Questions - Edexcel S1 - A-Level Maths

$x$	4	5	6	7	8	9
$\mathrm { P } ( X = x )$	$p$	0.1	$q$	$q$	0.3	0.2

It is known that $\mathrm { E } ( X ) = 6 \cdot 7$. Find

the values of $p$ and $q$,
the value of $a$ for which $\mathrm { E } ( 2 X + a ) = 0$,
$\operatorname { Var } ( X )$. \section*{STATISTICS 1 (A) TEST PAPER 9 Page 2}

Edexcel S1 Q6

The marks out of 75 obtained by a group of ten students in their first and second Statistics modules were as follows:

Student	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
Module 1 $( x )$	54	33	42	71	60	27	39	46	59	64
Module 2 $( y )$	50	22	44	58	42	19	35	46	55	60

Find $\sum x$ and $\sum y$. Given that $\sum x ^ { 2 } = 26353$ and $\sum x y = 22991$,
obtain the equation of the regression line of $y$ on $x$.
Estimate the Module 2 result of a student whose mark in Module 1 was Explain why one of these estimates is less reliable than the other. The equation of the regression line of $x$ on $y$ is $x = 0.921 y + 9.81$.
Deduce the product moment correlation coefficient between $x$ and $y$, and briefly interpret its value.

Edexcel S1 Q7

7. Among the families with two children in a large city, the probability that the elder child is a boy is $\frac { 5 } { 12 }$ and the probability that the younger child is a boy is $\frac { 9 } { 16 }$. The probability that the younger child is a girl, given that the elder child is a girl, is $\frac { 1 } { 4 }$.
One of the families is chosen at random. Using a tree diagram, or otherwise,

show that the probability that both children are boys is $\frac { 1 } { 8 }$. Find the probability that
one child is a boy and the other is a girl,
one child is a boy given that the other is a girl. If three of the families are chosen at random,
find the probability that exactly two of the families have two boys.
State an assumption that you have made in answering part (d).

Edexcel S1 Q1

Given that $\mathrm { P } ( A \cup B ) = 0.65 , \mathrm { P } ( A \cap B ) = 0.15$ and $\mathrm { P } ( A ) = 0.3$, determine, with explanation, whether or not the events $A$ and $B$ are
1. mutually exclusive,
2. independent.
3. (a) Give one example in each case of a quantity which could be modelled as
  1. a discrete random variable,
  2. a continuous random variable.
4. Name one discrete distribution and one continuous distribution, stating clearly which is which.
5. A regular tetrahedron has its faces numbered $1,2,3$ and 4 . It is weighted so that when it is thrown, the probability of each face being in contact with the table is inversely proportional to the number on that face. This number is represented by the random variable $X$.
6. Show that $\mathrm { P } ( X = 1 ) = \frac { 12 } { 25 }$ and find the probabilities of the other values of $X$.
7. Calculate the mean and the variance of $X$.
8. The random variable $X$ is normally distributed with mean 17 . The probability that $X$ is less than 16 is 0-3707.
9. Calculate the standard deviation of $X$.
10. In 75 independent observations of $X$, how many would you expect to be greater than 20?
11. The students in a large Sixth Form can choose to do exactly one of Community Service, Games or Private Study on Wednesday afternoons. The probabilities that a randomly chosen student does Games and Private Study are $\frac { 3 } { 8 }$ and $\frac { 1 } { 5 }$ respectively. It may be assumed that the number of students is large enough for these probabilities to be treated as constant.
12. Find the probability that a randomly chosen student does Community Service.
13. If two students are chosen at random, find the probability that they both do the same activity.
14. If three students are chosen at random, find the probability that exactly one of them does Games.
Two-fifths of the students are girls, and a quarter of these girls do Private Study.
Find the probability that a randomly chosen student who does Private Study is a boy. \section*{STATISTICS 1 (A)TEST PAPER 10 Page 2}

Edexcel S1 Q6

Two variables $x$ and $y$ are such that, for a sample of ten pairs of values,

$$\sum x = 104 \cdot 5 , \quad \sum y = 113 \cdot 6 , \quad \sum x ^ { 2 } = 1954 \cdot 1 , \sum y ^ { 2 } = 2100 \cdot 6 .$$ The regression line of $x$ on $y$ has gradient 0.8 . Find

$\sum x y$,
the equation of the regression line of $y$ on $x$,
the product moment correlation coefficient between $y$ and $x$.
Describe the kind of correlation indicated by your answer to (c).

