Questions - Edexcel S1 - A-Level Maths

$\mathrm { P } ( A \cap B )$,
$\mathrm { P } ( B )$,
$\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)$.

Edexcel S1 Q3

3. The random variable $X$ is such that $$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$ Find expressions in terms of $a$ and $b$ for

$\mathrm { E } ( 2 X + 3 )$,
$\quad \operatorname { Var } ( 2 X + 3 )$,
$\mathrm { E } \left( X ^ { 2 } \right)$.
Show that $$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$

Edexcel S1 Q4

An engineer tested a new material under extreme conditions in a wind tunnel. He recorded the number of microfractures, $n$, that formed and the wind speed, $v$ metres per second, for 8 different values of $v$ with all other conditions remaining constant. He then coded the data using $x = v - 700$ and $y = n - 20$ and calculated the following summary statistics.

$$\Sigma x = 100 , \quad \Sigma y = 23 , \quad \Sigma x ^ { 2 } = 215000 , \quad \Sigma x y = 11600 .$$

Find an equation of the regression line of $y$ on $x$.
Hence, find an equation of the regression line of $n$ on $v$.
Use your regression line to estimate the number of microfractures that would be formed if the material was tested in a wind speed of 900 metres per second with all other conditions remaining constant.
(2 marks)

Edexcel S1 Q5

5. An antiques shop recorded the value of items stolen to the nearest pound during each week for a year giving the data in the table below.

Value of goods stolen (£)	Number of weeks
0-199	31
200-399	6
400-599	3
600-799	4
800-999	5
1000-1999	2
2000-2999	1

Letting $x$ represent the mid-point of each group and using the coding $y = \frac { x - 699.5 } { 200 }$,

find $\sum$ fy.
estimate to the nearest pound the mean and standard deviation of the value of the goods stolen each week using your value for $\sum f y$ and $\sum f y ^ { 2 } = 424$.
(6 marks)
The median for these data is $\pounds 82$.
Explain why the manager of the shop might be reluctant to use either the mean or the median in summarising these data.
(3 marks)

Edexcel S1 Q6

6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.

Show that the probability that the first two contestants to be eliminated are both male is $\frac { 2 } { 15 }$.
Find the probability that more females than males are eliminated in the first three rounds of the game.
Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
(3 marks)

Edexcel S1 Q7

7. A cyber-cafe recorded how long each user stayed during one day giving the following results.

Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
Find the median and quartiles that this model would predict.
Comment on the suitability of the suggested model in the light of the new results.

Edexcel S1 Q2

Plot a scatter diagram showing these data. The student wanted to investigate further whether or not her data provided evidence of an increase in temperature in June each year. Using $Y$ for the number of years since 1993 and $T$ for the mean temperature, she calculated the following summary statistics. $$\Sigma Y = 28 , \quad \Sigma T = 182.5 , \quad \Sigma Y ^ { 2 } = 140 , \quad \Sigma T ^ { 2 } = 4173.93 , \quad \Sigma Y T = 644.7 .$$
Calculate the product moment correlation coefficient for these data.
Comment on your result in relation to the student's enquiry.

Edexcel S1 2022 January Q1

A factory produces shoes.

A quality control inspector at the factory checks a sample of 120 shoes for each of three types of defect. The Venn diagram represents the inspector's results. A represents the event that a shoe has defective stitching $B$ represents the event that a shoe has defective colouring $C$ represents the event that a shoe has defective soles
\includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-02_684_935_607_566} One of the shoes in the sample is selected at random.

Find the probability that it does not have defective soles.
Find $\mathrm { P } \left( A \cap B \cap C ^ { \prime } \right)$
Find $\mathrm { P } \left( A \cup B \cup C ^ { \prime } \right)$
Find the probability that the shoe has at most one type of defect.
Given the selected shoe has at most one type of defect, find the probability it has defective stitching. The random variable $X$ is the number of the events $A , B , C$ that occur for a randomly selected shoe.
Find $\mathrm { E } ( X )$ \section*{This is a copy of the Venn diagram for this question.} \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-05_684_940_388_566}

Edexcel S1 2022 January Q2

2. Tom's car holds 50 litres of petrol when the fuel tank is full. For each of 10 journeys, each starting with 50 litres of petrol in the fuel tank, Tom records the distance travelled, $d$ kilometres, and the amount of petrol used, $p$ litres. The summary statistics for the 10 journeys are given below. $$\sum d = 1029 \quad \sum p = 50.8 \quad \sum d p = 5240.8 \quad \mathrm {~S} _ { d d } = 344.9 \quad \mathrm {~S} _ { p p } = 0.576$$

