Questions S1 - A-Level Maths

Complete the tree diagram below to show the possible outcomes and associated probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_785_385_744_568}
\includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_1054_483_760_954} Shivani selects a ball and spins the appropriate coin.
Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.

Edexcel S1 Specimen Q3

Moderate -0.8

The discrete random variable $X$ has probability distribution given by

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

where $a$ is a constant.

Find the value of $a$.
Write down $\mathrm { E } ( X )$.
Find $\operatorname { Var } ( X )$. The random variable $Y = 6 - 2 X$
Find $\operatorname { Var } ( Y )$.
Calculate $\mathrm { P } ( X \geqslant Y )$.

Edexcel S1 Specimen Q4

Moderate -0.8

The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines $A , B$ and $C$.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{61983561-79f7-4883-8ae7-ab1f4955d444-12_396_912_411_523} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} One of these students is selected at random.

Show that the probability that the student reads more than one magazine is $\frac { 1 } { 6 }$.
Find the probability that the student reads $A$ or $B$ (or both).
Write down the probability that the student reads both $A$ and $C$. Given that the student reads at least one of the magazines,
find the probability that the student reads $C$.
Determine whether or not reading magazine $B$ and reading magazine $C$ are statistically independent.

Edexcel S1 Specimen Q5

Moderate -0.3

A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.

Hours	$1 - 10$	$11 - 20$	$21 - 25$	$26 - 30$	$31 - 40$	$41 - 59$
Frequency	6	15	11	13	8	3
Mid-point	5.5	15.5		28		50

Find the mid-points of the 21-25 hour and 31-40 hour groups. A histogram was drawn to represent these data. The 11-20 group was represented by a bar of width 4 cm and height 6 cm .
Find the width and height of the 26-30 group.
Estimate the mean and standard deviation of the time spent watching television by these students.
Use linear interpolation to estimate the median length of time spent watching television by these students. The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.
State, giving a reason, the skewness of these data.

Edexcel S1 Specimen Q6

Moderate -0.8

A travel agent sells flights to different destinations from Beerow airport. The distance $d$, measured in 100 km , of the destination from the airport and the fare $\pounds f$ are recorded for a random sample of 6 destinations.

Destination	$A$	$B$	$C$	$D$	$E$	$F$
$d$	2.2	4.0	6.0	2.5	8.0	5.0
$f$	18	20	25	23	32	28

$$\text { [You may use } \sum d ^ { 2 } = 152.09 \quad \sum f ^ { 2 } = 3686 \quad \sum f d = 723.1 \text { ] }$$

Using the axes below, complete a scatter diagram to illustrate this information.
Explain why a linear regression model may be appropriate to describe the relationship between $f$ and $d$.
Calculate $S _ { d d }$ and $S _ { f d }$
Calculate the equation of the regression line of $f$ on $d$ giving your answer in the form $f = a + b d$.
Give an interpretation of the value of $b$. Jane is planning her holiday and wishes to fly from Beerow airport to a destination $t \mathrm {~km}$ away. A rival travel agent charges 5 p per km.
Find the range of values of $t$ for which the first travel agent is cheaper than the rival.
\includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-20_967_1630_1722_164}

Edexcel S1 Specimen Q7

Moderate -0.3

The distances travelled to work, $D \mathrm {~km}$, by the employees at a large company are normally distributed with $D \sim \mathrm {~N} \left( 30,8 ^ { 2 } \right)$.
1. Find the probability that a randomly selected employee has a journey to work of more than 20 km .
2. Find the upper quartile, $Q _ { 3 }$, of $D$.
3. Write down the lower quartile, $Q _ { 1 }$, of $D$.
An outlier is defined as any value of $D$ such that $D < h$ or $D > k$ where $$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
Find the value of $h$ and the value of $k$. An employee is selected at random.
Find the probability that the distance travelled to work by this employee is an outlier.

END

Edexcel S1 2001 January Q1

Easy -1.3

The students in a class were each asked to write down how many CDs they owned. The student with the least number of CDs had 14 and all but one of the others owned 60 or fewer. The remaining student owned 65 . The quartiles for the class were 30,34 and 42 respectively.