Edexcel S1 Q7

7. The following table gives the weights, in grams, of 60 items delivered to a company in a day.

Weight (g)	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 80$
No. of items	2	11	18	12	9	6	2

Use interpolation to calculate estimated values of (i) the median weight,
(ii) the interquartile range,
(iii) the thirty-third percentile. Outliers are defined to be outside the range from $2 \cdot 5 Q _ { 1 } - 1 \cdot 5 Q _ { 3 }$ to $2 \cdot 5 Q _ { 3 } - 1 \cdot 5 Q _ { 1 }$.
Given that the lightest item weighed 3 g and the two heaviest weighed 65 g and 79 g , draw on graph paper an accurate box-and-whisker plot of the data. Indicate any outliers clearly.
Describe the skewness of the distribution. The mean weight was 32.0 g and the standard deviation of the weights was 14.9 g .
State, with a reason, whether you would choose to summarise the data by using the mean and standard deviation or the median and interquartile range. On another day, items were delivered whose weights ranged from 14 g to 58 g ; the median was 32 g , the lower quartile was 24 g and the interquartile range was 26 g .
Draw a further box plot for these data on the same diagram. Briefly compare the two sets of data using your plots.
( 6 marks)

Edexcel S1 Q1

An athlete believes that her times for running 200 metres in races are normally distributed with a mean of 22.8 seconds.
1. Given that her time is over 23.3 seconds in $20 \%$ of her races, calculate the variance of her times.
2. The record over this distance for women at her club is 21.82 seconds. According to her model, what is the chance that she will beat this record in her next race?
  (3 marks)
3. The events $A$ and $B$ are such that
$$\mathrm { P } ( A ) = \frac { 5 } { 16 } , \mathrm { P } ( B ) = \frac { 1 } { 2 } \text { and } \mathrm { P } ( A \mid B ) = \frac { 1 } { 4 }$$ Find
$\mathrm { P } ( A \cap B )$,
$\mathrm { P } \left( B ^ { \prime } \mid A \right)$,
$\mathrm { P } \left( A ^ { \prime } \cup B \right)$,
Determine, with a reason, whether or not the events $A$ and $B$ are independent.

Edexcel S1 Q3

3. A group of 60 children were each asked to choose an integer value between 1 and 9 inclusive. Their choices are summarised in the table below.

Value chosen	1	2	3	4	5	6	7	8	9
Number of children	3	4	5	10	12	13	7	4	2

Calculate the mean and standard deviation of the values chosen. It is suggested that the value chosen could be modelled by a discrete uniform distribution.
Write down the mean that this model would predict. Given also that the standard deviation according to this model would be 2.58,
explain why this model is not suitable and suggest why this is the case.

Edexcel S1 Q4

4. A six-sided die is biased such that there is an equal chance of scoring each of the numbers from 1 to 5 but a score of 6 is three times more likely than each of the other numbers.

Write down the probability distribution for the random variable, $X$, the score on a single throw of the die.
Show that $\mathrm { E } ( X ) = \frac { 33 } { 8 }$.
Find $\mathrm { E } ( 4 X - 1 )$.
Find $\operatorname { Var } ( X )$.

Edexcel S1 Q5

5. The number of patients attending a hospital trauma clinic each day was recorded over several months, giving the data in the table below.

Number of patients	$10 - 19$	$20 - 29$	$30 - 34$	$35 - 39$	$40 - 44$	$45 - 49$	$50 - 69$
Frequency	2	18	24	30	27	14	5

These data are represented by a histogram.
Given that the bar representing the 20-29 group is 2 cm wide and 7.2 cm high,

calculate the dimensions of the bars representing the groups
1. 30-34
2. 50-69
Use linear interpolation to estimate the median and quartiles of these data. The lowest and highest numbers of patients recorded were 14 and 67 respectively.
Represent these data with a boxplot drawn on graph paper and describe the skewness of the distribution.

Edexcel S1 Q6

6. Penshop have stores selling stationary in each of 6 towns. The population, $P$, in tens of thousands and the monthly turnover, $T$, in thousands of pounds for each of the shops are as recorded below.

Town	Abberton	Bember	Claster	Deller	Edgeton	Figland
$P$ (0000's)	3.2	7.6	5.2	9.0	8.1	4.8
T (£ 000's)	11.1	12.4	13.3	19.3	17.9	11.8

Represent these data on a scatter diagram with $T$ on the verical axis.
1. Which town's shop might appear to be underachieving given the populations of the towns?
2. Suggest two other factors that might affect each shop's turnover. You may assume that $$\Sigma P = 37.9 , \quad \Sigma T = 85.8 , \quad \Sigma P ^ { 2 } = 264.69 , \quad \Sigma T ^ { 2 } = 1286 , \quad \Sigma P T = 574.25 .$$
Find the equation of the regression line of $T$ on $P$.
Estimate the monthly turnover that might be expected if a shop were opened in Gratton, a town with a population of 68000.
Why might the management of Penshop be reluctant to use the regression line to estimate the monthly turnover they could expect if a shop were opened in Haggin, a town with a population of 172000 ?

Edexcel S1 Q1

An adult evening class has 14 students. The ages of these students have a mean of 31.2 years and a standard deviation of 7.4 years.

