Questions - Edexcel S1 - A-Level Maths

$l$	49.0	52.0	53.0	54.5	54.1	53.4	50.0	51.6	49.5	51.2
$w$	29	32	34	39	38	35	30	31	29	30

$$\text { (You may use } \mathrm { S } _ { l l } = 33.381 \quad \mathrm {~S} _ { w l } = 59.99 \quad \mathrm {~S} _ { w w } = 120.1 \text { ) }$$

Find the equation of the regression line of $w$ on $l$ in the form $w = a + b l$.
Use your regression line to estimate the weight of a newborn turtle of length 60 mm .
Comment on the reliability of your estimate giving a reason for your answer.

Edexcel S1 2009 June Q6

6. The discrete random variable $X$ has probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } a ( 3 - x ) & x = 0,1,2
b & x = 3 \end{array} \right.$$

Find $\mathrm { P } ( X = 2 )$ and complete the table below.
$x$ 0 1 2 3
$\mathrm { P } ( X = x )$ $3 a$ $2 a$ $b$
Given that $\mathrm { E } ( X ) = 1.6$
Find the value of $a$ and the value of $b$. Find
$\mathrm { P } ( 0.5 < X < 3 )$,
$\mathrm { E } ( 3 X - 2 )$.
Show that the $\operatorname { Var } ( X ) = 1.64$
Calculate $\operatorname { Var } ( 3 X - 2 )$.

Edexcel S1 2009 June Q7

7. (a) Given that $\mathrm { P } ( A ) = a$ and $\mathrm { P } ( B ) = b$ express $\mathrm { P } ( A \cup B )$ in terms of $a$ and $b$ when

$A$ and $B$ are mutually exclusive,
$A$ and $B$ are independent. Two events $R$ and $Q$ are such that
$\mathrm { P } \left( R \cap Q ^ { \prime } \right) = 0.15 , \quad \mathrm { P } ( Q ) = 0.35$ and $\mathrm { P } ( R \mid Q ) = 0.1$
Find the value of
(b) $\mathrm { P } ( R \cup Q )$,
(c) $\mathrm { P } ( R \cap Q )$,
(d) $\mathrm { P } ( R )$.

Edexcel S1 2009 June Q8

8. The lifetimes of bulbs used in a lamp are normally distributed. A company $X$ sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.

Find the probability of a bulb, from company $X$, having a lifetime of less than 830 hours.
In a box of 500 bulbs, from company $X$, find the expected number having a lifetime of less than 830 hours. A rival company $Y$ sells bulbs with a mean lifetime of 860 hours and $20 \%$ of these bulbs have a lifetime of less than 818 hours.
Find the standard deviation of the lifetimes of bulbs from company $Y$. Both companies sell the bulbs for the same price.
State which company you would recommend. Give reasons for your answer.

Edexcel S1 2010 June Q2

2. An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability $\frac { 2 } { 3 }$ of landing heads is spun.
When a blue ball is selected a fair coin is spun.

Complete the tree diagram below to show the possible outcomes and associated probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_787_395_734_548} \section*{Coin}
\includegraphics[max width=\textwidth, alt={}]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_1007_488_808_950}
Shivani selects a ball and spins the appropriate coin.
Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.

Edexcel S1 2010 June Q3

3. The discrete random variable $X$ has probability distribution given by

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

where $a$ is a constant.

Find the value of $a$.
Write down $\mathrm { E } ( X )$.
Find $\operatorname { Var } ( X )$. The random variable $Y = 6 - 2 X$
Find $\operatorname { Var } ( Y )$.
Calculate $\mathrm { P } ( X \geqslant Y )$.

Edexcel S1 2010 June Q4

4. The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines $A , B$ and $C$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-07_397_934_374_502} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} One of these students is selected at random.

Show that the probability that the student reads more than one magazine is $\frac { 1 } { 6 }$.
Find the probability that the student reads $A$ or $B$ (or both).
Write down the probability that the student reads both $A$ and $C$. Given that the student reads at least one of the magazines,
find the probability that the student reads $C$.
Determine whether or not reading magazine $B$ and reading magazine $C$ are statistically independent.

Edexcel S1 2010 June Q5

5. A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.

Hours	$1 - 10$	$11 - 20$	$21 - 25$	$26 - 30$	$31 - 40$	$41 - 59$
Frequency	6	15	11	13	8	3
Mid-point	5.5	15.5		28		50

Find the mid-points of the 21-25 hour and 31-40 hour groups. A histogram was drawn to represent these data. The $11 - 20$ group was represented by a bar of width 4 cm and height 6 cm .
Find the width and height of the 26-30 group.
Estimate the mean and standard deviation of the time spent watching television by these students.
Use linear interpolation to estimate the median length of time spent watching television by these students. The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.
State, giving a reason, the skewness of these data.

Edexcel S1 2010 June Q6

6. A travel agent sells flights to different destinations from Beerow airport. The distance $d$, measured in 100 km , of the destination from the airport and the fare $\pounds f$ are recorded for a random sample of 6 destinations.

