Questions S1 - A-Level Maths

Caravans			1	10 means 10	Totals
1	0				(2)
2	1	8			(4)
3	0	3	4	7	(8)
4	1	5	8	8	(9)
5	2	6	7		(5)
6	2				(3)

Find the three quartiles of these data. During the same month, the least number of caravans on Northcliffe caravan site was 31. The maximum number of caravans on this site on any night that month was 72 . The three quartiles for this site were 38,45 and 52 respectively.
On graph paper and using the same scale, draw box plots to represent the data for both caravan sites. You may assume that there are no outliers.
Compare and contrast these two box plots.
Give an interpretation to the upper quartiles of these two distributions.

Edexcel S1 2005 January Q3

Easy -1.3

3. The following table shows the height $x$, to the nearest cm , and the weight $y$, to the nearest kg , of a random sample of 12 students.

$x$	148	164	156	172	147	184	162	155	182	165	175	152
$y$	39	59	56	77	44	77	65	49	80	72	70	52

On graph paper, draw a scatter diagram to represent these data.
Write down, with a reason, whether the correlation coefficient between $x$ and $y$ is positive or negative. The data in the table can be summarised as follows. $$\Sigma x = 1962 , \quad \Sigma y = 740 , \quad \Sigma y ^ { 2 } = 47746 , \quad \Sigma x y = 122783 , \quad S _ { x x } = 1745 .$$
Find $S _ { x y }$. The equation of the regression line of $y$ on $x$ is $y = - 106.331 + b x$.
Find, to 3 decimal places, the value of $b$.
Find, to 3 significant figures, the mean $\bar { y }$ and the standard deviation $s$ of the weights of this sample of students.
Find the values of $\bar { y } \pm 1.96 s$.
Comment on whether or not you think that the weights of these students could be modelled by a normal distribution.

Edexcel S1 2005 January Q4

Easy -1.2

4. The random variable $X$ has probability function $$\mathrm { P } ( X = x ) = k x , \quad x = 1,2 , \ldots , 5$$

Show that $k = \frac { 1 } { 15 }$. Find
$\mathrm { P } ( X < 4 )$,
$\mathrm { E } ( X )$,
$\mathrm { E } ( 3 X - 4 )$.

Edexcel S1 2005 January Q5

Moderate -0.8

5. Articles made on a lathe are subject to three kinds of defect, $A , B$ or $C$. A sample of 1000 articles was inspected and the following results were obtained. \begin{displayquote} 31 had a type $A$ defect
37 had a type $B$ defect
42 had a type $C$ defect
11 had both type $A$ and type $B$ defects
13 had both type $B$ and type $C$ defects
10 had both type $A$ and type $C$ defects
6 had all three types of defect.

Draw a Venn diagram to represent these data. \end{displayquote} Find the probability that a randomly selected article from this sample had
no defects,
no more than one of these defects. An article selected at random from this sample had only one defect.
Find the probability that it was a type $B$ defect. Two different articles were selected at random from this sample.
Find the probability that both had type $B$ defects.

Edexcel S1 2005 January Q6

Easy -1.2

6. A discrete random variable is such that each of its values is assumed to be equally likely.

Write down the name of the distribution that could be used to model this random variable.
Give an example of such a distribution.
Comment on the assumption that each value is equally likely.
Suggest how you might refine the model in part (a).

Edexcel S1 2005 January Q7

Standard +0.3

7. The random variable $X$ is normally distributed with mean 79 and variance 144 . Find

$\mathrm { P } ( X < 70 )$,
$\mathrm { P } ( 64 < X < 96 )$. It is known that $\mathrm { P } ( 79 - a \leq X \leq 79 + b ) = 0.6463$. This information is shown in the figure below.
\includegraphics[max width=\textwidth, alt={}, center]{df898ff4-c3ef-400c-b4f7-f4df3757941d-6_581_983_818_590} Given that $\mathrm { P } ( X \geq 79 + b ) = 2 \mathrm { P } ( X \leq 79 - a )$,
show that the area of the shaded region is 0.1179 .
Find the value of $b$.

Edexcel S1 2006 January Q1

Easy -1.3

Over a period of time, the number of people $x$ leaving a hotel each morning was recorded. These data are summarised in the stem and leaf diagram below.