Outliers are defined to be any values outside the limits of $1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)$ below the lower quartile or above the upper quartile. On graph paper draw a box plot to represent these data, indicating clearly any outliers.
marks)

Edexcel S1 2001 January Q2

Easy -1.2

2. The random variable $X$ is normally distributed with mean 177.0 and standard deviation 6.4.

Find $\mathrm { P } ( 166 < X < 185 )$. It is suggested that $X$ might be a suitable random variable to model the height, in cm , of adult males.
Give two reasons why this is a sensible suggestion.
Explain briefly why mathematical models can help to improve our understanding of real-world problems.

Edexcel S1 2001 January Q3

Easy -1.2

3. A fair six-sided die is rolled. The random variable $Y$ represents the score on the uppermost, face.

Write down the probability function of $Y$.
State the name of the distribution of $Y$. Find the value of
$\mathrm { E } ( 6 Y + 2 )$,
$\operatorname { Var } ( 4 Y - 2 )$.

Edexcel S1 2001 January Q4

Easy -1.3

4. The employees of a company are classified as management, administration or production. The following table shows the number employed in each category and whether or not they live close to the company or some distance away.

An employee is chosen at random.
Find the probability that this employee

is an administrator,
lives close to the company, given that the employee is a manager. Of the managers, $90 \%$ are married, as are $60 \%$ of the administrators and $80 \%$ of the production employees.
Construct a tree diagram containing all the probabilities.
Find the probability that an employee chosen at random is married. An employee is selected at random and found to be married.
Find the probability that this employee is in production.

Edexcel S1 2001 January Q5

Moderate -0.3

5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.

Delay (mins)	Number of motorists
$4 - 6$	15
$7 - 8$	28
9	49
10	53
$11 - 12$	30
$13 - 15$	15
$16 - 20$	10

Using graph paper represent these data by a histogram.
Give a reason to justify the use of a histogram to represent these data.
Use interpolation to estimate the median of this distribution.
Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
Evaluate this coefficient for the above data.
Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.

Edexcel S1 2001 January Q6

Moderate -0.8

6. A local authority is investigating the cost of reconditioning its incinerators. Data from 10 randomly chosen incinerators were collected. The variables monitored were the operating time $x$ (in thousands of hours) since last reconditioning and the reconditioning cost $y$ (in $\pounds 1000$ ). None of the incinerators had been used for more than 3000 hours since last reconditioning. The data are summarised below, $$\Sigma x = 25.0 , \Sigma x ^ { 2 } = 65.68 , \Sigma y = 50.0 , \Sigma y ^ { 2 } = 260.48 , \Sigma x y = 130.64 .$$

Find $\mathrm { S } _ { x x } , \mathrm {~S} _ { x y } , \mathrm {~S} _ { y y }$.
Calculate the product moment correlation coefficient between $x$ and $y$.
Explain why this value might support the fitting of a linear regression model of the form $y = a + b x$.
Find the values of $a$ and $b$.
Give an interpretation of $a$.
Estimate
1. the reconditioning cost for an operating time of 2400 hours,
2. the financial effect of an increase of 1500 hours in operating time.
Suggest why the authority might be cautious about making a prediction of the reconditioning cost of an incinerator which had been operating for 4500 hours since its last reconditioning.

Edexcel S1 2002 January Q1

Easy -1.3

(a) Explain briefly what you understand by
1. a statistical experiment,
2. an event.
  (b) State one advantage and one disadvantage of a statistical model.
3. A meteorologist measured the number of hours of sunshine, to the nearest hour, each day for 100 days. The results are summarised in the table below.
Hours of sunshine Days
1 16
$2 - 4$ 32
$5 - 6$ 28
7 12
8 9
$9 - 11$ 2
12 1
(a) On graph paper, draw a histogram to represent these data.
(b) Calculate an estimate of the number of days that had between 6 and 9 hours of sunshine.

Edexcel S1 2002 January Q3

Moderate -0.8

3. A discrete random variable $X$ has the probability function shown in the table below.

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

Given that $\mathrm { E } ( X ) = \frac { 5 } { 6 }$, find $a$.
Find the exact value of Var ( $X$ ).
Find the exact value of $\mathrm { P } ( X \leq 15 )$.