A new student who is exactly 42 years old joins the class. Calculate the mean and standard deviation of the 15 students now in the group.

Edexcel S1 Q2

2. A tennis coach believes that taller players are generally capable of hitting faster serves. To investigate this hypothesis he collects data on the 20 adult male players he coaches. The height, $h$, in metres and the speed of each player's fastest serve, $v$, in miles per hour were recorded and summarised as follows: $$\Sigma h = 36.22 , \quad \Sigma v = 2275 , \quad \Sigma h ^ { 2 } = 65.7396 , \quad \Sigma v ^ { 2 } = 259853 , \quad \Sigma h v = 4128.03 .$$

Calculate the product moment correlation coefficient for these data.
Comment on the coach's hypothesis.

Edexcel S1 Q3

3. The events $A$ and $B$ are such that $$\mathrm { P } ( A ) = 0.2 \text { and } \mathrm { P } ( A \cup B ) = 0.6$$ Find

$\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)$,
$\quad \mathrm { P } \left( A ^ { \prime } \cap B \right)$. Given also that events $A$ and $B$ are independent, find
$\mathrm { P } ( B )$,
$\mathrm { P } \left( A ^ { \prime } \cup B ^ { \prime } \right)$.

Edexcel S1 Q4

4. The discrete random variable $X$ has the following probability distribution.

$x$	1	2	3	4	5
$\mathrm { P } ( X = x )$	0.1	0.35	$k$	0.15	$k$

Calculate

$k$,
$\mathrm { F } ( 2 )$,
$\mathrm { P } ( 1.3 < X \leq 3.8 )$,
$\mathrm { E } ( X )$,
$\operatorname { Var } ( 3 X + 2 )$.

Edexcel S1 Q5

5. For a project, a student asked 40 people to draw two straight lines with what they thought was an angle of $75 ^ { \circ }$ between them, using just a ruler and a pencil. She then measured the size of the angles that had been drawn and her data are summarised in this stem and leaf diagram.

Angle	( $6 \mid 4$ means $64 ^ { \circ }$ )	Totals
4	1	(1)
4		(0)
5	024	(3)
5	589	(3)
6	11334	(5)
6	55789	(5)
7	011233444	(9)
7	5667799	(7)
8	01134	(5)
8	56	(2)

Find the median and quartiles of these data. Given that any values outside of the limits $\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)$ and $\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)$ are to be regarded as outliers,
determine if there are any outliers in these data,
draw a box plot representing these data on graph paper,
describe the skewness of the distribution and suggest a reason for it.

Edexcel S1 Q6

6. The individual letters of the word STATISTICAL are written on 11 cards which are then shuffled. One card is selected at random. Find the probability that it is

a vowel,
a T, given that it is a consonant. The 11 cards are then shuffled again and the top three are turned over. Find the probability that
all three of the cards have a T on them,
at least two of the cards show a vowel.

Edexcel S1 Q7

7. The volume of liquid in bottles of sparkling water from one producer is believed to be normally distributed with a mean of 704 ml and a variance of $3.2 \mathrm { ml } ^ { 2 }$. Calculate the probability that a randomly chosen bottle from this producer contains

more than 706 ml ,
between 703 and 708 ml . The bottles are labelled as containing 700 ml .
In a delivery of 1200 bottles, how many could be expected to contain less than the stated 700 ml ? The bottling process can be adjusted so that the mean changes but the variance is unchanged.
What should the mean be changed to in order to have only a $0.1 \%$ chance of a bottle having less than 700 ml of sparkling water? Give your answer correct to 1 decimal place.

Edexcel S1 Q1

(a) Explain briefly what you understand by a statistical model.
(2 marks)
A zoologist is analysing data on the weights of adult female otters.
(b) Name a distribution that you think might be suitable for modelling such data.
(1 mark)
(c) Describe two features that you would expect to find in the distribution of the weights of adult female otters and that led to your choice in part (b).
(2 marks)
(d) Why might your choice in part (b) not be suitable for modelling the weights of all adult otters?
(1 mark)
For a geography project a student studied weather records kept by her school since 1993. To see if there was any evidence of global warming she worked out the mean temperature in degrees Celsius at noon for the month of June in each year.

Her results are shown in the table below.

$\left( { } ^ { \circ } \mathrm { C } \right)$

Edexcel S1 Q3

3. In a study of 120 pet-owners it was found that 57 owned at least one dog and of these 16 also owned at least one cat. There were 35 people in the group who didn't own any cats or dogs. As an incentive to take part in the study, one participant is chosen at random to win a year's free supply of pet food. Find the probability that the winner of this prize

owns a dog but does not own a cat,
owns a cat,
does not own a cat given that they do not own a dog.

