Questions S1 - A-Level Maths

Some children are playing a game involving throwing a ball into a bucket. Each child has 3 throws and the number of times the ball lands in the bucket, $x$, is recorded. Their results are given in the table below.

$x$	0	1	2	3
Frequency	16	36	24	4

Find $\bar { x }$
(1) Sandra decides to model the game by assuming that on each throw, the probability of the ball landing in the bucket is 0.4 for every child on every throw and that the throws are all independent. The random variable $S$ represents the number of times the ball lands in the bucket for a randomly selected child.
Find $\mathrm { P } ( S = 2 )$
Complete the table below to show the probability distribution for $S$.
$s$ 0 1 2 3
$\mathrm { P } ( S = s )$ 0.432 0.064
Ting believes that the probability of the ball landing in the bucket is not the same for each throw. He suggests that the probability will increase with each throw and uses the model $$p _ { i } = 0.15 i + 0.10$$ where $i = 1,2,3$ and $p _ { i }$ is the probability that the $i$ th throw of the ball, by any particular child, will land in the bucket.
The random variable $T$ represents the number of times the ball lands in the bucket for a randomly selected child using Ting’s model.
Show that
1. $\mathrm { P } ( T = 3 ) = 0.055$
2. $\mathrm { P } ( T = 1 ) = 0.45$
  (5)
Complete the table below to show the probability distribution for $T$, stating the exact probabilities in each case.
$t$ 0 1 2 3
$\mathrm { P } ( T = t )$ 0.45 0.055
State, giving your reasons, whether Sandra's model or Ting's model is the more appropriate for modelling this game.

Edexcel S1 2019 January Q6

18 marks Moderate -0.3

Following some school examinations, Chetna is studying the results of the 16 students in her class. The mark for paper $1 , x$, and the mark for paper $2 , y$, for each student are summarised in the following statistics.

$$\bar { x } = 35.75 \quad \bar { y } = 25.75 \quad \sigma _ { x } = 7.79 \quad \sigma _ { y } = 11.91 \quad \sum x y = 15837$$

Comment on the differences between the marks of the students on paper 1 and paper 2 Chetna decides to examine these data in more detail and plots the marks for each of the 16 students on the scatter diagram opposite.
1. Explain why the circled point $( 38,0 )$ is possibly an outlier.
2. Suggest a possible reason for this result. Chetna decides to omit the data point $( 38,0 )$ and examine the other 15 students' marks.
Find the value of $\bar { x }$ and the value of $\bar { y }$ for these 15 students. For these 15 students
1. explain why $\sum x y$ is still 15837
2. show that $\mathrm { S } _ { x y } = 1169.8$ For these 15 students, Chetna calculates $\mathrm { S } _ { x x } = 965.6$ and $\mathrm { S } _ { y y } = 1561.7$ correct to 1 decimal place.
Calculate the product moment correlation coefficient for these 15 students.
Calculate the equation of the line of regression of $y$ on $x$ for these 15 students, giving your answer in the form $y = a + b x$ The product moment correlation coefficient between $x$ and $y$ for all 16 students is 0.746
Explain how your calculation in part (e) supports Chetna's decision to omit the point $( 38,0 )$ before calculating the equation of the linear regression line.
(1)
Estimate the mark in the second paper for a student who scored 38 marks in the first paper.
\includegraphics[max width=\textwidth, alt={}]{d3f4450d-60eb-49b6-be1b-d2fcfad0451f-17_1127_1146_301_406}

\includegraphics[max width=\textwidth, alt={}]{d3f4450d-60eb-49b6-be1b-d2fcfad0451f-20_2630_1828_121_121}

Edexcel S1 2021 January Q1

5 marks Moderate -0.8

The Venn diagram shows the events $A , B$ and $C$ and their associated probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{4f034b9a-94c8-42f2-bd77-9adec277aba6-02_584_1061_296_445}

Find

$\mathrm { P } \left( B ^ { \prime } \right)$
$\mathrm { P } ( A \cup C )$
$\mathrm { P } \left( A \mid B ^ { \prime } \right)$