Calculate the product moment correlation coefficient between $d$ and $p$ The amount of petrol remaining in the fuel tank for each journey, $w$ litres, is recorded.
1. Write down an equation for $w$ in terms of $p$
2. Hence, write down the value of the product moment correlation coefficient between $w$ and $p$
Write down the value of the product moment correlation coefficient between $d$ and $w$

Edexcel S1 2022 January Q3

The stem and leaf diagram shows the number of deliveries made by Pat each day for 24 days

\begin{table}[h]

\captionsetup{labelformat=empty} \caption{Key: 10 $\mathbf { 8 }$ represents 108 deliveries}

10	8	9										(2)
11	0	3	6	6	6	8	8	9	9	9	9	(11)
12	4	5	5	5	5	5	5	8				(8)
13	$a$	$b$	$c$									(3)

\end{table} where $a$, $b$ and $c$ are positive integers with $a < b < c$
An outlier is defined as any value greater than $1.5 \times$ interquartile range above the upper quartile. Given that there is only one outlier for these data,

show that $c = 9$ The number of deliveries made by Pat each day is represented by $d$
The data in the stem and leaf diagram are coded using $$x = d - 125$$ and the following summary statistics are obtained $$\sum x = - 96 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1306$$
Find the mean number of deliveries.
Find the standard deviation of the number of deliveries. One of these 24 days is selected at random. The random variable $D$ represents the number of deliveries made by Pat on this day. The random variable $X = D - 125$
Find $\mathrm { P } ( D > 118 \mid X < 0 )$

Edexcel S1 2022 January Q4

The random variable $W$ has a discrete uniform distribution where

$$\mathrm { P } ( W = w ) = \frac { 1 } { 5 } \quad \text { for } w = 1,2,3,4,5$$

Find $\mathrm { P } ( 2 \leqslant W < 3.5 )$ The discrete random variable $X = 5 - 2 W$
Find $\mathrm { E } ( X )$
Find $\mathrm { P } ( X < W )$ The discrete random variable $\mathrm { Y } = \frac { 1 } { W }$
Find
1. the probability distribution of $Y$
2. $\operatorname { Var } ( Y )$, showing your working.
Find $\operatorname { Var } ( 2 - 3 Y )$

Edexcel S1 2022 January Q5

Jia writes a computer program that randomly generates values from a normal distribution. He sets the mean as 40 and the standard deviation as 2.4
1. Find the probability that a particular value generated by the computer program is less than 37
Jia changes the mean to $m$ but leaves the standard deviation as 2.4
The computer program then randomly generates 2 independent values from this normal distribution. The probability that both of these values are greater than 32 is 0.16
Find the value of $m$, giving your answer to 2 decimal places. Jia now changes the mean to 4 and the standard deviation to 8
The computer program then randomly generates 5 independent values from this normal distribution.
Find the probability that at least one of these values is negative.

Edexcel S1 2022 January Q6

Students on a psychology course were given a pre-test at the start of the course and a final exam at the end of the course. The teacher recorded the number of marks achieved on the pre-test, $p$, and the number of marks achieved on the final exam, $f$, for 34 students and displayed them on the scatter diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-22_1121_1136_447_438}

The equation of the least squares regression line for these data is found to be $$f = 10.8 + 0.748 p$$ For these students, the mean number of marks on the pre-test is 62.4

Use the regression model to find the mean number of marks on the final exam.
Give an interpretation of the gradient of the regression line. Considering the equation of the regression line, Priya says that she would expect someone who scored 0 marks on the pre-test to score 10.8 marks on the final exam.
Comment on the reliability of Priya's statement.
Write down the number of marks achieved on the final exam for the student who exceeded the expectation of the regression model by the largest number of marks.
Find the range of values of $p$ for which this regression model, $f = 10.8 + 0.748 p$, predicts a greater number of marks on the final exam than on the pre-test. Later the teacher discovers an error in the recorded data. The student who achieved a score of 98 on the pre-test, scored 92 not 29 on the final exam. The summary statistics used for the model $f = 10.8 + 0.748 p$ are corrected to include this information and a new least squares regression line is found. Given the original summary statistics were, $$n = 34 \quad \sum p = 2120 \quad \sum p f = 133486 \quad \mathrm {~S} _ { p p } = 15573.76 \quad \mathrm {~S} _ { p f } = 11648.35$$
calculate the gradient of the new regression line. Show your working clearly.

