Questions S1 - A-Level Maths

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
8	15	25	29	31	31	34	36	34	26	15	8

Construct a sorted stem and leaf diagram to represent these data, taking stem values of $0,10 , \ldots$.
Write down the median of these data.
The mean of these data is 24.3. Would the mean or the median be a better measure of central tendency of the data? Briefly explain your answer.

OCR MEI S1 Q3

3 marks Easy -1.2

3 The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas.

8	3	9
9	3	5	6	6	6	8	9	9
10	0	2	2	3	4	6	9
11	2	4	7	8
12	3	4	5
13	2

Key: 11 | 2 represents 1120 grams

Find the mode and the median of the data.
Identify the type of skewness of the distribution.

OCR MEI S1 Q4

8 marks Easy -1.2

4 A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$\begin{array} { l l l l l l l l l l l l l l l } 73 & 67 & 75 & 64 & 52 & 63 & 75 & 81 & 77 & 72 & 68 & 74 & 79 & 72 & 71 \end{array}$$

Construct a sorted stem and leaf diagram to represent these data, taking stem values of $50,60 , \ldots$.
Write down the median and midrange of the data.
Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer.

OCR MEI S1 Q5

5 marks Easy -1.8

5 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.

Number of people	1	2	3	4
Frequency	50	31	16	5

Write down the median and mode of these data.
Draw a vertical line diagram for these data.
State the type of skewness of the distribution.

OCR MEI S1 Q6

7 marks Easy -1.2

6 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a $\pounds 10$ prize, 20 of them have a $\pounds 100$ prize, one of them has a $\pounds 5000$ prize and all of the rest have no prize. This information is summarised in the frequency table below.

Prize money	$\pounds 0$	$\pounds 10$	$\pounds 100$	$\pounds 5000$
Frequency	9929	50	20	1

Find the mean and standard deviation of the prize money per ticket.
I buy two of these tickets at random. Find the probability that I win either two $\pounds 10$ prizes or two $\pounds 100$ prizes.

OCR MEI S1 Q7

20 marks Moderate -0.8

7 The histogram shows the age distribution of people living in Inner London in 2001.
\includegraphics[max width=\textwidth, alt={}, center]{aabf9d8b-5f91-4a3b-bcf8-e46cb45127c4-4_805_1372_392_401} Data sourced from he 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}

State the type of skewness shown by the distribution.
Use the histogram to estimate the number of people aged under 25.
The table below shows the cumulative frequency distribution.
Age 20 30 40 50 65 100
Cumulative frequency (thousands) 660 1240 1810 $a$ 2490 2770
(A) Use the histogram to find the value of $a$.
(B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
Age ( $x$ years) $0 \leqslant x < 20$ $20 \leqslant x < 30$ $30 \leqslant x < 40$ $40 \leqslant x < 50$ $50 \leqslant x < 65$ $65 \leqslant x < 100$
Frequency (thousands) 1120 650 770 590 680 610
Illustrate these data by means of a histogram.
Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
[0pt] [4]

OCR MEI S1 Q7

4 marks Moderate -0.8

7 The histogram shows the age distribution of people living in Inner London in 2001.
\includegraphics[max width=\textwidth, alt={}, center]{93bbc0cf-d3ad-4bc2-a6c6-36a3b8e394a9-4_805_1372_392_401} Data sourced from he 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}

State the type of skewness shown by the distribution.
Use the histogram to estimate the number of people aged under 25.
The table below shows the cumulative frequency distribution.
Age 20 30 40 50 65 100
Cumulative frequency (thousands) 660 1240 1810 $a$ 2490 2770
(A) Use the histogram to find the value of $a$.
(B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
Age ( $x$ years) $0 \leqslant x < 20$ $20 \leqslant x < 30$ $30 \leqslant x < 40$ $40 \leqslant x < 50$ $50 \leqslant x < 65$ $65 \leqslant x < 100$
Frequency (thousands) 1120 650 770 590 680 610
Illustrate these data by means of a histogram.
Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
[0pt] [4]

OCR MEI S1 Q2

7 marks Moderate -0.8

2 The total annual emissions of carbon dioxide, $x$ tonnes per person, for 13 European countries are given below.

Find the mean, median and midrange of these data.
Comment on how useful each of these is as a measure of central tendency for these data, giving a brief reason for each of your answers.

OCR MEI S1 Q3

8 marks Easy -1.2

3 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.

