Questions - Edexcel S1 - A-Level Maths

$x$	1	2	3
$\mathrm {~F} ( x )$	0.4	0.65	1

\includegraphics[max width=\textwidth, alt={}, center]{4cf4f2d7-d912-4b65-a666-caa37009661a-04_24_37_182_2010}

Edexcel S1 2013 June Q3

3. An agriculturalist is studying the yields, $y \mathrm {~kg}$, from tomato plants. The data from a random sample of 70 tomato plants are summarised below.

Yield ( $y \mathrm {~kg}$ )	Frequency (f)	Yield midpoint ( $x \mathrm {~kg}$ )
$0 \leqslant y < 5$	16	2.5
$5 \leqslant y < 10$	24	7.5
$10 \leqslant y < 15$	14	12.5
$15 \leqslant y < 25$	12	20
$25 \leqslant y < 35$	4	30

$$\text { (You may use } \sum \mathrm { f } x = 755 \text { and } \sum \mathrm { f } x ^ { 2 } = 12037.5 \text { ) }$$ A histogram has been drawn to represent these data. The bar representing the yield $5 \leqslant y < 10$ has a width of 1.5 cm and a height of 8 cm .

Calculate the width and the height of the bar representing the yield $15 \leqslant y < 25$
Use linear interpolation to estimate the median yield of the tomato plants.
Estimate the mean and the standard deviation of the yields of the tomato plants.
Describe, giving a reason, the skewness of the data.
Estimate the number of tomato plants in the sample that have a yield of more than 1 standard deviation above the mean.

Edexcel S1 2013 June Q4

The time, in minutes, taken to fly from London to Malaga has a normal distribution with mean 150 minutes and standard deviation 10 minutes.
1. Find the probability that the next flight from London to Malaga takes less than 145 minutes.
The time taken to fly from London to Berlin has a normal distribution with mean 100 minutes and standard deviation $d$ minutes. Given that $15 \%$ of the flights from London to Berlin take longer than 115 minutes,
find the value of the standard deviation $d$. The time, $X$ minutes, taken to fly from London to another city has a normal distribution with mean $\mu$ minutes. Given that $\mathrm { P } ( X < \mu - 15 ) = 0.35$
find $\mathrm { P } ( X > \mu + 15 \mid X > \mu - 15 )$.

Edexcel S1 2013 June Q5

5. A researcher believes that parents with a short family name tended to give their children a long first name. A random sample of 10 children was selected and the number of letters in their family name, $x$, and the number of letters in their first name, $y$, were recorded. The data are summarised as: $$\sum x = 60 , \quad \sum y = 61 , \quad \sum y ^ { 2 } = 393 , \quad \sum x y = 382 , \quad \mathrm {~S} _ { x x } = 28$$

Find $\mathrm { S } _ { y y }$ and $\mathrm { S } _ { x y }$
Calculate the product moment correlation coefficient, $r$, between $x$ and $y$.
State, giving a reason, whether or not these data support the researcher's belief. The researcher decides to add a child with family name "Turner" to the sample.
Using the definition $\mathrm { S } _ { x x } = \sum ( x - \bar { x } ) ^ { 2 }$, state the new value of $\mathrm { S } _ { x x }$ giving a reason for your answer. Given that the addition of the child with family name "Turner" to the sample leads to an increase in $\mathrm { S } _ { y y }$
use the definition $\mathrm { S } _ { x y } = \sum ( x - \bar { x } ) ( y - \bar { y } )$ to determine whether or not the value of $r$ will increase, decrease or stay the same. Give a reason for your answer.

Edexcel S1 2013 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4cf4f2d7-d912-4b65-a666-caa37009661a-11_606_1131_210_411} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The Venn diagram in Figure 1 shows three events $A , B$ and $C$ and the probabilities associated with each region of $B$. The constants $p , q$ and $r$ each represent probabilities associated with the three separate regions outside $B$. The events $A$ and $B$ are independent.