Number leaving				3		2 means 32	Totals
2	7	9	9				(3)
3	2	2	3	5	6		(5)
4	0	1	4	8	9		(5)
5	2	3	3	6	6	6	(7)
6	0	1	4	5			(4)
7	2	3					(2)
8	1						(1)

For these data,

write down the mode,
find the values of the three quartiles. Given that $\Sigma x = 1335$ and $\Sigma x ^ { 2 } = 71801$, find
the mean and the standard deviation of these data. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } } \text {. }$$
Evaluate this measure to show that these data are negatively skewed.
Give two other reasons why these data are negatively skewed.

Edexcel S1 2006 January Q2

Moderate -0.8

2. The random variable $X$ has probability distribution

$x$	1	2	3	4	5
$\mathrm { P } ( X = x )$	0.10	$p$	0.20	$q$	0.30

Given that $\mathrm { E } ( X ) = 3.5$, write down two equations involving $p$ and $q$. Find
the value of $p$ and the value of $q$,
$\operatorname { Var } ( X )$,
$\operatorname { Var } ( 3 - 2 X )$.

Edexcel S1 2006 January Q3

Easy -1.2

3. A manufacturer stores drums of chemicals. During storage, evaporation takes place. A random sample of 10 drums was taken and the time in storage, $x$ weeks, and the evaporation loss, $y \mathrm { ml }$, are shown in the table below.

$x$	3	5	6	8	10	12	13	15	16	18
$y$	36	50	53	61	69	79	82	90	88	96

On graph paper, draw a scatter diagram to represent these data.
Give a reason to support fitting a regression model of the form $y = a + b x$ to these data.
Find, to 2 decimal places, the value of $a$ and the value of $b$. $$\text { (You may use } \Sigma x ^ { 2 } = 1352 , \Sigma y ^ { 2 } = 53112 \text { and } \Sigma x y = 8354 \text {.) }$$
Give an interpretation of the value of $b$.
Using your model, predict the amount of evaporation that would take place after
1. 19 weeks,
2. 35 weeks.
Comment, with a reason, on the reliability of each of your predictions.

Edexcel S1 2006 January Q4

Easy -1.2

4. A bag contains 9 blue balls and 3 red balls. A ball is selected at random from the bag and its colour is recorded. The ball is not replaced. A second ball is selected at random and its colour is recorded.

Draw a tree diagram to represent the information. Find the probability that
the second ball selected is red,
both balls selected are red, given that the second ball selected is red.

Edexcel S1 2006 January Q5

Easy -1.8

5. (a) Write down two reasons for using statistical models.
(b) Give an example of a random variable that could be modelled by

a normal distribution,
a discrete uniform distribution.

Edexcel S1 2006 January Q6

Standard +0.3

6. For the events $A$ and $B$, $$\mathrm { P } \left( A \cap B ^ { \prime } \right) = 0.32 , \mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.11 \text { and } \mathrm { P } ( A \cup B ) = 0.65$$

Draw a Venn diagram to illustrate the complete sample space for the events $A$ and $B$.
Write down the value of $\mathrm { P } ( A )$ and the value of $\mathrm { P } ( B )$.
Find $\mathrm { P } \left( A \mid B ^ { \prime } \right)$.
Determine whether or not $A$ and $B$ are independent.

Edexcel S1 2006 January Q7

Moderate -0.8

7. The heights of a group of athletes are modelled by a normal distribution with mean 180 cm and a standard deviation 5.2 cm . The weights of this group of athletes are modelled by a normal distribution with mean 85 kg and standard deviation 7.1 kg . Find the probability that a randomly chosen athlete

is taller than 188 cm ,
weighs less than 97 kg .
(2)
Assuming that for these athletes height and weight are independent, find the probability that a randomly chosen athlete is taller than 188 cm and weighs more than 97 kg .
Comment on the assumption that height and weight are independent.

Edexcel S1 2007 January Q1

Moderate -0.8

As part of a statistics project, Gill collected data relating to the length of time, to the nearest minute, spent by shoppers in a supermarket and the amount of money they spent. Her data for a random sample of 10 shoppers are summarised in the table below, where $t$ represents time and $\pounds m$ the amount spent over $\pounds 20$.