On the insert, draw a cumulative frequency diagram to illustrate the data.
Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
State the type of skewness of the distribution of the data.

OCR MEI S1 Q5

20 marks Moderate -0.3

5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{056d3e9a-088d-4c97-9546-7cecb59b8727-3_815_1628_505_304}

Calculate the number of pears which are between 90 and 100 mm long.
Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
Calculate an estimate of the standard deviation.
Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
Name the type of skewness of the distribution.
Illustrate the data using a cumulative frequency diagram.

OCR MEI S1 Q6

6 marks Easy -1.2

6 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.

Time $( t$ minutes $)$	$0 \leqslant t < 5$	$5 \leqslant t < 10$	$10 \leqslant t < 20$	$20 \leqslant t < 30$	$30 \leqslant t < 40$	$40 \leqslant t < 60$
Frequency	34	153	188	73	27	5

Illustrate these data by means of a histogram.
Identify the type of skewness of the distribution.

OCR MEI S1 Q5

22 marks Moderate -0.3

5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{99c502aa-2c9f-461d-9dc0-ed55e3df32a2-3_815_1628_505_304}

Calculate the number of pears which are between 90 and 100 mm long.
Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
Calculate an estimate of the standard deviation.
Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
Name the type of skewness of the distribution.
Illustrate the data using a cumulative frequency diagram.

Edexcel S1 2014 January Q2

8 marks Moderate -0.8

2. A rugby club coach uses club records to take a random sample of 15 players from 1990 and an independent random sample of 15 players from 2010. The body weight of each player was recorded to the nearest kg and the results from 2010 are summarised in the table below.

Body weight (kg)	75-79	80-84	85-89	90-94	95-99	100-104	105-109
Number of Players (2010)	1	2	2	4	3	2	1

Find the estimated values in kg of the summary statistics $a$, $b$ and $c$ in the table below.
Estimate in 1990 Estimate in 2010
Mean 83.0 $a$
Median 82.0 $b$
Variance 44.0 $c$
Give your answers to 3 significant figures. The rugby coach claims that players’ body weight increased between 1990 and 2010.
Using the table in part (a), comment on the rugby coach's claim. \includegraphics[max width=\textwidth, alt={}, center]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-05_104_97_2613_1784}

Edexcel S1 2014 January Q3

11 marks Moderate -0.8

3. Jean works for an insurance company. She randomly selects 8 people and records the price of their car insurance, $\pounds p$, and the time, $t$ years, since they passed their driving test. The data is shown in the table below.

$t$	10	13	17	18	22	24	25	27
$p$	720	650	430	490	500	390	280	300

$$\text { (You may use } \bar { t } = 19.5 , \bar { p } = 470 , S _ { t p } = - 6080 , S _ { t t } = 254 , S _ { p p } = 169200 \text { ) }$$

On the graph below draw a scatter diagram for these data.
Comment on the relationship between $p$ and $t$.
Find the equation of the regression line of $p$ on $t$.
Use your regression equation to estimate the price of car insurance for someone who passed their driving test 20 years ago. Jack passed his test 39 years ago and decides to use Jean's data to predict the price of his car insurance.
Comment on Jack's decision. Give a reason for your answer.
\includegraphics[max width=\textwidth, alt={}, center]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-06_951_1365_1603_294}

Edexcel S1 2014 January Q4

11 marks Moderate -0.3

4. A discrete random variable $X$ has the probability distribution given in the table below, where $a$ and $b$ are constants.

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

Given $\mathrm { E } ( X ) = \frac { 9 } { 5 }$

1. find two simultaneous equations for $a$ and $b$,
2. show that $a = \frac { 1 } { 20 }$ and find the value of $b$.
Specify the cumulative distribution function $\mathrm { F } ( x )$ for $x = - 1,0,1$, 2 and 3
Find $\mathrm { P } ( X < 2.5 )$.
Find $\operatorname { Var } ( 3 - 2 X )$.
\includegraphics[max width=\textwidth, alt={}, center]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-13_90_68_2613_1877}

Edexcel S1 2014 January Q5

10 marks Standard +0.3

5. A group of 100 students are asked if they like folk music, rock music or soul music. \begin{displayquote} All students who like folk music also like rock music No students like both rock music and soul music 75 students do not like soul music 12 students who like rock music do not like folk music 30 students like folk music

Draw a Venn diagram to illustrate this information.
State two of these types of music that are mutually exclusive. \end{displayquote} Find the probability that a randomly chosen student
does not like folk music, rock music or soul music,
likes rock music,
likes folk music or soul music. Given that a randomly chosen student likes rock music,
find the probability that he or she also likes folk music.