Edexcel S1 2021 January Q2

9 marks Easy -1.3

2. The stem and leaf diagram below shows the ages (in years) of the residents in a care home.

Age													Key: $4 \mid 3$ is an age of 43
4	3											$( 1 )$
5	4
6	2	3	5	6	8	8	8	9	9			$( 1 )$
7	1	1	3	4	4	6	6	6	8	8	9	$( 9 )$
8	0	0	2	7	8	8	9					$( 11 )$
9	3	7

Find the median age of the residents.
Find the interquartile range (IQR) of the ages of the residents. An outlier is defined as a value that is either
more than $1.5 \times ( \mathrm { IQR } )$ below the lower quartile or more than $1.5 \times ( \mathrm { IQR } )$ above the upper quartile.
Determine any outliers in these data. Show clearly any calculations that you use.
On the grid on page 5, draw a box plot to summarise these data.
Ages

Edexcel S1 2021 January Q3

13 marks Moderate -0.3

3. The weights of packages that arrive at a factory are normally distributed with a mean of 18 kg and a standard deviation of 5.4 kg

Find the probability that a randomly selected package weighs less than 10 kg The heaviest 15\% of packages are moved around the factory by Jemima using a forklift truck.
Find the weight, in kg , of the lightest of these packages that Jemima will move. One of the packages not moved by Jemima is selected at random.
Find the probability that it weighs more than 18 kg A delivery of 4 packages is made to the factory. The weights of the packages are independent.
Find the probability that exactly 2 of them will be moved by Jemima.

Edexcel S1 2021 January Q4

16 marks Moderate -0.8

4. A spinner can land on the numbers $10,12,14$ and 16 only and the probability of the spinner landing on each number is the same.
The random variable $X$ represents the number that the spinner lands on when it is spun once.

State the name of the probability distribution of $X$.
1. Write down the value of $\mathrm { E } ( X )$
2. Find $\operatorname { Var } ( X )$ A second spinner can land on the numbers 1, 2, 3, 4 and 5 only. The random variable $Y$ represents the number that this spinner lands on when it is spun once. The probability distribution of $Y$ is given in the table below
  $y$ 1 2 3 4 5
  $\mathrm { P } ( Y = y )$ $\frac { 4 } { 30 }$ $\frac { 9 } { 30 }$ $\frac { 6 } { 30 }$ $\frac { 5 } { 30 }$ $\frac { 6 } { 30 }$
Find
1. $\mathrm { E } ( Y )$
2. $\operatorname { Var } ( Y )$ The random variable $W = a X + b$, where $a$ and $b$ are constants and $a > 0$ Given that $\mathrm { E } ( W ) = \mathrm { E } ( Y )$ and $\operatorname { Var } ( W ) = \operatorname { Var } ( Y )$
find the value of $a$ and the value of $b$. Each of the two spinners is spun once.
Find $\mathrm { P } ( W = Y )$

Edexcel S1 2021 January Q5

17 marks Moderate -0.8

A company director wants to introduce a performance-related pay structure for her managers. A random sample of 15 managers is taken and the annual salary, $y$ in $\pounds 1000$, was recorded for each manager. The director then calculated a performance score, $x$, for each of these managers.
The results are shown on the scatter diagram in Figure 1 on the next page.
1. Describe the correlation between performance score and annual salary.
The results are also summarised in the following statistics. $$\sum x = 465 \quad \sum y = 562 \quad \mathrm {~S} _ { x x } = 2492 \quad \sum y ^ { 2 } = 23140 \quad \sum x y = 19428$$
1. Show that $\mathrm { S } _ { x y } = 2006$
2. Find $\mathrm { S } _ { y y }$
Find the product moment correlation coefficient between performance score and annual salary. The director believes that there is a linear relationship between performance score and annual salary.
State, giving a reason, whether or not these data are consistent with the director's belief.
Calculate the equation of the 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.
Give an interpretation of the value of $b$.
Plot your regression line on the scatter diagram in Figure 1 The director hears that one of the managers in the sample seems to be underperforming.
On the scatter diagram, circle the point that best identifies this manager. The director decides to use this regression line for the new performance related pay structure.
1. Estimate, to 3 significant figures, the new salary of a manager with a performance score of 30 \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{4f034b9a-94c8-42f2-bd77-9adec277aba6-15_1390_1408_299_187} \captionsetup{labelformat=empty} \caption{Figure 1}
  \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{4f034b9a-94c8-42f2-bd77-9adec277aba6-17_2654_99_115_9} Annual salary (£1000) \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Only use this scatter diagram if you need to redraw your line.} \includegraphics[alt={},max width=\textwidth]{4f034b9a-94c8-42f2-bd77-9adec277aba6-17_1378_1143_402_468}
  \end{figure}

