people said they use computer every day.
Find the value of $p$ One person is selected at random. Given that this person uses a computer every day,
find the probability that this person is under 50 years old.
\href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

\(x\)	- 1	0	1	2	3
\(\mathrm { P } ( X = x )\)	\(a\)	\(\frac { 1 } { 10 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 3 } { 10 }\)	\(b\)

\(x\)	1	2	3	4	5
\(\mathrm { p } ( x )\)	0.10	\(a\)	0.28	\(c\)	0.24
\(\mathrm {~F} ( x )\)	0.10	0.26	\(b\)	0.76	\(d\)

\(x\)	- 2	1	3	4	6
\(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 12 }\)	\(\frac { 1 } { 6 }\)

	Estimate in 1990	Estimate in 2010
Mean	83.0	\(a\)
Median	82.0	\(b\)
Variance	44.0	\(c\)

\(v\)	0.60	8.10	4.30	0.40	1.60	6.40	2.50	5.10
\(s\)	1.84	6.73	5.95	1.30	2.45	7.46	4.82	6.25

\(t\)	20	35	40	25	45	70	75	90
\(h\)	88	85	77	75	71	66	60	54


END

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