Edexcel S1 2014 January Q6

9 marks Moderate -0.3

6. A manufacturer has a machine that fills bags with flour such that the weight of flour in a bag is normally distributed. A label states that each bag should contain 1 kg of flour.

The machine is set so that the weight of flour in a bag has mean 1.04 kg and standard deviation 0.17 kg . Find the proportion of bags that weigh less than the stated weight of 1 kg . The manufacturer wants to reduce the number of bags which contain less than the stated weight of 1 kg . At first she decides to adjust the mean but not the standard deviation so that only $5 \%$ of the bags filled are below the stated weight of 1 kg .
Find the adjusted mean. The manufacturer finds that a lot of the bags are overflowing with flour when the mean is adjusted, so decides to adjust the standard deviation instead to make the machine more accurate. The machine is set back to a mean of 1.04 kg . The manufacturer wants $1 \%$ of bags to be under 1 kg .
Find the adjusted standard deviation. Give your answer to 3 significant figures.

Edexcel S1 2014 January Q7

9 marks Moderate -0.3

7. In a large college, $\frac { 3 } { 5 }$ of the students are male, $\frac { 3 } { 10 }$ of the students are left handed and $\frac { 1 } { 5 }$ of the male students are left handed. A student is chosen at random.

Given that the student is left handed, find the probability that the student is male.
Given that the student is female, find the probability that she is left handed.
Find the probability that the randomly chosen student is male and right handed. Two students are chosen at random.
Find the probability that one student is left handed and one is right handed.

Edexcel S1 2014 January Q8

10 marks Moderate -0.8

8. A manager records the number of hours of overtime claimed by 40 staff in a month. The histogram in Figure 1 represents the results. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-26_1107_1513_406_210} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

Calculate the number of staff who have claimed less than 10 hours of overtime in the month.
Estimate the median number of hours of overtime claimed by these 40 staff in the month.
Estimate the mean number of hours of overtime claimed by these 40 staff in the month. The manager wants to compare these data with overtime data he collected earlier to find out if the overtime claimed by staff has decreased.
State, giving a reason, whether the manager should use the median or the mean to compare the overtime claimed by staff.
(2)

Edexcel S1 2015 January Q1

10 marks Moderate -0.8

The discrete random variable $X$ has probability function $\mathrm { p } ( x )$ and cumulative distribution function $\mathrm { F } ( x )$ given in the table below.

$x$	1	2	3	4	5
$\mathrm { p } ( x )$	0.10	$a$	0.28	$c$	0.24
$\mathrm {~F} ( x )$	0.10	0.26	$b$	0.76	$d$

Write down the value of $d$
Find the values of $a$, $b$ and $c$
Write down the value of $\mathrm { P } ( X > 4 )$ Two independent observations, $X _ { 1 }$ and $X _ { 2 }$, are taken from the distribution of $X$.
Find the probability that $X _ { 1 }$ and $X _ { 2 }$ are both odd. Given that $X _ { 1 }$ and $X _ { 2 }$ are both odd,
find the probability that the sum of $X _ { 1 }$ and $X _ { 2 }$ is 6 Give your answer to 3 significant figures.

Edexcel S1 2015 January Q2

9 marks Moderate -0.8

A sports teacher recorded the number of press-ups done by his students in two minutes. He recorded this information for a Year 7 class and for a Year 11 class.

The back-to-back stem and leaf diagram shows this information.

Totals	Year 7 class		Year 11 class	Totals
(6)	876554	1
(10)	9776544422	2	0569	(4)
(7)	8754330	3	34588	(5)
(5)	99722	4	05679	(5)
(3)	840	5	03556677799	(11)
		6	0333348	(7)

Key: $2 | 4 | 0$ means 42 press-ups for a Year 7 student and 40 press-ups for a Year 11 student

Find the median number of press-ups for each class. For the Year 11 class, the lower quartile is 38 and the upper quartile is 59
Find the lower quartile and the upper quartile for the Year 7 class.
Use the medians and quartiles to describe the skewness of each of the two distributions.
Give two reasons why the normal distribution should not be used to model the number of press-ups done by the Year 11 class.