Edexcel S1 2021 January Q6

15 marks Moderate -0.3

A disc of radius 1 cm is rolled onto a horizontal grid of rectangles so that the disc is equally likely to land anywhere on the grid. Each rectangle is 5 cm long and 3 cm wide. There are no gaps between the rectangles and the grid is sufficiently large so that no discs roll off the grid.

If the disc lands inside a rectangle without covering any part of the edges of the rectangle then a prize is won. By considering the possible positions for the centre of the disc,

show that the probability of winning a prize on any particular roll is $\frac { 1 } { 5 }$ A group of 15 students each roll the disc onto the grid twenty times and record the number of times, $x$, that each student wins a prize. Their results are summarised as follows $$\sum x = 61 \quad \sum x ^ { 2 } = 295$$
Find the standard deviation of the number of prizes won per student. A second group of 12 students each roll the disc onto the grid twenty times and the mean number of prizes won per student is 3.5 with a standard deviation of 2
Find the mean and standard deviation of the number of prizes won per student for the whole group of 27 students. The 27 students also recorded the number of times that the disc covered a corner of a rectangle and estimated the probability to be 0.2216 (to 4 decimal places).
Explain how this probability could be used to find an estimate for the value of $\pi$ and state the value of your estimate.

Edexcel S1 2023 January Q1

10 marks Moderate -0.3

The histogram shows the times taken, $t$ minutes, by each of 100 people to swim 500 metres.
\includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-02_986_1070_342_424}
1. Use the histogram to complete the frequency table for the times taken by the 100 people to swim 500 metres.
Time taken ( $\boldsymbol { t }$ minutes) $5 - 10$ $10 - 14$ $14 - 18$ $18 - 25$ $25 - 40$
Frequency ( $\boldsymbol { f }$ ) 10 16 24
Estimate the number of people who took less than 16 minutes to swim 500 metres.
Find an estimate for the mean time taken to swim 500 metres. Given that $\sum f t ^ { 2 } = 41033$
find an estimate for the standard deviation of the times taken to swim 500 metres. Given that $Q _ { 3 } = 23$
use linear interpolation to estimate the interquartile range of the times taken to swim 500 metres.

Edexcel S1 2023 January Q2

10 marks Moderate -0.3

Two bags, $\boldsymbol { X }$ and $\boldsymbol { Y }$, each contain green marbles (G) and blue marbles (B) only.

Bag $\boldsymbol { X }$ contains 5 green marbles and 4 blue marbles
Bag $\boldsymbol { Y }$ contains 6 green marbles and 5 blue marbles

A marble is selected at random from bag $\boldsymbol { X }$ and placed in bag $\boldsymbol { Y }$
A second marble is selected at random from bag $\boldsymbol { X }$ and placed in bag $\boldsymbol { Y }$
A third marble is then selected, this time from bag $\boldsymbol { Y }$

Use this information to complete the tree diagram shown on page 7
Find the probability that the 2 marbles selected from bag $\boldsymbol { X }$ are of different colours.
Find the probability that all 3 marbles selected are the same colour. Given that all three marbles selected are the same colour,
find the probability that they are all green. 2nd Marble (from bag $\boldsymbol { X }$ ) \section*{3rd Marble (from bag Y)} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{1st Marble (from bag $\boldsymbol { X }$ )} \includegraphics[alt={},max width=\textwidth]{c316fa29-dedc-4890-bd82-31eb0bb819f9-07_1694_1312_484_310}
\end{figure}

Edexcel S1 2023 January Q3

11 marks Standard +0.3

The probability distribution of the discrete random variable $X$ is given by

$x$	2	3	4
$\mathrm { P } ( X = x )$	$a$	0.4	$0.6 - a$

where $a$ is a constant.

