Questions - Edexcel S1 - A-Level Maths

Find $S _ { x x }$.
Calculate, to 3 significant figures, the product moment correlation coefficient between $x$ and $y$.
Give an interpretation of your coefficient.

Edexcel S1 2001 June Q3

3. The continuous random variable $Y$ is normally distributed with mean 100 and variance 256 .

Find $\mathrm { P } ( Y < 80 )$.
Find $k$ such that $\mathrm { P } ( 100 - k \leq Y \leq 100 + k ) = 0.516$.

Edexcel S1 2001 June Q4

4. The discrete random variable $X$ has the probability function shown in the table below.

$x$	- 2	- 1	0	1	2	3
$\mathrm { P } ( X = x )$	0.1	$\alpha$	0.3	0.2	0.1	0.1

Find

$\alpha$,
$\mathrm { P } ( - 1 < X \leq 2 )$,
$\mathrm { F } ( - 0.4 )$,
$\mathrm { E } ( 3 X + 4 )$,
$\operatorname { Var } ( 2 X + 3 )$.

Edexcel S1 2001 June Q5

5. A market researcher asked 100 adults which of the three newspapers $A , B , C$ they read. The results showed that $30 \operatorname { read } A , 26$ read $B , 21$ read $C , 5$ read both $A$ and $B , 7$ read both $B$ and $C , 6$ read both $C$ and $A$ and 2 read all three.

Draw a Venn diagram to represent these data. One of the adults is then selected at random.
Find the probability that she reads
at least one of the newspapers,
only $A$,
only one of the newspapers,
$A$ given that she reads only one newspaper.

Edexcel S1 2001 June Q6

6. Three swimmers Alan, Diane and Gopal record the number of lengths of the swimming pool they swim during each practice session over several weeks. The stem and leaf diagram below shows the results for Alan.

Lengths	20 means 20
2	0122	$( 4 )$
2	55667789	$( 7 )$
3	012224	$( 5 )$
3	566679	$( 5 )$
4	01333333444	$( 10 )$
4	5556667788999	$( 12 )$
5	000	$( 3 )$

Find the three quartiles for Alan's results. The table below summarises the results for Diane and Gopal.
Diane Gopal
Smallest value 35 25
Lower quartile 37 34
Median 42 42
Upper quartile 53 50
Largest value 65 57
Using the same scale and on the same sheet of graph paper draw box plots to represent the data for Alan, Diane and Gopal.
Compare and contrast the three box plots.

Edexcel S1 2001 June Q7

7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors $y$. She also recorded the
number of hours $x$ that the pupil had practised each week. The data are shown in the table below.

$x$	12	15	7	11	1	8	4	6	9	3
$y$	8	4	13	8	18	12	15	14	12	16

Plot these data on a scatter diagram.
Find the equation of the regression line of $y$ on $x$ in the form $y = a + b x$. $$\text { (You may use } \left. \Sigma x ^ { 2 } = 746 , \Sigma x y = 749 . \right)$$
Give an interpretation of the slope and the intercept of your regression line.
State whether or not you think the regression model is reasonable
1. for the range of $x$-values given in the table,
2. for all possible $x$-values. In each case justify your answer either by giving a reason for accepting the model or by suggesting an alternative model. END

Edexcel S1 2002 June Q1

An unbiased die has faces numbered 1 to 6 inclusive. The die is rolled and the number that appears on the uppermost face is recorded.
1. State the probability of not recording a 6 in one roll of the die.
The die is thrown until a 6 is recorded.
Find the probability that a 6 occurs for the first time on the third roll of the die.
(3)

Edexcel S1 2002 June Q2

2. Statistical models can be used to describe real world problems. Explain the process involved in the formulation of a statistical model.
(4)

