Questions S1 - A-Level Maths

Time	Mid-point, $x$	Frequency, f
2-8	5	2
9-12		7
13-15	14	5
16-18	17	8
19-22	20.5	4
23-30	26.5	4

$$\text { (You may use } \sum \mathrm { f } x ^ { 2 } = 8603.75 \text { ) }$$

Write down the mid-point for the 9-12 interval.
Use linear interpolation to estimate the median time taken by the students.
Estimate the mean and standard deviation of the times taken by the students. The teacher suggested that a normal distribution could be used to model the times taken by the students to complete the sudoku.
Give a reason to support the use of a normal distribution in this case. On another occasion the teacher calculated the quartiles for the times taken by the students to complete a different sudoku and found $$Q _ { 1 } = 8.5 \quad Q _ { 2 } = 13.0 \quad Q _ { 3 } = 21.0$$
Describe, giving a reason, the skewness of the times on this occasion.

Edexcel S1 2011 June Q6

Jake and Kamil are sometimes late for school.

The events $J$ and $K$ are defined as follows
$J =$ the event that Jake is late for school
$K =$ the event that Kamil is late for school
$\mathrm { P } ( J ) = 0.25 , \mathrm { P } ( J \cap K ) = 0.15$ and $\mathrm { P } \left( J ^ { \prime } \cap K ^ { \prime } \right) = 0.7$ On a randomly selected day, find the probability that

at least one of Jake or Kamil are late for school,
Kamil is late for school. Given that Jake is late for school,
find the probability that Kamil is late. The teacher suspects that Jake being late for school and Kamil being late for school are linked in some way.
Determine whether or not $J$ and $K$ are statistically independent.
Comment on the teacher's suspicion in the light of your calculation in (d).

Edexcel S1 2011 June Q7

A teacher took a random sample of 8 children from a class. For each child the teacher recorded the length of their left foot, $f \mathrm {~cm}$, and their height, $h \mathrm {~cm}$. The results are given in the table below.

$f$	23	26	23	22	27	24	20	21
$h$	135	144	134	136	140	134	130	132

(You may use $\sum f = 186 \quad \sum h = 1085 \quad \mathrm {~S} _ { f f } = 39.5 \quad \mathrm {~S} _ { h h } = 139.875 \quad \sum f h = 25291$ )

Calculate $\mathrm { S } _ { f h }$
Find the equation of the regression line of $h$ on $f$ in the form $h = a + b f$. Give the value of $a$ and the value of $b$ correct to 3 significant figures.
Use your equation to estimate the height of a child with a left foot length of 25 cm .
Comment on the reliability of your estimate in (c), giving a reason for your answer. The left foot length of the teacher is 25 cm .
Give a reason why the equation in (b) should not be used to estimate the teacher's height.

Edexcel S1 2011 June Q8

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

$s$	0	1	2	4	5
$\mathrm { P } ( S = s )$	$p$	0.25	0.25	0.20	0.20

Find the value of $p$.
Find $\mathrm { E } ( S )$.
Show that $\mathrm { E } \left( S ^ { 2 } \right) = 9.45$
Find $\operatorname { Var } ( S )$. Tom and Jess play a game with this spinner. The spinner is spun repeatedly and $S$ counters are awarded on the outcome of each spin. If $S$ is even then Tom receives the counters and if $S$ is odd then Jess receives them. The first player to collect 10 or more counters is the winner.
Find the probability that Jess wins after 2 spins.
Find the probability that Tom wins after exactly 3 spins.
Find the probability that Jess wins after exactly 3 spins.

Edexcel S1 2012 June Q1

A discrete random variable $X$ has the probability function

$$\mathrm { P } ( X = x ) = \begin{cases} k ( 1 - x ) ^ { 2 } & x = - 1,0,1 \text { and } 2
0 & \text { otherwise } \end{cases}$$

Show that $k = \frac { 1 } { 6 }$
Find $\mathrm { E } ( X )$
Show that $\mathrm { E } \left( X ^ { 2 } \right) = \frac { 4 } { 3 }$
Find $\operatorname { Var } ( 1 - 3 X )$

Edexcel S1 2012 June Q2

2. A bank reviews its customer records at the end of each month to find out how many customers have become unemployed, $u$, and how many have had their house repossessed, $h$, during that month. The bank codes the data using variables $x = \frac { u - 100 } { 3 }$ and $y = \frac { h - 20 } { 7 }$ The results for the 12 months of 2009 are summarised below. $$\sum x = 477 \quad S _ { x x } = 5606.25 \quad \sum y = 480 \quad S _ { y y } = 4244 \quad \sum x y = 23070$$

Calculate the value of the product moment correlation coefficient for $x$ and $y$.
Write down the product moment correlation coefficient for $u$ and $h$. The bank claims that an increase in unemployment among its customers is associated with an increase in house repossessions.
State, with a reason, whether or not the bank's claim is supported by these data.

Edexcel S1 2012 June Q3

3. A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells. He collects a random sample of egg shells from each of 6 different nests and tests for pollutant level, $p$, and measures the thinning of the shell, $t$. The results are shown in the table below.