Destination	$A$	$B$	$C$	$D$	$E$	$F$
$d$	2.2	4.0	6.0	2.5	8.0	5.0
$f$	18	20	25	23	32	28

$$\text { [You may use } \sum d ^ { 2 } = 152.09 \quad \sum f ^ { 2 } = 3686 \quad \sum f d = 723.1 \text { ] }$$

Using the axes below, complete a scatter diagram to illustrate this information.
Explain why a linear regression model may be appropriate to describe the relationship between $f$ and $d$.
Calculate $S _ { d d }$ and $S _ { f d }$
Calculate the equation of the regression line of $f$ on $d$ giving your answer in the form $f = a + b d$.
Give an interpretation of the value of $b$. Jane is planning her holiday and wishes to fly from Beerow airport to a destination $t \mathrm {~km}$ away. A rival travel agent charges 5 p per km.
Find the range of values of $t$ for which the first travel agent is cheaper than the rival.
\includegraphics[max width=\textwidth, alt={}, center]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-11_1013_1701_1718_116}

Edexcel S1 2010 June Q7

7. The distances travelled to work, $D \mathrm {~km}$, by the employees at a large company are normally distributed with $D \sim \mathrm {~N} \left( 30,8 ^ { 2 } \right)$.

Find the probability that a randomly selected employee has a journey to work of more than 20 km .
Find the upper quartile, $Q _ { 3 }$, of $D$.
Write down the lower quartile, $Q _ { 1 }$, of $D$. An outlier is defined as any value of $D$ such that $D < h$ or $D > k$ where $$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
Find the value of $h$ and the value of $k$. An employee is selected at random.
Find the probability that the distance travelled to work by this employee is an outlier.

Edexcel S1 2011 June Q1

On a particular day the height above sea level, $x$ metres, and the mid-day temperature, $y ^ { \circ } \mathrm { C }$, were recorded in 8 north European towns. These data are summarised below

$$\mathrm { S } _ { x x } = 3535237.5 \quad \sum y = 181 \quad \sum y ^ { 2 } = 4305 \quad \mathrm {~S} _ { x y } = - 23726.25$$

Find $\mathrm { S } _ { y y }$
Calculate, to 3 significant figures, the product moment correlation coefficient for these data.
Give an interpretation of your coefficient. A student thought that the calculations would be simpler if the height above sea level, $h$, was measured in kilometres and used the variable $h = \frac { x } { 1000 }$ instead of $x$.
Write down the value of $\mathrm { S } _ { h h }$
Write down the value of the correlation coefficient between $h$ and $y$.

Edexcel S1 2011 June Q2

The random variable $X \sim \mathrm {~N} \left( \mu , 5 ^ { 2 } \right)$ and $\mathrm { P } ( X < 23 ) = 0.9192$
1. Find the value of $\mu$.
2. Write down the value of $\mathrm { P } ( \mu < X < 23 )$.
3. The discrete random variable $Y$ has probability distribution
$y$ 1 2 3 4
$\mathrm { P } ( Y = y )$ $a$ $b$ 0.3 $c$
where $a , b$ and $c$ are constants. The cumulative distribution function $\mathrm { F } ( y )$ of $Y$ is given in the following table
$y$ 1 2 3 4
$\mathrm {~F} ( y )$ 0.1 0.5 $d$ 1.0
where $d$ is a constant.
Find the value of $a$, the value of $b$, the value of $c$ and the value of $d$.
Find $\mathrm { P } ( 3 Y + 2 \geqslant 8 )$.

Edexcel S1 2011 June Q4

4. Past records show that the times, in seconds, taken to run 100 m by children at a school can be modelled by a normal distribution with a mean of 16.12 and a standard deviation of 1.60 A child from the school is selected at random.

Find the probability that this child runs 100 m in less than 15 s . On sports day the school awards certificates to the fastest $30 \%$ of the children in the 100 m race.
Estimate, to 2 decimal places, the slowest time taken to run 100 m for which a child will be awarded a certificate.

Edexcel S1 2011 June Q5

5. A class of students had a sudoku competition. The time taken for each student to complete the sudoku was recorded to the nearest minute and the results are summarised in the table below.

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

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

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

\(y\)	1	2	3	4
\(\mathrm { P } ( Y = y )\)	\(a\)	\(b\)	0.3	\(c\)

\(l\)	49.0	52.0	53.0	54.5	54.1	53.4	50.0	51.6	49.5	51.2
\(w\)	29	32	34	39	38	35	30	31	29	30

Hours	\(1 - 10\)	\(11 - 20\)	\(21 - 25\)	\(26 - 30\)	\(31 - 40\)	\(41 - 59\)
Frequency	6	15	11	13	8	3
Mid-point	5.5	15.5		28		50

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

Destination	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)
\(d\)	2.2	4.0	6.0	2.5	8.0	5.0
\(f\)	18	20	25	23	32	28

\(y\)	1	2	3	4
\(\mathrm {~F} ( y )\)	0.1	0.5	\(d\)	1.0

\(p\)	3	8	30	25	15	12
\(t\)	1	3	9	10	5	6

Questions — Edexcel S1 (574 questions)

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