Edexcel S1 2002 June Q3

3. For the events $A$ and $B$,

explain in words the meaning of the term $\mathrm { P } \left( \begin{array} { l l } B & A \end{array} \right)$,
sketch a Venn diagram to illustrate the relationship $\mathrm { P } \left( \begin{array} { l l } B & A \end{array} \right) = 0$. Three companies operate a bus service along a busy main road. Amber buses run $50 \%$ of the service and $2 \%$ of their buses are more than 5 minutes late. Blunder buses run $30 \%$ of the service and $10 \%$ of their buses are more than 5 minutes late. Clipper buses run the remainder of the service and only $1 \%$ of their buses run more than 5 minutes late. Jean is waiting for a bus on the main road.
Find the probability that the first bus to arrive is an Amber bus that is more than 5 minutes late. Let $A , B$ and $C$ denote the events that Jean catches an Amber bus, a Blunder bus and a Clipper bus respectively. Let $L$ denote the event that Jean catches a bus that is more than 5 minutes late.
Draw a Venn diagram to represent the events $A , B , \mathrm { C }$ and $L$. Calculate the probabilities associated with each region and write them in the appropriate places on the Venn diagram.
Find the probability that Jean catches a bus that is more than 5 minutes late.

Edexcel S1 2002 June Q4

4. A discrete random variable $X$ takes only positive integer values. It has a cumulative distribution function $\mathrm { F } ( x ) = \mathrm { P } ( X \leq x )$ defined in the table below.

$X$	1	2	3	4	5	6	7	8
$\mathrm {~F} ( x )$	0.1	0.2	0.25	0.4	0.5	0.6	0.75	1

Determine the probability function, $\mathrm { P } ( X = x )$, of $X$.
Calculate $\mathrm { E } ( X )$ and show that $\operatorname { Var } ( X ) = 5.76$.
Given that $Y = 2 X + 3$, find the mean and variance of $Y$.

Edexcel S1 2002 June Q5

5. A random variable $X$ has a normal distribution.

Describe two features of the distribution of $X$. A company produces electronic components which have life spans that are normally distributed. Only $1 \%$ of the components have a life span less than 3500 hours and $2.5 \%$ have a life span greater than 5500 hours.
Determine the mean and standard deviation of the life spans of the components. The company gives warranty of 4000 hours on the components.
Find the proportion of components that the company can expect to replace under the warranty.

Edexcel S1 2002 June Q6

6. The labelling on bags of garden compost indicates that the bags weigh 20 kg . The weights of a random sample of 50 bags are summarised in the table below.

Weight in kg	Frequency
14.6-14.8	1
14.8-18.0	0
18.0-18.5	5
18.5-20.0	6
20.0-20.2	22
20.2-20.4	15
20.4-21.0	1

On graph paper, draw a histogram of these data.
Using the coding $y = 10$ (weight in $\mathrm { kg } - 14$ ), find an estimate for the mean and standard deviation of the weight of a bag of compost.
[0pt] [Use $\Sigma f y ^ { 2 } = 171$ 503.75]
Using linear interpolation, estimate the median. The company that produces the bags of compost wants to improve the accuracy of the labelling. The company decides to put the average weight in kg on each bag.
Write down which of these averages you would recommend the company to use. Give a reason for your answer.

Edexcel S1 2002 June Q7

7. An ice cream seller believes that there is a relationship between the temperature on a summer day and the number of ice creams sold. Over a period of 10 days he records the temperature at 1 p.m., $t ^ { \circ } \mathrm { C }$, and the number of ice creams sold, $c$, in the next hour. The data he collects is summarised in the table below.

$t$	$c$
13	24
22	55
17	35
20	45
10	20
15	30
19	39
12	19
18	36
23	54

[Use $\left. \Sigma t ^ { 2 } = 3025 , \Sigma c ^ { 2 } = 14245 , \Sigma c t = 6526 .\right]$

Calculate the value of the product moment correlation coefficient between $t$ and $c$.
State whether or not your value supports the use of a regression equation to predict the number of ice creams sold. Give a reason for your answer.
Find the equation of the least squares regression line of $c$ on $t$ in the form $c = a + b t$.
Interpret the value of $b$.
Estimate the number of ice creams sold between 1 p.m. and 2 p.m. when the temperature at 1 p.m. is $16 ^ { \circ } \mathrm { C }$.
(3)
At 1 p.m. on a particular day, the highest temperature for 50 years was recorded. Give a reason why you should not use the regression equation to predict ice cream sales on that day.
(1)

Edexcel S1 2004 June Q1

A fair die has six faces numbered $1,2,2,3,3$ and 3 . The die is rolled twice and the number showing on the uppermost face is recorded each time.