Edexcel S1 2002 January Q4

Moderate -0.8

4. A contractor bids for two building projects. He estimates that the probability of winning the first project is 0.5 , the probability of winning the second is 0.3 and the probability of winning both projects is 0.2 .

Find the probability that he does not win either project.
Find the probability that he wins exactly one project.
Given that he does not win the first project, find the probability that he wins the second.
By calculation, determine whether or not winning the first contract and winning the second contract are independent events.

Edexcel S1 2002 January Q5

Standard +0.3

5. The duration of the pregnancy of a certain breed of cow is normally distributed with mean $\mu$ days and standard deviation $\sigma$ days. Only $2.5 \%$ of all pregnancies are shorter than 235 days and $15 \%$ are longer than 286 days.

Show that $\mu - 235 = 1.96 \sigma$.
Obtain a second equation in $\mu$ and $\sigma$.
Find the value of $\mu$ and the value of $\sigma$.
Find the values between which the middle $68.3 \%$ of pregnancies lie.

Edexcel S1 2002 January Q6

Easy -1.2

6. Hospital records show the number of babies born in a year. The number of babies delivered by 15 male doctors is summarised by the stem and leaf diagram below.

Babies	(4 5 means 45)	Totals
0		(0)
1	9	(1)
2	1677	(4)
3	22348	(5)
4	5	(1)
5	1	(1)
6	0	(1)
7		(0)
8	67	(2)

Find the median and inter-quartile range of these data.
Given that there are no outliers, draw a box plot on graph paper to represent these data. Start your scale at the origin.
Calculate the mean and standard deviation of these data. The records also contain the number of babies delivered by 10 female doctors.
34 30 20 15 6
32 26 19 11 4
The quartiles are 11, 19.5 and 30 .
Using the same scale as in part (b) and on the same graph paper draw a box plot for the data for the 10 female doctors.
Compare and contrast the box plots for the data for male and female doctors.

Edexcel S1 2002 January Q7

Moderate -0.3

7. A number of people were asked to guess the calorific content of 10 foods. The
mean $s$ of the guesses for each food and the true calorific content $t$ are given in the table below.

Food	$t$	$s$
Packet of biscuits	170	420
1 potato	90	160
1 apple	80	110
Crisp breads	10	70
Chocolate bar	260	360
1 slice white bread	75	135
1 slice brown bread	60	115
Portion of beef curry	270	350
Portion of rice pudding	165	390
Half a pint of milk	160	200

[You may assume that $\Sigma t = 1340 , \Sigma s = 2310 , \Sigma t s = 396775 , \Sigma t ^ { 2 } = 246050 , \Sigma s ^ { 2 } = 694650$.]

Draw a scatter diagram, indicating clearly which is the explanatory (independent) and which is the response (dependent) variable.
Calculate, to 3 significant figures, the product moment correlation coefficient for the above data.
State, with a reason, whether or not the value of the product moment correlation coefficient changes if all the guesses are 50 calories higher than the values in the table. The mean of the guesses for the portion of rice pudding and for the packet of biscuits are outside the linear relation of the other eight foods.
Find the equation of the regression line of $s$ on $t$ excluding the values for rice pudding and biscuits.
[0pt] [You may now assume that $S _ { t s } = 72587 , S _ { t t } = 63671.875 , \bar { t } = 125.625 , \bar { s } = 187.5$.]
Draw the regression line on your scatter diagram.
State, with a reason, what the effect would be on the regression line of including the values for a portion of rice pudding and a packet of biscuits. \section*{END}

Edexcel S1 2003 January Q1

Easy -1.3

The total amount of time a secretary spent on the telephone in a working day was recorded to the nearest minute. The data collected over 40 days are summarised in the table below.

Time (mins)	$90 - 139$	$140 - 149$	$150 - 159$	$160 - 169$	$170 - 179$	$180 - 229$
No. of days	8	10	10	4	4	4

Draw a histogram to illustrate these data

Edexcel S1 2003 January Q2

Easy -1.2

2. A car dealer offers purchasers a three year warranty on a new car. He sells two models, the Zippy and the Nifty. For the first 50 cars sold of each model the number of claims under the warranty is shown in the table below.