Find the value of $p$. Given that $\mathrm { P } ( B \mid C ) = \frac { 5 } { 11 }$
find the value of $q$ and the value of $r$.
Find $\mathrm { P } ( A \cup C \mid B )$.

Edexcel S1 2013 June Q7

7. The score $S$ when a spinner is spun has the following probability distribution.

$s$	0	1	2	4	5
$\mathrm { P } ( S = s )$	0.2	0.2	0.1	0.3	0.2

Find $\mathrm { E } ( S )$.
Show that $\mathrm { E } \left( S ^ { 2 } \right) = 10.4$
Hence find $\operatorname { Var } ( S )$.
Find
1. $\mathrm { E } ( 5 S - 3 )$,
2. $\operatorname { Var } ( 5 S - 3 )$.
Find $\mathrm { P } ( 5 S - 3 > S + 3 )$ The spinner is spun twice.
The score from the first spin is $S _ { 1 }$ and the score from the second spin is $S _ { 2 }$
The random variables $S _ { 1 }$ and $S _ { 2 }$ are independent and the random variable $X = S _ { 1 } \times S _ { 2 }$
Show that $\mathrm { P } \left( \left\{ S _ { 1 } = 1 \right\} \cap X < 5 \right) = 0.16$
Find $\mathrm { P } ( X < 5 )$.

Edexcel S1 2013 June Q1

A meteorologist believes that there is a relationship between the height above sea level, $h \mathrm {~m}$, and the air temperature, $t ^ { \circ } \mathrm { C }$. Data is collected at the same time from 9 different places on the same mountain. The data is summarised in the table below.

$h$	1400	1100	260	840	900	550	1230	100	770
$t$	3	10	20	9	10	13	5	24	16

[You may assume that $\sum h = 7150 , \sum t = 110 , \sum h ^ { 2 } = 7171500 , \sum t ^ { 2 } = 1716$, $\sum t h = 64980$ and $\mathrm { S } _ { t t } = 371.56$ ]

Calculate $\mathrm { S } _ { t h }$ and $\mathrm { S } _ { h h }$. Give your answers to 3 significant figures.
Calculate the product moment correlation coefficient for this data.
State whether or not your value supports the use of a regression equation to predict the air temperature at different heights on this mountain. Give a reason for your answer.
Find the equation of the regression line of $t$ on $h$ giving your answer in the form $t = a + b h$.
Interpret the value of $b$.
Estimate the difference in air temperature between a height of 500 m and a height of 1000 m .

Edexcel S1 2013 June Q2

The marks of a group of female students in a statistics test are summarised in Figure 1

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6faf2dd2-a114-40b7-88ae-4a75dbfb4706-04_629_1102_342_429} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

Write down the mark which is exceeded by $75 \%$ of the female students. The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.
Mark (2|6 means 26) Totals
1 4 (1)
2 6 (1)
3 447 (3)
4 066778 (6)
5 001113677 (9)
6 223338 (6)
7 008 (3)
8 5 (1)
9 0 (1)
Find the median and interquartile range of the marks of the male students. An outlier is a mark that is
either more than $1.5 \times$ interquartile range above the upper quartile or more than $1.5 \times$ interquartile range below the lower quartile.
In the space provided on Figure 1 draw a box plot to represent the marks of the male students, indicating clearly any outliers.
Compare and contrast the marks of the male and the female students.