Find the probability that the sum of the two numbers recorded is at least 5 .
(5)

Edexcel S1 2004 June Q2

2. A researcher thinks there is a link between a person's height and level of confidence. She measured the height $h$, to the nearest cm , of a random sample of 9 people. She also devised a test to measure the level of confidence $c$ of each person. The data are shown in the table below.

$h$	179	169	187	166	162	193	161	177	168
$c$	569	561	579	561	540	598	542	565	573

[You may use $\Sigma h ^ { 2 } = 272094 , \Sigma c ^ { 2 } = 2878966 , \Sigma h c = 884484$ ]

Draw a scatter diagram to illustrate these data.
Find exact values of $S _ { h c } S _ { h h }$ and $S _ { c c }$.
Calculate the value of the product moment correlation coefficient for these data.
Give an interpretation of your correlation coefficient.
Calculate the equation of the regression line of $c$ on $h$ in the form $c = a + b h$.
Estimate the level of confidence of a person of height 180 cm .
State the range of values of $h$ for which estimates of $c$ are reliable.

Edexcel S1 2004 June Q3

3. A discrete random variable $X$ has a probability function as shown in the table below, where $a$ and $b$ are constants.

$x$	0	1	2	3
$\mathrm { P } ( X = x )$	0.2	0.3	$b$	$a$

Given that $\mathrm { E } ( X ) = 1.7$,

find the value of $a$ and the value of $b$. Find
$\mathrm { P } ( 0 < X < 1.5 )$,
$\mathrm { E } ( 2 X - 3 )$.
Show that $\operatorname { Var } ( X ) = 1.41$.
Evaluate $\operatorname { Var } ( 2 X - 3 )$.

Edexcel S1 2004 June Q4

4. The attendance at college of a group of 18 students was recorded for a 4-week period. The number of students actually attending each of 16 classes are shown below.

18	18	17	17
16	17	16	18
18	14	17	18
15	17	18	16

1. Calculate the mean and the standard deviation of the number of students attending these classes.
2. Express the mean as a percentage of the 18 students in the group. In the same 4-week period, the attendance of a different group of 20, students is shown below.
  20 16 18 19
  15 14 14 15
  18 15 16 17
  16 18 15 14
Construct a back-to-back stem and leaf diagram to represent the attendance in both groups.
Find the mode, median and inter-quartile range for each group of students. The mean percentage attendance and standard deviation for the second group of students are 81.25 and 1.82 respectively.
Compare and contrast the attendance of these 2 groups of students.

Edexcel S1 2004 June Q5

5. A health club lets members use, on each visit, its facilities for as long as they wish. The club's records suggest that the length of a visit can be modelled by a normal distribution with mean 90 minutes. Only $20 \%$ of members stay for more than 125 minutes.

Find the standard deviation of the normal distribution.
Find the probability that a visit lasts less than 25 minutes. The club introduce a closing time of 10:00 pm. Tara arrives at the club at 8:00 pm.
Explain whether or not this normal distribution is still a suitable model for the length of her visit.

Edexcel S1 2004 June Q6

6. Three events $A , B$ and $C$ are defined in the sample space $S$. The events $A$ and $B$ are mutually exclusive and $A$ and $C$ are independent.

Draw a Venn diagram to illustrate the relationships between the 3 events and the sample space. Given that $\mathrm { P } ( A ) = 0.2 , \mathrm { P } ( B ) = 0.4$ and $\mathrm { P } ( A \cup C ) = 0.7$, find
$\mathrm { P } ( A C )$,
$\mathrm { P } ( A \cup B )$,
$\mathrm { P } ( C )$. END

Edexcel S1 2005 June Q1

The scatter diagrams below were drawn by a student.

$$\begin{aligned} & y \underset { x } { \begin{array} { l l l l } & &
+ & & &
+ & + & + &
+ & + & + \end{array} } \end{aligned}$$ The student calculated the value of the product moment correlation coefficient for each of the sets of data. The values were $$\begin{array} { l l l } 0.68 & - 0.79 & 0.08 \end{array}$$ Write down, with a reason, which value corresponds to which scatter diagram.
(6)

Edexcel S1 2005 June Q2

2. The following table summarises the distances, to the nearest km , that 134 examiners travelled to attend a meeting in London.