Edexcel S1 Q4

4. An internet service provider runs a series of television adverts at weekly intervals. To investigate the effectiveness of the adverts the company recorded the viewing figures in millions, $v$, for the programme in which the advert was shown, and the number of new customers, $c$, who signed up for their service the next day. The results are summarised as follows. $$\bar { v } = 4.92 , \quad \bar { c } = 104.4 , \quad S _ { v c } = 594.05 , \quad S _ { v v } = 85.44 .$$

Calculate the equation of the regression line of $c$ on $v$ in the form $c = a + b v$.
Give an interpretation of the constants $a$ and $b$ in this context.
Estimate the number of customers that will sign up with the company the day after an advert is shown during a programme watched by 3.7 million viewers.
State two other factors besides viewing figures that will affect the success of an advert in gaining new customers for the company.

Edexcel S1 Q5

5. The time taken in minutes, $T$, for a mechanic to service a bicycle follows a normal distribution with a mean of 25 minutes and a variance of 16 minutes $^ { 2 }$. Find

$\mathrm { P } ( T < 28 )$,
$\quad \mathrm { P } ( | T - 25 | < 5 )$. One afternoon the mechanic has 3 bicycles to service.
Find the probability that he will take less than 23 minutes on each of the three bicycles.
(4 marks)

Edexcel S1 Q6

6. The number of people visiting a new art gallery each day is recorded over a three-month period and the results are summarised in the table below.

Number of visitors	Number of days
400-459	3
460-479	8
480-499	13
500-519	12
520-539	18
540-559	11
560-599	9
600-699	5

Draw a histogram on graph paper to illustrate these data. In order to calculate summary statistics for the data it is coded using $y = \frac { x - 509.5 } { 10 }$, where $x$ is the mid-point of each class.
Find $\sum$ fy. You may assume that $\sum f y ^ { 2 } = 2041$.
Using these values for $\sum f y$ and $\sum f y ^ { 2 }$, calculate estimates of the mean and standard deviation of the number of visitors per day.
(6 marks)

Edexcel S1 Q7

7. A bag contains 4 red and 2 blue balls, all of the same size. A ball is selected at random and removed from the bag. This is repeated until a blue ball is pulled out of the bag. The random variable $B$ is the number of balls that have been removed from the bag.

Show that $\mathrm { P } ( B = 2 ) = \frac { 4 } { 15 }$.
Find the probability distribution of $B$.
Find $\mathrm { E } ( B )$. The bag and the same 6 balls are used in a game at a funfair. One ball is removed from the bag at a time and a contestant wins 50 pence if one of the first two balls picked out is blue.
What are the expected winnings from playing this game once? For $\pounds 1$, a contestant gets to play the game three times.
What is the expected profit or loss from the three games?

Edexcel S1 Q1

(a) Draw two separate scatter diagrams, each with eight points, to illustrate the relationship between $x$ and $y$ in the cases where they have a product moment correlation coefficient equal to
1. exactly + 1 ,
2. about ${ } ^ { - } 0.4$.
  (b) Explain briefly how the conclusion you would draw from a product moment correlation coefficient of + 0.3 would vary according to the number of pairs of data used in its calculation.
  (2 marks)
3. A histogram was drawn to show the distribution of age in completed years of the participants on an outward-bound course.
There were 32 people aged 30-34 years on the course. The height of the rectangle representing this group was 19.2 cm and it was 1 cm in width. Given that there were 28 people aged 35-39 years,
(a) find the height of the rectangle representing this group. Given that the height of the rectangle representing people aged 40-59 years was 2.7 cm ,
(b) find the number of people on the course in this age group.

\(x\)	4	5	6	7	8	9
\(\mathrm { P } ( X = x )\)	\(p\)	0.1	\(q\)	\(q\)	0.3	0.2

Weight (g)	\(0 - 10\)	\(10 - 20\)	\(20 - 30\)	\(30 - 40\)	\(40 - 50\)	\(50 - 60\)	\(60 - 80\)
No. of items	2	11	18	12	9	6	2

\(x\)	1	2	3	4	5
\(\mathrm { P } ( X = x )\)	0.1	0.35	\(k\)	0.15	\(k\)

Student	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Module 1 \(( x )\)	54	33	42	71	60	27	39	46	59	64
Module 2 \(( y )\)	50	22	44	58	42	19	35	46	55	60

Number of patients	\(10 - 19\)	\(20 - 29\)	\(30 - 34\)	\(35 - 39\)	\(40 - 44\)	\(45 - 49\)	\(50 - 69\)
Frequency	2	18	24	30	27	14	5

Angle	( \(6 \mid 4\) means \(64 ^ { \circ }\) )	Totals
4	1	(1)
4		(0)
5	024	(3)
5	589	(3)
6	11334	(5)
6	55789	(5)
7	011233444	(9)
7	5667799	(7)
8	01134	(5)
8	56	(2)

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