Edexcel S1 2013 June Q3

3. In a company the 200 employees are classified as full-time workers, part-time workers or contractors.
The table below shows the number of employees in each category and whether they walk to work or use some form of transport.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Walk	Transport
Full-time worker	2	8
Part-time worker	35	75
Contractor	30	50

The events $F , H$ and $C$ are that an employee is a full-time worker, part-time worker or contractor respectively. Let $W$ be the event that an employee walks to work. An employee is selected at random.
Find

$\mathrm { P } ( H )$
$\mathrm { P } \left( [ F \cap W ] ^ { \prime } \right)$
$\mathrm { P } ( W \mid C )$ Let $B$ be the event that an employee uses the bus.
Given that $10 \%$ of full-time workers use the bus, $30 \%$ of part-time workers use the bus and $20 \%$ of contractors use the bus,
draw a Venn diagram to represent the events $F , H , C$ and $B$,
find the probability that a randomly selected employee uses the bus to travel to work.

Edexcel S1 2013 June Q4

4. The following table summarises the times, $t$ minutes to the nearest minute, recorded for a group of students to complete an exam.

Time (minutes) $t$	$11 - 20$	$21 - 25$	$26 - 30$	$31 - 35$	$36 - 45$	$46 - 60$
Number of students f	62	88	16	13	11	10

$$\text { [You may use } \sum \mathrm { f } t ^ { 2 } = 134281.25 \text { ] }$$

Estimate the mean and standard deviation of these data.
Use linear interpolation to estimate the value of the median.
Show that the estimated value of the lower quartile is 18.6 to 3 significant figures.
Estimate the interquartile range of this distribution.
Give a reason why the mean and standard deviation are not the most appropriate summary statistics to use with these data. The person timing the exam made an error and each student actually took 5 minutes less than the times recorded above. The table below summarises the actual times.
Time (minutes) $t$ $6 - 15$ $16 - 20$ $21 - 25$ $26 - 30$ $31 - 40$ $41 - 55$
Number of students f 62 88 16 13 11 10
Without further calculations, explain the effect this would have on each of the estimates found in parts (a), (b), (c) and (d).

Edexcel S1 2013 June Q5

A biased die with six faces is rolled. The discrete random variable $X$ represents the score on the uppermost face. The probability distribution of $X$ is shown in the table below.

$x$	1	2	3	4	5	6
$\mathrm { P } ( X = x )$	$a$	$a$	$a$	$b$	$b$	0.3

Given that $\mathrm { E } ( X ) = 4.2$ find the value of $a$ and the value of $b$.
Show that $\mathrm { E } \left( X ^ { 2 } \right) = 20.4$
Find $\operatorname { Var } ( 5 - 3 X )$ A biased die with five faces is rolled. The discrete random variable $Y$ represents the score which is uppermost. The cumulative distribution function of $Y$ is shown in the table below.
$y$ 1 2 3 4 5
$\mathrm {~F} ( y )$ $\frac { 1 } { 10 }$ $\frac { 2 } { 10 }$ $3 k$ $4 k$ $5 k$
Find the value of $k$.
Find the probability distribution of $Y$. Each die is rolled once. The scores on the two dice are independent.
Find the probability that the sum of the two scores equals 2

Edexcel S1 2013 June Q6

The weight, in grams, of beans in a tin is normally distributed with mean $\mu$ and standard deviation 7.8

Given that $10 \%$ of tins contain less than 200 g , find

the value of $\mu$
the percentage of tins that contain more than 225 g of beans. The machine settings are adjusted so that the weight, in grams, of beans in a tin is normally distributed with mean 205 and standard deviation $\sigma$.
Given that $98 \%$ of tins contain between 200 g and 210 g find the value of $\sigma$.

Edexcel S1 2014 June Q1

The discrete random variable $X$ has probability distribution

$x$	- 4	- 2	1	3	5
$\mathrm { P } ( X = x )$	0.4	$p$	0.05	0.15	$p$

Show that $p = 0.2$ Find
$\mathrm { E } ( X )$
$\mathrm { F } ( 0 )$
$\mathrm { P } ( 3 X + 2 > 5 )$ Given that $\operatorname { Var } ( X ) = 13.35$
find the possible values of $a$ such that $\operatorname { Var } ( a X + 3 ) = 53.4$