Edexcel S1 2015 January Q3

9 marks Moderate -0.8

The table shows the price of a bottle of milk, $m$ pence, and the price of a loaf of bread, $b$ pence, for 8 different years.

$m$	29	29	35	39	41	43	44	46
$b$	75	83	91	121	120	126	119	126

(You may use $\mathrm { S } _ { b b } = 3083.875$ and $\mathrm { S } _ { m m } = 305.5$ )

Find the exact value of $\sum b m$
Find $\mathrm { S } _ { b m }$
Calculate the product moment correlation coefficient between $b$ and $m$
Interpret the value of the correlation coefficient. A ninth year is added to the data set. In this year the price of the bottle of milk is 46 pence and the price of a loaf of bread is 175 pence.
Without further calculation, state whether the value of the product moment correlation coefficient will increase, decrease or stay the same when all nine years are used. Give a reason for your answer.

Edexcel S1 2015 January Q4

9 marks Standard +0.3

4. Events $A$ and $B$ are shown in the Venn diagram below
where $x , y , 0.10$ and 0.32 are probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{c58f3e88-2dbc-40d6-a966-a5765a7c67ba-08_467_798_408_575}

Find an expression in terms of $x$ for
1. $\mathrm { P } ( A )$
2. $\mathrm { P } ( B \mid A )$
Find an expression in terms of $x$ and $y$ for $\mathrm { P } ( A \cup B )$ Given that $\mathrm { P } ( A ) = 2 \mathrm { P } ( B )$
find the value of $x$ and the value of $y$

Edexcel S1 2015 January Q5

13 marks Easy -1.3

The resting heart rate, $h$ beats per minute (bpm), and average length of daily exercise, $t$ minutes, of a random sample of 8 teachers are shown in the table below.

$t$	20	35	40	25	45	70	75	90
$h$	88	85	77	75	71	66	60	54

State, with a reason, which variable is the response variable. The equation of the least squares regression line of $h$ on $t$ is $$h = 93.5 - 0.43 t$$
Give an interpretation of the gradient of this regression line.
Find the value of $\bar { t }$ and the value of $\bar { h }$
Show that the point $( \bar { t } , \bar { h } )$ lies on the regression line.
Estimate the resting heart rate of a teacher with an average length of daily exercise of 1 hour.
Comment, giving a reason, on the reliability of the estimate in part (e). The resting heart rate of teachers is assumed to be normally distributed with mean 73 bpm and standard deviation 8 bpm . The middle $95 \%$ of resting heart rates of teachers lies between $a$ and $b$
Find the value of $a$ and the value of $b$.

Edexcel S1 2015 January Q6

14 marks Moderate -0.8

The random variable $X$ has probability function

$$\mathrm { P } ( X = x ) = \frac { x ^ { 2 } } { k } \quad x = 1,2,3,4$$

Show that $k = 30$
Find $\mathrm { P } ( X \neq 4 )$
Find the exact value of $\mathrm { E } ( X )$
Find the exact value of $\operatorname { Var } ( X )$ Given that $Y = 3 X - 1$
find $\mathrm { E } \left( Y ^ { 2 } \right)$

Time \(( t\) minutes \()\)	\(0 \leqslant t < 5\)	\(5 \leqslant t < 10\)	\(10 \leqslant t < 20\)	\(20 \leqslant t < 30\)	\(30 \leqslant t < 40\)	\(40 \leqslant t < 60\)
Frequency	34	153	188	73	27	5

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

\(x\)	1	2	3	4	5
\(\mathrm { p } ( x )\)	0.10	\(a\)	0.28	\(c\)	0.24
\(\mathrm {~F} ( x )\)	0.10	0.26	\(b\)	0.76	\(d\)

Prize money	\(\pounds 0\)	\(\pounds 10\)	\(\pounds 100\)	\(\pounds 5000\)
Frequency	9929	50	20	1

Age	20	30	40	50	65	100
Cumulative frequency (thousands)	660	1240	1810	\(a\)	2490	2770

Age ( \(x\) years)	\(0 \leqslant x < 20\)	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 65\)	\(65 \leqslant x < 100\)
Frequency (thousands)	1120	650	770	590	680	610

	Estimate in 1990	Estimate in 2010
Mean	83.0	\(a\)
Median	82.0	\(b\)
Variance	44.0	\(c\)

\(t\)	20	35	40	25	45	70	75	90
\(h\)	88	85	77	75	71	66	60	54

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Edexcel S1 2014 January Q2

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