Find, in terms of $a , \mathrm { E } ( X )$
Find the range of the possible values of $\mathrm { E } ( X )$ Given that $\operatorname { Var } ( X ) = 0.56$
find the possible values of $a$

Edexcel S1 2023 January Q4

13 marks Standard +0.3

1. In the Venn diagram below, $A$ and $B$ represent events and $p , q , r$ and $s$ are probabilities.
  \includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-12_400_789_347_639}

$$\mathrm { P } ( A ) = \frac { 7 } { 25 } \quad \mathrm { P } ( B ) = \frac { 1 } { 5 } \quad \mathrm { P } \left[ \left( A \cap B ^ { \prime } \right) \cup \left( A ^ { \prime } \cap B \right) \right] = \frac { 8 } { 25 }$$

Use algebra to show that $2 p + 2 q + 2 r = \frac { 4 } { 5 }$
Find the value of $p$, the value of $q$, the value of $r$ and the value of $s$
(ii) Two events, $C$ and $D$, are such that $$\mathrm { P } ( C ) = \frac { x } { x + 5 } \quad \mathrm { P } ( D ) = \frac { 5 } { x }$$ where $x$ is a positive constant.
By considering $\mathrm { P } ( C ) + \mathrm { P } ( D )$ show that $C$ and $D$ cannot be mutually exclusive.

Edexcel S1 2023 January Q5

17 marks Moderate -0.3

The lengths, $L \mathrm {~mm}$, of housefly wings are normally distributed with $L \sim \mathrm {~N} \left( 4.5,0.4 ^ { 2 } \right)$
1. Find the probability that a randomly selected housefly has a wing length of less than 3.86 mm .
2. Find
  1. the upper quartile ( $Q _ { 3 }$ ) of $L$
  2. the lower quartile ( $Q _ { 1 }$ ) of $L$
A value that is greater than $Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$ or smaller than $Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$ is defined as an outlier.
Find these two outlier limits. A housefly is selected at random.
Using standardisation, show that the probability that this housefly is not an outlier is 0.993 to 3 decimal places. Given that this housefly is not an outlier,
showing your working, find the probability that the wing length of this housefly is greater than 5 mm .

Edexcel S1 2023 January Q6

14 marks Moderate -0.3

A research student is investigating the maximum weight, $y$ grams, of sugar that will dissolve in 100 grams of water at various temperatures, $x ^ { \circ } \mathrm { C }$, where $10 \leqslant x \leqslant 80$

The research student calculated the regression line of $y$ on $x$ and found it to be $$y = 151.2 + 2.72 x$$

Give an interpretation of the gradient of the regression line.
Use the regression line to estimate the maximum weight of sugar that will dissolve in 100 grams of water when the temperature is $90 ^ { \circ } \mathrm { C }$.
Comment on the reliability of your estimate, giving a reason for your answer. Using the regression line of $y$ on $x$ and the following summary statistics $$\sum y = 3119 \quad \sum y ^ { 2 } = 851093 \quad \sum x ^ { 2 } = 24500 \quad n = 12$$
show that the product moment correlation coefficient for these data is 0.988 to 3 decimal places. The research student's supervisor plotted the original data on a scatter diagram, shown on page 23 With reference to both the scatter diagram and the correlation coefficient,
discuss the suitability of a linear regression model to describe the relationship between $x$ and $y$.
\includegraphics[max width=\textwidth, alt={}]{c316fa29-dedc-4890-bd82-31eb0bb819f9-23_990_1138_205_356}

Edexcel S1 2024 January Q1

8 marks Moderate -0.8

The histogram below shows the distribution of the heights, to the nearest cm , of 408 plants.
\includegraphics[max width=\textwidth, alt={}, center]{86446ce3-496a-4f02-9566-9b207bac9efa-02_1001_1473_340_296}
1. Use the histogram to complete the following table.
Height $( h$ cm) $5 \leqslant h < 9$ $9 \leqslant h < 13$ $13 \leqslant h < 15$ $15 \leqslant h < 17$ $17 \leqslant h < 25$
Frequency 32 152 120
Use interpolation to estimate the median. The mean height of these plants is 13.2 cm correct to one decimal place.
Describe the skew of these data. Give a reason for your answer. Two of these plants are chosen at random.
Estimate the probability that both of their heights are between 8 cm and 14 cm

Edexcel S1 2024 January Q2

12 marks Moderate -0.3

The average minimum monthly temperature, $x$ degrees Fahrenheit ( ${ } ^ { \circ } \mathrm { F }$ ), and the average maximum monthly temperature, $y$ degrees Fahrenheit ( ${ } ^ { \circ } \mathrm { F }$ ), in Kolkata were recorded for 12 months.