Edexcel S1 2022 January Q7

7. A bag contains $n$ marbles of which 7 are green. From the bag, 3 marbles are selected at random.
The random variable $X$ represents the number of green marbles selected.
The cumulative distribution function of $X$ is given by

$x$	0	1	2	3
$\mathrm {~F} ( x )$	$a$	$b$	$\frac { 37 } { 38 }$	1

Show that $n ( n - 1 ) ( n - 2 ) = 7980$
Verify that $n = 21$ satisfies the equation in part (a). Given that $n = 21$
find the exact value of $a$ and the exact value of $b$

\includegraphics[max width=\textwidth, alt={}]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-28_2655_1947_114_116}

Edexcel S1 2017 June Q1

Nina weighed a random sample of 50 carrots from her shop and recorded the weight, in grams to the nearest gram, for each carrot. The results are summarised below.

Weight of carrot	Frequency (f)	Weight midpoint $( \boldsymbol { x }$ grams $)$
$45 - 54$	5	49.5
$55 - 59$	10	57
$60 - 64$	22	62
$65 - 74$	13	69.5

$$\text { (You may use } \sum \mathrm { f } x ^ { 2 } = 192102.5 \text { ) }$$

Use linear interpolation to estimate the median weight of these carrots.
Find an estimate for the mean weight of these carrots.
Find an estimate for the standard deviation of the weights of these carrots. A carrot is selected at random from Nina's shop.
Estimate the probability that the weight of this carrot is more than 70 grams.

Edexcel S1 2017 June Q2

2. The box plot shows the times, $t$ minutes, it takes a group of office workers to travel to work.
\includegraphics[max width=\textwidth, alt={}, center]{7d45bacd-20ac-49b4-8f3f-613edf3739f9-04_365_1237_351_356}

Find the range of the times.
Find the interquartile range of the times.
Using the quartiles, describe the skewness of these data. Give a reason for your answer. Chetna believes that house prices will be higher if the time to travel to work is shorter. She asks a random sample of these office workers for their house prices $\pounds x$, where $x$ is measured in thousands, and obtains the following statistics $$\mathrm { S } _ { x x } = 5514 \quad \mathrm {~S} _ { x t } = 10 \quad \mathrm {~S} _ { t t } = 1145.6$$
Calculate the product moment correlation coefficient between $x$ and $t$.
State, giving a reason, whether or not your correlation coefficient supports Chetna's belief. Adam and Betty are part of the group of office workers and they have both moved house. Adam's time to travel to work changes from 32 minutes to 36 minutes. Betty's time to travel to work changes from 38 minutes to 58 minutes. Outliers are defined as values that are more than 1.5 times the interquartile range above the upper quartile.
Showing all necessary calculations, determine how the box plot of times to travel to work will change and draw a new box plot on the grid on page 5. \includegraphics[max width=\textwidth, alt={}, center]{7d45bacd-20ac-49b4-8f3f-613edf3739f9-05_499_1413_2122_180}

Edexcel S1 2017 June Q3

At a school athletics day, the distances, in metres, achieved by students in the long jump are modelled by the normal distribution with mean 3.3 m and standard deviation 0.6 m
1. Find an estimate for the proportion of students who jump less than 2.5 m
The long jump competition consists of 2 jumps. All the students can take part in the first jump and the $40 \%$ who jump the greatest distance in their first jump qualify for the second jump.
Find an estimate for the minimum distance achieved in the first jump in order to qualify for the second jump.
Give your answer correct to 4 significant figures.
Find an estimate for the median distance achieved in the first jump by those who qualify for the second jump. The distance of the second jump is independent of the distance of the first jump and is modelled with the same normal distribution. Students who jump a distance greater than 4.1 m in their second jump receive a certificate. At the start of the long jump competition, a student is selected at random.
Find the probability that this student will receive a certificate.

Edexcel S1 2017 June Q4

4．The partially completed tree diagram，where $p$ and $q$ are probabilities，gives information about Andrew＇s journey to work each day．
\includegraphics[max width=\textwidth, alt={}, center]{7d45bacd-20ac-49b4-8f3f-613edf3739f9-12_661_794_395_511}
$R$ represents the event that it is raining
W represents the event that Andrew walks to work
$B$ represents the event that Andrew takes the bus to work
$C$ represents the event that Andrew cycles to work Given that $\mathrm { P } ( B ) = 0.26$
（a）find the value of $p$ Given also that $\mathrm { P } \left( R ^ { \prime } \mid W \right) = 0.175$
（b）find the value of $q$
（c）Find the probability that Andrew cycles to work． Given that Andrew did not cycle to work on Friday，
（d）find the probability that it was raining on Friday．