Distance (km)	Number of examiners
41-45	4
46-50	19
51-60	53
61-70	37
71-90	15
91-150	6

Give a reason to justify the use of a histogram to represent these data.
Calculate the frequency densities needed to draw a histogram for these data.
(DO NOT DRAW THE HISTOGRAM)
Use interpolation to estimate the median $Q _ { 2 }$, the lower quartile $Q _ { 1 }$, and the upper quartile $Q _ { 3 }$ of these data. The mid-point of each class is represented by $x$ and the corresponding frequency by $f$. Calculations then give the following values $$\Sigma f _ { x } = 8379.5 \quad \text { and } \quad \Sigma f _ { x ^ { 2 } } = 557489.75$$
Calculate an estimate of the mean and an estimate of the standard deviation for these data. One coefficient of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
Evaluate this coefficient and comment on the skewness of these data.
Give another justification of your comment in part (e).

Edexcel S1 2005 June Q3

A long distance lorry driver recorded the distance travelled, $m$ miles, and the amount of fuel used, $f$ litres, each day. Summarised below are data from the driver's records for a random sample of 8 days.

The data are coded such that $x = m - 250$ and $y = f - 100$. $$\Sigma x = 130 \quad \Sigma y = 48 \quad \Sigma x y = 8880 \quad \mathrm {~S} _ { x x } = 20487.5$$

Find the equation of the regression line of $y$ on $x$ in the form $y = a + b x$.
Hence find the equation of the regression line of $f$ on $m$.
Predict the amount of fuel used on a journey of 235 miles.

Edexcel S1 2005 June Q4

4. Aeroplanes fly from City $A$ to City $B$. Over a long period of time the number of minutes delay in take-off from City $A$ was recorded. The minimum delay was 5 minutes and the maximum delay was 63 minutes. A quarter of all delays were at most 12 minutes, half were at most 17 minutes and $75 \%$ were at most 28 minutes. Only one of the delays was longer than 45 minutes. An outlier is an observation that falls either $1.5 \times$ (interquartile range) above the upper quartile or $1.5 \times$ (interquartile range) below the lower quartile.

On the graph paper opposite draw a box plot to represent these data.
Comment on the distribution of delays. Justify your answer.
Suggest how the distribution might be interpreted by a passenger who frequently flies from City $A$ to City $B$.
\includegraphics[max width=\textwidth, alt={}, center]{9698650f-ef85-468d-a703-1b40df7f9d02-07_1190_1487_278_223}

Edexcel S1 2005 June Q5

5. The random variable $X$ has probability function $$P ( X = x ) = \begin{cases} k x , & x = 1,2,3
k ( x + 1 ) , & x = 4,5 \end{cases}$$ where $k$ is a constant.

Find the value of $k$.
Find the exact value of $\mathrm { E } ( X )$.
Show that, to 3 significant figures, $\operatorname { Var } ( X ) = 1.47$.
Find, to 1 decimal place, $\operatorname { Var } ( 4 - 3 X )$.

Edexcel S1 2005 June Q6

A scientist found that the time taken, $M$ minutes, to carry out an experiment can be modelled by a normal random variable with mean 155 minutes and standard deviation 3.5 minutes.

Find

$\mathrm { P } ( M > 160 )$.
$\mathrm { P } ( 150 \leqslant M \leqslant 157 )$.
the value of $m$, to 1 decimal place, such that $\mathrm { P } ( M \leqslant m ) = 0.30$.

\(x\)	0	1	2	3
\(\mathrm { P } ( X = x )\)	0.2	0.3	\(b\)	\(a\)

\(x\)	- 2	- 1	0	1	2	3
\(\mathrm { P } ( X = x )\)	0.1	\(\alpha\)	0.3	0.2	0.1	0.1

	Diane	Gopal
Smallest value	35	25
Lower quartile	37	34
Median	42	42
Upper quartile	53	50
Largest value	65	57

\(h\)	179	169	187	166	162	193	161	177	168
\(c\)	569	561	579	561	540	598	542	565	573

\(x\)	12	15	7	11	1	8	4	6	9	3
\(y\)	8	4	13	8	18	12	15	14	12	16

\(t\)	\(c\)
13	24
22	55
17	35
20	45
10	20
15	30
19	39
12	19
18	36
23	54

Questions — Edexcel S1 (574 questions)

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