Edexcel S1 2014 June Q2

The discrete random variable $X$ has probability distribution

$$\mathrm { P } ( X = x ) = \frac { 1 } { 10 } \quad x = 1,2,3 , \ldots 10$$

Write down the name given to this distribution.
Write down the value of
1. $\mathrm { P } ( X = 10 )$
2. $\mathrm { P } ( X < 10 )$ The continuous random variable $Y$ has the normal distribution $\mathrm { N } \left( 10,2 ^ { 2 } \right)$
Write down the value of
1. $\mathrm { P } ( Y = 10 )$
2. $\mathrm { P } ( Y < 10 )$

Edexcel S1 2014 June Q3

3. A large company is analysing how much money it spends on paper in its offices every year. The number of employees, $x$, and the amount of money spent on paper, $p$ ( $\pounds$ hundreds), in 8 randomly selected offices are given in the table below.

$x$	8	9	12	14	7	3	16	19
$p$ (£ hundreds)	40.5	36.1	30.4	39.4	32.6	31.1	43.4	45.7

$$\text { (You may use } \sum x ^ { 2 } = 1160 \quad \sum p = 299.2 \quad \sum p ^ { 2 } = 11422 \quad \sum x p = 3449.5 \text { ) }$$

Show that $S _ { p p } = 231.92$ and find the value of $S _ { x x }$ and the value of $S _ { x p }$
Calculate the product moment correlation coefficient between $x$ and $p$. The equation of the regression line of $p$ on $x$ is given in the form $p = a + b x$.
Show that, to 3 significant figures, $b = 0.824$ and find the value of $a$.
Estimate the amount of money spent on paper in an office with 10 employees.
Explain the effect each additional employee has on the amount of money spent on paper. Later the company realised it had made a mistake in adding up its costs, $p$. The true costs were actually half of the values recorded. The product moment correlation coefficient and the equation of the linear regression line are recalculated using this information.
Write down the new value of
1. the product moment correlation coefficient,
2. the gradient of the regression line.

Edexcel S1 2014 June Q4

$\quad A$ and $B$ are two events such that

$$\mathrm { P } ( B ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \mid B ) = \frac { 2 } { 5 } \quad \mathrm { P } ( A \cup B ) = \frac { 13 } { 20 }$$

Find $\mathrm { P } ( A \cap B )$.
Draw a Venn diagram to show the events $A , B$ and all the associated probabilities. Find
$\mathrm { P } ( A )$
$\mathrm { P } ( B \mid A )$
$\mathrm { P } \left( A ^ { \prime } \cap B \right)$

Edexcel S1 2014 June Q5

The table shows the time, to the nearest minute, spent waiting for a taxi by each of 80 people one Sunday afternoon.

Write down the upper class boundary for the $2 - 4$ minute interval. A histogram is drawn to represent these data. The height of the tallest bar is 6 cm .
Calculate the height of the second tallest bar.
Estimate the number of people with a waiting time between 3.5 minutes and 7 minutes.
Use linear interpolation to estimate the median, the lower quartile and the upper quartile of the waiting times.
Describe the skewness of these data, giving a reason for your answer.

Edexcel S1 2014 June Q6

6. The time taken, in minutes, by children to complete a mathematical puzzle is assumed to be normally distributed with mean $\mu$ and standard deviation $\sigma$. The puzzle can be completed in less than 24 minutes by $80 \%$ of the children. For $5 \%$ of the children it takes more than 28 minutes to complete the puzzle.

Show this information on the Normal curve below.
Write down the percentage of children who take between 24 minutes and 28 minutes to complete the puzzle.
1. Find two equations in $\mu$ and $\sigma$.
2. Hence find, to 3 significant figures, the value of $\mu$ and the value of $\sigma$. A child is selected at random.
Find the probability that the child takes less than 12 minutes to complete the puzzle.
\includegraphics[max width=\textwidth, alt={}, center]{ca8418eb-4d35-40f4-af40-77503327ae52-11_314_1255_1375_356}

Edexcel S1 2014 June Q7

7. In a large company, 78\% of employees are car owners,
$30 \%$ of these car owners are also bike owners,
85\% of those who are not car owners are bike owners.