Some of the summary statistics are given below. $$\sum x = 862 \quad \sum x ^ { 2 } = 62802 \quad \mathrm {~S} _ { y y } = 413.67 \quad S _ { x y } = 512.67 \quad n = 12$$

1. Calculate the mean of the 12 values of the average minimum
  monthly temperature.
2. Show that the standard deviation of the 12 values of the average minimum monthly temperature is $8.57 ^ { \circ } \mathrm { F }$ to 3 significant figures.
Calculate the product moment correlation coefficient between $x$ and $y$ For comparative purposes with a UK city, it was necessary to convert the temperatures from degrees Fahrenheit ( ${ } ^ { \circ } \mathrm { F }$ ) to degrees Celsius ( ${ } ^ { \circ } \mathrm { C }$ ). The formula used was $$c = \frac { 5 } { 9 } ( f - 32 )$$ where $f$ is the temperature in ${ } ^ { \circ } \mathrm { F }$ and $c$ is the temperature in ${ } ^ { \circ } \mathrm { C }$
Use this formula and the values from part (a) to calculate, in ${ } ^ { \circ } \mathrm { C }$, the mean and the standard deviation of the 12 values of the average minimum monthly temperature in Kolkata.
Give your answers to 3 significant figures. Given that
- $u$ is the equivalent temperature in ${ } ^ { \circ } \mathrm { C }$ of $x$
- $\quad v$ is the equivalent temperature in ${ } ^ { \circ } \mathrm { C }$ of $y$
- state, giving a reason, the product moment correlation coefficient between $u$ and $v$

Edexcel S1 2024 January Q3

8 marks Easy -1.3

In a sixth form college each student in Year 12 and Year 13 is either left-handed (L) or right-handed (R).

The partially completed tree diagram, where $p$ is a probability, gives information about these students.
\includegraphics[max width=\textwidth, alt={}, center]{86446ce3-496a-4f02-9566-9b207bac9efa-10_960_981_477_543}

Complete the tree diagram, in terms of $p$ where necessary. The probability that a student is left-handed is 0.11
Find the value of $p$
Find the probability that a student selected at random is in Year 12 and left-handed. Given that a student is right-handed,
find the probability that the student is in Year 12

Edexcel S1 2024 January Q4

12 marks Moderate -0.8

A French test and a Spanish test were sat by 11 students.

The table below shows their marks.

Student	A	B	C	D	E	F	G	H	I	J	K
French mark ( f )	24	30	32	32	36	36	40	44	50	60	68
Spanish mark ( $\boldsymbol { s }$ )	16	90	24	28	32	36	38	44	48	48	68

Greg says that if these points were plotted on a scatter diagram, then the point $( 30,90 )$ would be an outlier because 90 is an outlier for the Spanish marks. An outlier is defined as a value that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { or smaller than } Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$

Show that 90 is an outlier for the Spanish marks. Ignoring the point (30, 90), Greg calculated the following summary statistics. $$\sum f = 422 \quad \sum s = 382 \quad S _ { f f } = 1667.6 \quad S _ { f s } = 1735.6$$
Use these summary statistics to show that the equation of the least squares regression line of $s$ on $f$ for the remaining 10 students is $$s = - 5.72 + 1.04 f$$ where the values of the intercept and gradient are given to 3 significant figures. You must show your working.
Give an interpretation of the gradient of the regression line. Two further students sat the French test but missed the Spanish test.
Using the equation given in part (b), estimate
1. a Spanish mark for the student who scored 55 marks in their French test,
2. a Spanish mark for the student who scored 18 marks in their French test.
State, giving a reason, which of the two estimates found in part (d) would be the more reliable estimate.