Edexcel S1 2017 June Q5

Tomas is studying the relationship between temperature and hours of sunshine in Seapron. He records the midday temperature, $t ^ { \circ } \mathrm { C }$, and the hours of sunshine, $s$ hours, for a random sample of 9 days in October. He calculated the following statistics

$$\sum s = 15 \quad \sum s ^ { 2 } = 44.22 \quad \sum t = 127 \quad \mathrm {~S} _ { t t } = 10.89$$

Calculate $\mathrm { S } _ { s s }$ Tomas calculated the product moment correlation coefficient between $s$ and $t$ to be 0.832 correct to 3 decimal places.
State, giving a reason, whether or not this correlation coefficient supports the use of a linear regression model to describe the relationship between midday temperature and hours of sunshine.
State, giving a reason, why the hours of sunshine would be the explanatory variable in a linear regression model between midday temperature and hours of sunshine.
Find $\mathrm { S } _ { s t }$
Calculate a suitable linear regression equation to model the relationship between midday temperature and hours of sunshine.
Calculate the standard deviation of $s$ Tomas uses this model to estimate the midday temperature in Seapron for a day in October with 5 hours of sunshine.
State the value of Tomas' estimate. Given that the values of $s$ are all within 2 standard deviations of the mean,
comment, giving your reason, on the reliability of this estimate.

Edexcel S1 2017 June Q6

A biased coin has probability 0.4 of showing a head. In an experiment, the coin is spun until a head appears. If a head has not appeared after 4 spins, the coin is not spun again. The random variable $X$ represents the number of times the coin is spun.

For example, $X = 3$ if the first two spins do not show a head but the third spin does show a head. The coin would not then be spun a fourth time since the coin has already shown a head.

Show that $\mathrm { P } ( X = 3 ) = 0.144$ The table gives some values for the probability distribution of $X$
$x$ 1 2 3 4
$\mathrm { P } ( X = x )$ 0.24 0.144
1. Write down the value of $\mathrm { P } ( X = 1 )$
2. Find $\mathrm { P } ( X = 4 )$
Find $\mathrm { E } ( X )$
Find $\operatorname { Var } ( X )$ The random variable $H$ represents the number of heads obtained when the coin is spun in the experiment.
Explain why $H$ can only take the values 0 and 1 and find the probability distribution of $H$.
Write down the value of
1. $\mathrm { P } ( \{ X = 3 \} \cap \{ H = 0 \} )$
2. $\mathrm { P } ( \{ X = 4 \} \cap \{ H = 0 \} )$ The random variable $S = X + H$
Find the probability distribution of $S$

Edexcel S1 2017 October Q1

At the start of a course, an instructor asked a group of 80 apprentices to estimate the length of a piece of pipe. The error (true length - estimated length) was recorded in centimetres. The results are summarised in the box plot below.
\includegraphics[max width=\textwidth, alt={}, center]{77ae01cd-2b58-48ab-889f-272e27ecf99d-02_291_1445_397_246}

Find the range for these data.
Find the interquartile range for these data.

One month later, the instructor asked the 80 apprentices to estimate the length of a different piece of pipe and recorded their errors. The results are summarised in the table below.

Error ( $\boldsymbol { e }$ cm)	Number of apprentices
$- 40 < e \leqslant - 16$	2
$- 16 < e \leqslant - 8$	18
$- 8 < e \leqslant 0$	33
$0 < e \leqslant 8$	14
$8 < e \leqslant 16$	10
$16 < e \leqslant 40$	3

Use linear interpolation to estimate the median error for these data.
Show that the upper quartile for these data, to the nearest centimetre, is 4 . For these data, the lower quartile is - 8 and the five worst errors were $- 25 , - 21,18,23,28$ An outlier is a value that falls either more than $1.5 \times$ (interquartile range) above the upper quartile or more than $1.5 \times$ (interquartile range) below the lower quartile.
1. Show that there are only 2 outliers for these data.
2. Draw a box plot for these data on the grid on page 3.
State, giving reasons, whether or not the apprentices' ability to estimate the length of a piece of pipe has improved over the first month of the course. \includegraphics[max width=\textwidth, alt={}, center]{77ae01cd-2b58-48ab-889f-272e27ecf99d-03_412_1520_2222_173}

Edexcel S1 2017 October Q2

The Venn diagram, where $w , x , y$ and $z$ are probabilities, shows the probabilities of a group of students buying each of 3 magazines.