Draw a tree diagram to represent this information. An employee is selected at random.
Find the probability that the employee is a car owner or a bike owner but not both. Another employee is selected at random. Given that this employee is a bike owner,
find the probability that the employee is a car owner. Two employees are selected at random.
Find the probability that only one of them is a bike owner.

Edexcel S1 2014 June Q1

A random sample of 35 homeowners was taken from each of the villages Greenslax and Penville and their ages were recorded. The results are summarised in the back-to-back stem and leaf diagram below.

Totals	Greenslax											Penville					Totals
(2)						8	7	2	5	5	6	7	8	8	9				(7)
(3)					9	8	7	3	1	1	1	2	3	4	4	5	6	9	(11)
(4)				4	4	4	0	4	0	1	2	4	7						(5)
(5)			6	6	5	2	2	5	0	0	5	5	5						(5)
(7)	8	6	5	4	2	1	1	6	2	5	6	6							(4)
(8)	8	6	6	4	3	1	1	7	0	5									(2)
(5)			9	8	4	3	2	8											(0)
(1)							4	9	9										(1)

Key: 7 | 3 | 1 means 37 years for Greenslax and 31 years for Penville
Some of the quartiles for these two distributions are given in the table below.

	Greenslax	Penville
Lower quartile, $Q _ { 1 }$	$a$	31
Median, $Q _ { 2 }$	64	39
Upper quartile, $Q _ { 3 }$	$b$	55

Find the value of $a$ and the value of $b$. An outlier is a value that falls either $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 }
& \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
On the graph paper opposite draw a box plot to represent the data from Penville. Show clearly any outliers.
State the skewness of each distribution. Justify your answers.
\includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-03_930_1237_1800_367}

Edexcel S1 2014 June Q2

2. The mark, $x$, scored by each student who sat a statistics examination is coded using $$y = 1.4 x - 20$$ The coded marks have mean 60.8 and standard deviation 6.60 Find the mean and the standard deviation of $x$.
\includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-04_99_97_2613_1784}

Edexcel S1 2014 June Q3

3. The table shows data on the number of visitors to the UK in a month, $v$ (1000s), and the amount of money they spent, $m$ ( $\pounds$ millions), for each of 8 months.

Number of visitors

$v ( 1000 \mathrm {~s} )$

Amount of money spent

$m ( \pounds$ millions $)$

You may use
$S _ { v v } = 42587.5 \quad S _ { v m } = 31512.5 \quad S _ { m m } = 25187.5 \quad \sum v = 19390 \quad \sum m = 10610$

Find the product moment correlation coefficient between $m$ and $v$.
Give a reason to support fitting a regression model of the form $m = a + b v$ to these data.
Find the value of $b$ correct to 3 decimal places.
Find the equation of the regression line of $m$ on $v$.
Interpret your value of $b$.
Use your answer to part (d) to estimate the amount of money spent when the number of visitors to the UK in a month is 2500000
Comment on the reliability of your estimate in part (f). Give a reason for your answer.

Edexcel S1 2014 June Q4

In a factory, three machines, $J , K$ and $L$, are used to make biscuits.

Machine $J$ makes $25 \%$ of the biscuits. Machine $K$ makes $45 \%$ of the biscuits. The rest of the biscuits are made by machine $L$.
It is known that $2 \%$ of the biscuits made by machine $J$ are broken, $3 \%$ of the biscuits made by machine $K$ are broken and 5\% of the biscuits made by machine $L$ are broken.