Edexcel S1 2024 January Q5

7 marks Standard +0.8

The distance an athlete can throw a discus is normally distributed with mean 40 m and standard deviation 4 m
1. Using standardisation, show that the probability that this athlete throws the discus less than 38.8 m is 0.3821
This athlete enters a discus competition.
To qualify for the final, they have 3 attempts to throw the discus a distance of more than 38.8 m
Once they qualify, they do not use any of their remaining attempts.
Given that they qualified for the final and that throws are independent,
find the probability that this athlete qualified for the final on their second throw with a distance of more than 44 m

Edexcel S1 2024 January Q6

9 marks Standard +0.3

The events $A$ and $B$ satisfy

$$\mathrm { P } ( A ) = x \quad \mathrm { P } ( B ) = y \quad \mathrm { P } ( A \cup B ) = 0.65 \quad \mathrm { P } ( B \mid A ) = 0.3$$

Show that $$14 x + 20 y = 13$$ The events $B$ and $C$ are mutually exclusive such that $$\mathrm { P } ( B \cup C ) = 0.85 \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } x + y$$
1. Find a second equation in $x$ and $y$
2. Hence find the value of $x$ and the value of $y$
Determine whether or not $A$ and $B$ are statistically independent. You must show your working clearly.

Edexcel S1 2024 January Q7

10 marks Moderate -0.3

The cumulative distribution of a discrete random variable $X$ is given by

$x$	1	2	3	4
$\mathrm {~F} ( x )$	$\frac { 1 } { 13 }$	$\frac { 2 k - 1 } { 26 }$	$\frac { 3 ( k + 1 ) } { 26 }$	$\frac { k + 4 } { 8 }$

where $k$ is a positive constant.

Show that $k = 4$
Find the probability distribution of the discrete random variable $X$
Using your answer to part (b), write down the mode of $X$
Calculate $\operatorname { Var } ( 13 X - 6 )$

Edexcel S1 2024 January Q8

9 marks Standard +0.8

The random variable $X$ is normally distributed with mean $\mu$ and variance 36

Given that $$\mathrm { P } ( \mu - 2 k < X < \mu + 2 k ) = 0.6$$

find the value of $k$ The random variable $Y$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ Given that $$2 \mu = 3 \sigma ^ { 2 } \quad \text { and } \quad \mathrm { P } \left( \mathrm { Y } > \frac { 3 } { 2 } \mu \right) = 0.0668$$
find the value of $\mu$ and the value of $\sigma$

Edexcel S1 2014 June Q1

12 marks Moderate -0.8

A medical researcher is studying the relationship between age ( $x$ years) and volume of blood ( $y \mathrm { ml }$ ) pumped by each contraction of the heart. The researcher obtained the following data from a random sample of 8 patients.

Age (x)	20	25	30	45	55	60	65	70
Volume (y)	74	76	77	72	68	67	64	62

[You may use $\sum x = 370 , \mathrm {~S} _ { x x } = 2587.5 , \sum y = 560 , \sum y ^ { 2 } = 39418 , \mathrm {~S} _ { x y } = - 710$ ]

Calculate $\mathrm { S } _ { y y }$
Calculate the product moment correlation coefficient for these data.
Interpret your value of the correlation coefficient. The researcher believes that a linear regression model may be appropriate to describe these data.
State, giving a reason, whether or not your value of the correlation coefficient supports the researcher's belief.
Find the equation of the regression line of $y$ on $x$, giving your answer in the form $y = a + b x$ Jack is a 40-year-old patient.
1. Use your regression line to estimate the volume of blood pumped by each contraction of Jack's heart.
2. Comment, giving a reason, on the reliability of your estimate.

Edexcel S1 2014 June Q2

14 marks Moderate -0.8

The table below shows the distances (to the nearest km ) travelled to work by the 50 employees in an office.

Distance (km)	Frequency (f)	Distance midpoint (x)
0-2	16	1.25
3-5	12	4
6-10	10	8
11-20	8	15.5
21-40	4	30.5

$$\text { [You may use } \left. \sum \mathrm { f } x = 394 , \quad \sum \mathrm { f } x ^ { 2 } = 6500 \right]$$ A histogram has been drawn to represent these data.
The bar representing the distance of $3 - 5$ has a width of 1.5 cm and a height of 6 cm .