$t$ (minutes)	£m
15	-3
23	17
5	-19
16	4
30	12
6	-9
32	27
23	6
35	20
27	6

Write down the actual amount spent by the shopper who was in the supermarket for 15 minutes.
Calculate $S _ { t t } , S _ { m m }$ and $S _ { t m }$. $$\text { (You may use } \Sigma t ^ { 2 } = 5478 \Sigma m ^ { 2 } = 2101 \Sigma t m = 2485 \text { ) }$$
Calculate the value of the product moment correlation coefficient between $t$ and $m$.
Write down the value of the product moment correlation coefficient between $t$ and the actual amount spent. Give a reason to justify your value. On another day Gill collected similar data. For these data the product moment correlation coefficient was 0.178
Give an interpretation to both of these coefficients.
Suggest a practical reason why these two values are so different.

Edexcel S1 2007 January Q2

Moderate -0.8

In a factory, machines $A , B$ and $C$ are all producing metal rods of the same length. Machine $A$ produces $35 \%$ of the rods, machine $B$ produces $25 \%$ and the rest are produced by machine $C$. Of their production of rods, machines $A , B$ and $C$ produce $3 \% , 6 \%$ and $5 \%$ defective rods respectively.
1. Draw a tree diagram to represent this information.
2. Find the probability that a randomly selected rod is
  1. produced by machine $A$ and is defective,
  2. is defective.
3. Given that a randomly selected rod is defective, find the probability that it was produced by machine $C$.

Edexcel S1 2007 January Q3

Moderate -0.8

The random variable $X$ has probability function

$$\mathrm { P } ( X = x ) = \frac { ( 2 x - 1 ) } { 36 } \quad x = 1,2,3,4,5,6$$

Construct a table giving the probability distribution of $X$. Find
$\mathrm { P } ( 2 < X \leqslant 5 )$,
the exact value of $\mathrm { E } ( X )$.
Show that $\operatorname { Var } ( X ) = 1.97$ to 3 significant figures.
Find $\operatorname { Var } ( 2 - 3 X )$.

Edexcel S1 2007 January Q4

Moderate -0.8

Summarised below are the distances, to the nearest mile, travelled to work by a random sample of 120 commuters.

Distance (to the nearest mile)	Number of commuters
0-9	10
10-19	19
20-29	43
30-39	25
40-49	8
50-59	6
60-69	5
70-79	3
80-89	1

For this distribution,

describe its shape,
use linear interpolation to estimate its median. The mid-point of each class was represented by $x$ and its corresponding frequency by $f$ giving $$\Sigma f x = 3550 \text { and } \Sigma f x ^ { 2 } = 138020$$
Estimate the mean and the standard deviation of this distribution. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
Evaluate this coefficient for this distribution.
State whether or not the value of your coefficient is consistent with your description in part (a). Justify your answer.
State, with a reason, whether you should use the mean or the median to represent the data in this distribution.
State the circumstance under which it would not matter whether you used the mean or the median to represent a set of data.

Edexcel S1 2007 January Q5

Easy -1.2

A teacher recorded, to the nearest hour, the time spent watching television during a particular week by each child in a random sample. The times were summarised in a grouped frequency table and represented by a histogram.

One of the classes in the grouped frequency distribution was 20-29 and its associated frequency was 9. On the histogram the height of the rectangle representing that class was 3.6 cm and the width was 2 cm .

Give a reason to support the use of a histogram to represent these data.
Write down the underlying feature associated with each of the bars in a histogram.
Show that on this histogram each child was represented by $0.8 \mathrm {~cm} ^ { 2 }$. The total area under the histogram was $24 \mathrm {~cm} ^ { 2 }$.
Find the total number of children in the group.

Edexcel S1 2007 January Q6

Easy -2.0

(a) Give two reasons to justify the use of statistical models.

It has been suggested that there are 7 stages involved in creating a statistical model. They are summarised below, with stages 3 , 4 and 7 missing. Stage 1. The recognition of a real-world problem. Stage 2. A statistical model is devised. Stage 3. Stage 4. Stage 5. Comparisons are made against the devised model. Stage 6. Statistical concepts are used to test how well the model describes the real-world problem. Stage 7.
(b) Write down the missing stages.

Edexcel S1 2007 January Q7

Moderate -0.8

The measure of intelligence, IQ, of a group of students is assumed to be Normally distributed with mean 100 and standard deviation 15.
1. Find the probability that a student selected at random has an IQ less than 91.
The probability that a randomly selected student has an IQ of at least $100 + k$ is 0.2090 .
Find, to the nearest integer, the value of $k$.