Draw a tree diagram to illustrate all the possible outcomes and associated probabilities. A biscuit is selected at random.
Calculate the probability that the biscuit is made by machine $J$ and is not broken.
Calculate the probability that the biscuit is broken.
Given that the biscuit is broken, find the probability that it was not made by machine $K$.

Edexcel S1 2014 June Q5

5. The discrete random variable $X$ has the probability function $$\mathrm { P } ( X = x ) = \begin{cases} k x & x = 2,4,6
k ( x - 2 ) & x = 8
0 & \text { otherwise } \end{cases}$$ where $k$ is a constant.

Show that $k = \frac { 1 } { 18 }$
Find the exact value of $\mathrm { F } ( 5 )$.
Find the exact value of $\mathrm { E } ( X )$.
Find the exact value of $\mathrm { E } \left( X ^ { 2 } \right)$.
Calculate $\operatorname { Var } ( 3 - 4 X )$ giving your answer to 3 significant figures.

Edexcel S1 2014 June Q6

6. The times, in seconds, spent in a queue at a supermarket by 85 randomly selected customers, are summarised in the table below.

Time (seconds)	Number of customers, $f$
0-30	2
30-60	10
60-70	17
70-80	25
80-100	25
100-150	6

A histogram was drawn to represent these data. The $30 - 60$ group was represented by a bar of width 1.5 cm and height 1 cm .

Find the width and the height of the $70 - 80$ group.
Use linear interpolation to estimate the median of this distribution. Given that $x$ denotes the midpoint of each group in the table and $$\sum f x = 6460 \quad \sum f x ^ { 2 } = 529400$$
calculate an estimate for
1. the mean,
2. the standard deviation,
  for the above data. One measure of skewness is given by $$\text { coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
Evaluate this coefficient and comment on the skewness of these data.

\(x\)	1	2	3	4	5	6
\(\mathrm { P } ( X = x )\)	\(a\)	\(a\)	\(a\)	\(b\)	\(b\)	0.3

\(y\)	1	2	3	4	5
\(\mathrm {~F} ( y )\)	\(\frac { 1 } { 10 }\)	\(\frac { 2 } { 10 }\)	\(3 k\)	\(4 k\)	\(5 k\)

Yield ( \(y \mathrm {~kg}\) )	Frequency (f)	Yield midpoint ( \(x \mathrm {~kg}\) )
\(0 \leqslant y < 5\)	16	2.5
\(5 \leqslant y < 10\)	24	7.5
\(10 \leqslant y < 15\)	14	12.5
\(15 \leqslant y < 25\)	12	20
\(25 \leqslant y < 35\)	4	30

Mark	(2\|6 means 26)	Totals
1	4	(1)
2	6	(1)
3	447	(3)
4	066778	(6)
5	001113677	(9)
6	223338	(6)
7	008	(3)
8	5	(1)
9	0	(1)

Time (minutes) \(t\)	\(11 - 20\)	\(21 - 25\)	\(26 - 30\)	\(31 - 35\)	\(36 - 45\)	\(46 - 60\)
Number of students f	62	88	16	13	11	10

Time (minutes) \(t\)	\(6 - 15\)	\(16 - 20\)	\(21 - 25\)	\(26 - 30\)	\(31 - 40\)	\(41 - 55\)
Number of students f	62	88	16	13	11	10

\(x\)	- 4	- 2	1	3	5
\(\mathrm { P } ( X = x )\)	0.4	\(p\)	0.05	0.15	\(p\)

\(x\)	8	9	12	14	7	3	16	19
\(p\) (£ hundreds)	40.5	36.1	30.4	39.4	32.6	31.1	43.4	45.7

	Greenslax	Penville
Lower quartile, \(Q _ { 1 }\)	\(a\)	31
Median, \(Q _ { 2 }\)	64	39
Upper quartile, \(Q _ { 3 }\)	\(b\)	55

Time (seconds)	Number of customers, \(f\)
0-30	2
30-60	10
60-70	17
70-80	25
80-100	25
100-150	6

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