A represents the event that a student buys magazine $A$ and $\mathrm { P } ( A ) = 0.60$
$B$ represents the event that a student buys magazine $B$ and $\mathrm { P } ( B ) = 0.15$
$C$ represents the event that a student buys magazine $C$ and $\mathrm { P } ( C ) = 0.35$
\includegraphics[max width=\textwidth, alt={}, center]{77ae01cd-2b58-48ab-889f-272e27ecf99d-06_504_755_641_596}

State which two of the three events $A$, $B$ and $C$ are mutually exclusive. The events $A$ and $C$ are independent.
Show that $w = 0.21$
Find the value of $x$, the value of $y$ and the value of $z$.
Find the probability that a student selected at random buys only one of these magazines.
Find the probability that a student selected at random buys magazine $B$ or magazine $C$.
Find $\mathrm { P } ( A \mid [ B \cup C ] )$

Edexcel S1 2017 October Q3

3. Hei and Tang are designing some pieces of art. They collected a large number of sticks. The random variable $L$ represents the length of a stick in centimetres and has a normal distribution with mean $\mu$ and standard deviation $\sigma$. They sorted the sticks into lengths and painted them.
They found that $60 \%$ of the sticks were longer than 45 cm and these were painted red, whilst $15 \%$ of the sticks were shorter than 35 cm and these were painted blue. The remaining sticks were painted yellow.

Show that $\mu$ and $\sigma$ satisfy $$45 + 0.2533 \sigma = \mu$$
Find a second equation in $\mu$ and $\sigma$.
Hence find the value of $\mu$ and the value of $\sigma$.
Find
1. $\mathrm { P } ( L > 35 \mid L < 45 )$
2. $\mathrm { P } ( L < 45 \mid L > 35 )$ Hei created her piece of art using a random selection of blue and yellow sticks.
  Tang created his piece of art using a random selection of red and yellow sticks.
  Hei and Tang each used the same number of sticks to create their piece of art.
  George is viewing Hei's and Tang's pieces of art. He finds a yellow stick on the floor that has fallen from one of these pieces.
With reference to your answers to part (d), state, giving a reason, whether the stick is more likely to have fallen from Hei's or Tang's piece of art.

Edexcel S1 2017 October Q4

The following incomplete tree diagram shows the relationships between the event $A$ and the event $B$.
\includegraphics[max width=\textwidth, alt={}, center]{77ae01cd-2b58-48ab-889f-272e27ecf99d-14_799_839_351_548}

Given that $\mathrm { P } ( B ) = \frac { 9 } { 20 }$

find $\mathrm { P } ( A )$ and complete the tree diagram,
find $\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)$.

Edexcel S1 2017 October Q5

A company wants to pay its employees according to their performance at work. Last year's performance score $x$ and annual salary $y$, in thousands of dollars, were recorded for a random sample of 10 employees of the company.

The performance scores were $$\begin{array} { l l l l l l l l l l } 15 & 24 & 32 & 39 & 41 & 18 & 16 & 22 & 34 & 42 \end{array}$$ (You may use $\sum x ^ { 2 } = 9011$ )

Find the mean and the variance of these performance scores. The corresponding $y$ values for these 10 employees are summarised by $$\sum y = 306.1 \quad \text { and } \quad \mathrm { S } _ { y y } = 546.3$$
Find the mean and the variance of these $y$ values. The regression line of $y$ on $x$ based on this sample is $$y = 12.0 + 0.659 x$$
Find the product moment correlation coefficient for these data.
State, giving a reason, whether or not the value of the product moment correlation coefficient supports the use of a regression line to model the relationship between performance score and annual salary. The company decides to use this regression model to determine future salaries.
Find the proposed annual salary, in dollars, for an employee who has a performance score of 35

\(x\)	0	1	2	3
\(\mathrm {~F} ( x )\)	\(a\)	\(b\)	\(\frac { 37 } { 38 }\)	1

Error ( \(\boldsymbol { e }\) cm)	Number of apprentices
\(- 40 < e \leqslant - 16\)	2
\(- 16 < e \leqslant - 8\)	18
\(- 8 < e \leqslant 0\)	33
\(0 < e \leqslant 8\)	14
\(8 < e \leqslant 16\)	10
\(16 < e \leqslant 40\)	3

Weight of carrot	Frequency (f)	Weight midpoint \(( \boldsymbol { x }\) grams \()\)
\(45 - 54\)	5	49.5
\(55 - 59\)	10	57
\(60 - 64\)	22	62
\(65 - 74\)	13	69.5

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