$p$	3	8	30	25	15	12
$t$	1	3	9	10	5	6

[You may use $\sum p ^ { 2 } = 1967$ and $\sum p t = 694$ ]

Draw a scatter diagram on the axes on page 7 to represent these data.
Explain why a linear regression model may be appropriate to describe the relationship between $p$ and $t$.
Calculate the value of $S _ { p t }$ and the value of $S _ { p p }$.
Find the equation of the regression line of $t$ on $p$, giving your answer in the form $t = a + b p$.
Plot the point ( $\bar { p } , \bar { t }$ ) and draw the regression line on your scatter diagram. The scientist reviews similar studies and finds that pollutant levels above 16 are likely to result in the death of a chick soon after hatching.
Estimate the minimum thinning of the shell that is likely to result in the death of a chick. \includegraphics[max width=\textwidth, alt={}, center]{0593544d-392d-465b-b922-c9cb1435abb5-05_1257_1568_301_173}

Edexcel S1 2012 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0593544d-392d-465b-b922-c9cb1435abb5-06_611_1127_237_447} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows how 25 people travelled to work.
Their travel to work is represented by the events $$\begin{array} { l l } B & \text { bicycle }
T & \text { train }
W & \text { walk } \end{array}$$

Write down 2 of these events that are mutually exclusive. Give a reason for your answer.
Determine whether or not $B$ and $T$ are independent events. One person is chosen at random.
Find the probability that this person
walks to work,
travels to work by bicycle and train.
Given that this person travels to work by bicycle, find the probability that they will also take the train.

Edexcel S1 2012 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0593544d-392d-465b-b922-c9cb1435abb5-08_1031_1239_116_354} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A policeman records the speed of the traffic on a busy road with a 30 mph speed limit. He records the speeds of a sample of 450 cars. The histogram in Figure 2 represents the results.

Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample.
Estimate the value of the mean speed of the cars in the sample.
Estimate, to 1 decimal place, the value of the median speed of the cars in the sample.
Comment on the shape of the distribution. Give a reason for your answer.
State, with a reason, whether the estimate of the mean or the median is a better representation of the average speed of the traffic on the road.

Edexcel S1 2012 June Q6

The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm .
1. Find the probability that a randomly chosen adult female is taller than 150 cm .
  (3)
Sarah is a young girl. She visits her doctor and is told that she is at the 60th percentile for height.
Assuming that Sarah remains at the 60th percentile, estimate her height as an adult. The heights of an adult male population are normally distributed with standard deviation 9.0 cm . Given that $90 \%$ of adult males are taller than the mean height of adult females,
find the mean height of an adult male.

Edexcel S1 2012 June Q7

A manufacturer carried out a survey of the defects in their soft toys. It is found that the probability of a toy having poor stitching is 0.03 and that a toy with poor stitching has a probability of 0.7 of splitting open. A toy without poor stitching has a probability of 0.02 of splitting open.
1. Draw a tree diagram to represent this information.
2. Find the probability that a randomly chosen soft toy has exactly one of the two defects, poor stitching or splitting open.
  (3)
The manufacturer also finds that soft toys can become faded with probability 0.05 and that this defect is independent of poor stitching or splitting open. A soft toy is chosen at random.
Find the probability that the soft toy has none of these 3 defects.
Find the probability that the soft toy has exactly one of these 3 defects.

Edexcel S1 2013 June Q1

Sammy is studying the number of units of gas, $g$, and the number of units of electricity, $e$, used in her house each week. A random sample of 10 weeks use was recorded and the data for each week were coded so that $x = \frac { g - 60 } { 4 }$ and $y = \frac { e } { 10 }$. The results for the coded data are summarised below

$$\sum x = 48.0 \quad \sum y = 58.0 \quad \mathrm {~S} _ { x x } = 312.1 \quad \mathrm {~S} _ { y y } = 2.10 \quad \mathrm {~S} _ { x y } = 18.35$$

Find the equation of the regression line of $y$ on $x$ in the form $y = a + b x$. Give the values of $a$ and $b$ correct to 3 significant figures.
Hence find the equation of the regression line of $e$ on $g$ in the form $e = c + d g$. Give the values of $c$ and $d$ correct to 2 significant figures.
Use your regression equation to estimate the number of units of electricity used in a week when 100 units of gas were used.
（a）Find the probability distribution of $X$ ．
（b）Write down the value of $\mathrm { F } ( 1.8 )$ ．
（a）Find the probability distribution of $X$ ．勤

Edexcel S1 2013 June Q2

2．The discrete random variable $X$ takes the values 1,2 and 3 and has cum
function $\mathrm { F } ( x )$ given by

$x$	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 $\operatorname { Var } ( a X + 3 ) = 53.4$

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

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

Time	Mid-point, \(x\)	Frequency, f
2-8	5	2
9-12		7
13-15	14	5
16-18	17	8
19-22	20.5	4
23-30	26.5	4

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	\(p\)

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