Calculate the width and height of the bar representing the distance of 6-10
Use linear interpolation to estimate the median distance travelled to work.
1. Show that an estimate of the mean distance travelled to work is 7.88 km .
2. Estimate the standard deviation of the distances travelled to work.
Describe, giving a reason, the skewness of these data. Peng starts to work in this office as the $51 ^ { \text {st } }$ employee.
She travels a distance of 7.88 km to work.
Without carrying out any further calculations, state, giving a reason, what effect Peng's addition to the workforce would have on your estimates of the
1. mean,
2. median,
3. standard deviation
  of the distances travelled to work.

Edexcel S1 2014 June Q3

7 marks Easy -1.3

A biased four-sided die has faces marked $1,3,5$ and 7 . The random variable $X$ represents the score on the die when it is rolled. The cumulative distribution function of $X , \mathrm {~F} ( x )$, is given in the table below.

$x$	1	3	5	7
$\mathrm {~F} ( x )$	0.2	0.5	0.9	1

Find the probability distribution of $X$
Find $\mathrm { P } ( 2 < X \leqslant 6 )$
Write down the value of $\mathrm { F } ( 4 )$

Age													Key: \(4 \mid 3\) is an age of 43
4	3											\(( 1 )\)
5	4
6	2	3	5	6	8	8	8	9	9			\(( 1 )\)
7	1	1	3	4	4	6	6	6	8	8	9	\(( 9 )\)
8	0	0	2	7	8	8	9					\(( 11 )\)
9	3	7

\(y\)	1	2	3	4	5
\(\mathrm { P } ( Y = y )\)	\(\frac { 4 } { 30 }\)	\(\frac { 9 } { 30 }\)	\(\frac { 6 } { 30 }\)	\(\frac { 5 } { 30 }\)	\(\frac { 6 } { 30 }\)

Time taken ( \(\boldsymbol { t }\) minutes)	\(5 - 10\)	\(10 - 14\)	\(14 - 18\)	\(18 - 25\)	\(25 - 40\)
Frequency ( \(\boldsymbol { f }\) )	10	16	24

\(x\)	2	3	4
\(\mathrm { P } ( X = x )\)	\(a\)	0.4	\(0.6 - a\)

Height \(( h\) cm)	\(5 \leqslant h < 9\)	\(9 \leqslant h < 13\)	\(13 \leqslant h < 15\)	\(15 \leqslant h < 17\)	\(17 \leqslant h < 25\)
Frequency	32	152	120

Student	A	B	C	D	E	F	G	H	I	J	K
French mark ( f )	24	30	32	32	36	36	40	44	50	60	68
Spanish mark ( \(\boldsymbol { s }\) )	16	90	24	28	32	36	38	44	48	48	68

\(x\)	1	2	3	4
\(\mathrm {~F} ( x )\)	\(\frac { 1 } { 13 }\)	\(\frac { 2 k - 1 } { 26 }\)	\(\frac { 3 ( k + 1 ) } { 26 }\)	\(\frac { k + 4 } { 8 }\)

\(s\)	0	1	2	3
\(\mathrm { P } ( S = s )\)		0.432		0.064

\(t\)	0	1	2	3
\(\mathrm { P } ( T = t )\)		0.45		0.055

Questions S1 (1967 questions)

Browse by module

Browse by board

Edexcel S1 2019 January Q5

Edexcel S1 2019 January Q6

Edexcel S1 2021 January Q1

Edexcel S1 2021 January Q2

Edexcel S1 2021 January Q3

Edexcel S1 2021 January Q4

Edexcel S1 2021 January Q5

Edexcel S1 2021 January Q6

Edexcel S1 2023 January Q1

Edexcel S1 2023 January Q2

Edexcel S1 2023 January Q3

Edexcel S1 2023 January Q4

Edexcel S1 2023 January Q5

Edexcel S1 2023 January Q6

Edexcel S1 2024 January Q1

Edexcel S1 2024 January Q2

Edexcel S1 2024 January Q3

Edexcel S1 2024 January Q4

Edexcel S1 2024 January Q5

Edexcel S1 2024 January Q6

Edexcel S1 2024 January Q7

Edexcel S1 2024 January Q8

Edexcel S1 2014 June Q1

Edexcel S1 2014 June Q2

Edexcel S1 2014 June Q3