Edexcel S1 2008 January Q1

Moderate -0.3

A personnel manager wants to find out if a test carried out during an employee's interview and a skills assessment at the end of basic training is a guide to performance after working for the company for one year.

The table below shows the results of the interview test of 10 employees and their performance after one year.

Employee	A	$B$	C	D	$E$	$F$	G	$H$	I	J
Interview test, $x$ \%.	65	71	79	77	85	78	85	90	81	62
Performance after one year, $y \%$.	65	74	82	64	87	78	61	65	79	69

$$\text { [You may use } \sum x ^ { 2 } = 60475 , \sum y ^ { 2 } = 53122 , \sum x y = 56076 \text { ] }$$

Showing your working clearly, calculate the product moment correlation coefficient between the interview test and the performance after one year. The product moment correlation coefficient between the skills assessment and the performance after one year is - 0.156 to 3 significant figures.
Use your answer to part (a) to comment on whether or not the interview test and skills assessment are a guide to the performance after one year. Give clear reasons for your answers.

Edexcel S1 2008 January Q2

Easy -1.3

2. Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below.

$$\text { [You may use } \sum x ^ { 2 } = 724 \text { 961] }$$

Find the mean and standard deviation of the level of cotinine in a patient's blood.
Find the median, upper and lower quartiles of these data. A doctor suspects that some of his patients have been smoking more than a packet of cigarettes per day. He decides to use $\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)$ to determine if any of the cotinine results are far enough away from the upper quartile to be outliers.
Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly. Research suggests that cotinine levels in the blood form a skewed distribution.
One measure of skewness is found using $\frac { \left( Q _ { 1 } - 2 Q _ { 2 } + Q _ { 3 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }$.
Evaluate this measure and describe the skewness of these data.

Edexcel S1 2008 January Q3

Easy -1.2

3. The histogram in Figure 1 shows the time taken, to the nearest minute, for 140 runners to complete a fun run. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{af84d17b-5308-4b1e-99b5-40c5df5bf01e-06_1027_1509_367_258} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Use the histogram to calculate the number of runners who took between 78.5 and 90.5 minutes to complete the fun run.

Edexcel S1 2008 January Q4

Moderate -0.8

4. A second hand car dealer has 10 cars for sale. She decides to investigate the link between the age of the cars, $x$ years, and the mileage, $y$ thousand miles. The data collected from the cars are shown in the table below.

[You may assume that $\sum x = 41 , \sum y = 406 , \sum x ^ { 2 } = 188 , \sum x y = 1818.5$ ]

Find $S _ { x x }$ and $S _ { x y }$.
Find the equation of the least squares regression line in the form $y = a + b x$. Give the values of $a$ and $b$ to 2 decimal places.
Give a practical interpretation of the slope $b$.
Using your answer to part (b), find the mileage predicted by the regression line for a 5 year old car.
$\_\_\_\_$}

Edexcel S1 2008 January Q5

Easy -1.2

5. The following shows the results of a wine tasting survey of 100 people. \begin{displayquote} 96 like wine $A$,
93 like wine $B$,
96 like wine $C$,
92 like $A$ and $B$,
91 like $B$ and $C$,
93 like $A$ and $C$,
90 like all three wines.

Draw a Venn Diagram to represent these data. \end{displayquote} Find the probability that a randomly selected person from the survey likes
none of the three wines,
wine $A$ but not wine $B$,
any wine in the survey except wine $C$,
exactly two of the three kinds of wine. Given that a person from the survey likes wine $A$,
find the probability that the person likes wine $C$.

\(x\)	1	2	3	4	5
\(\mathrm { P } ( X = x )\)	0.10	\(p\)	0.20	\(q\)	0.30

\(x\)	148	164	156	172	147	184	162	155	182	165	175	152
\(y\)	39	59	56	77	44	77	65	49	80	72	70	52

Employee	A	\(B\)	C	D	\(E\)	\(F\)	G	\(H\)	I	J
Interview test, \(x\) \%.	65	71	79	77	85	78	85	90	81	62
Performance after one year, \(y \%\).	65	74	82	64	87	78	61	65	79	69

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Edexcel S1 2005 January Q2

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