	Claim	No claim
Zippy	35	15
Nifty	40	10

One of the purchasers is chosen at random. Let $A$ be the event that no claim is made by the purchaser under the warranty and $B$ the event that the car purchased is a Nifty.

Find $\mathrm { P } ( A \cap B )$.
Find $\mathrm { P } \left( A ^ { \prime } \right)$. Given that the purchaser chosen does not make a claim under the warranty,
find the probability that the car purchased is a Zippy.
Show that making a claim is not independent of the make of the car purchased. Comment on this result.

Edexcel S1 2003 January Q3

Standard +0.3

3. A drinks machine dispenses coffee into cups. A sign on the machine indicates that each cup contains 50 ml of coffee. The machine actually dispenses a mean amount of 55 ml per cup and $10 \%$ of the cups contain less than the amount stated on the sign. Assuming that the amount of coffee dispensed into each cup is normally distributed find

the standard deviation of the amount of coffee dispensed per cup in ml ,
the percentage of cups that contain more than 61 ml . Following complaints, the owners of the machine make adjustments. Only $2.5 \%$ of cups now contain less than 50 ml . The standard deviation of the amount dispensed is reduced to 3 ml . Assuming that the amount of coffee dispensed is still normally distributed,
find the new mean amount of coffee per cup.
(4)

Edexcel S1 2003 January Q4

Easy -1.2

4. A restaurant owner is concerned about the amount of time customers have to wait before being served. He collects data on the waiting times, to the nearest minute, of 20 customers. These data are listed below.

15,	14,	16,	15,	17,	16,	15,	14,	15,	16,
17,	16,	15,	14,	16,	17,	15,	25,	18,	16

Find the median and inter-quartile range of the waiting times. An outlier is an observation that falls either $1.5 \times$ (inter-quartile range) above the upper quartile or $1.5 \times$ (inter-quartile range) below the lower quartile.
Draw a boxplot to represent these data, clearly indicating any outliers.
Find the mean of these data.
Comment on the skewness of these data. Justify your answer.

Edexcel S1 2003 January Q5

Standard +0.3

5. The discrete random variable $X$ has probability function $$\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) , & x = 0,1,2 \\ k ( x - 2 ) , & x = 3 \\ 0 , & \text { otherwise } \end{cases}$$ where $k$ is a positive constant.

Show that $k = 0.25$.
Find $\mathrm { E } ( X )$ and show that $\mathrm { E } \left( X ^ { 2 } \right) = 2.5$.
Find $\operatorname { Var } ( 3 X - 2 )$. Two independent observations $X _ { 1 }$ and $X _ { 2 }$ are made of $X$.
Show that $\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0$.
Find the complete probability function for $X _ { 1 } + X _ { 2 }$.
Find $\mathrm { P } \left( 1.3 \leq X _ { 1 } + X _ { 2 } \leq 3.2 \right)$.

Edexcel S1 2003 January Q6

Moderate -0.3

6. The chief executive of Rex cars wants to investigate the relationship between the number of new car sales and the amount of money spent on advertising. She collects data from company records on the number of new car sales, $c$, and the cost of advertising each year, $p$ (£000). The data are shown in the table below.

Year	Number of new car sale, $c$	Cost of advertising (£000), $p$
1990	4240	120
1991	4380	126
1992	4420	132
1993	4440	134
1994	4430	137
1995	4520	144
1996	4590	148
1997	4660	150
1998	4700	153
1999	4790	158

Using the coding $x = ( p - 100 )$ and $y = \frac { 1 } { 10 } ( c - 4000 )$, draw a scatter diagram to represent these data. Explain why $x$ is the explanatory variable.
Find the equation of the least squares regression line of $y$ on $x$. $$\text { [Use } \left. \Sigma x = 402 , \Sigma y = 517 , \Sigma x ^ { 2 } = 17538 \text { and } \Sigma x y = 22611 . \right]$$
Deduce the equation of the least squares regression line of $c$ on $p$ in the form $c = a + b p$.
Interpret the value of $a$.
Predict the number of extra new cars sales for an increase of $\pounds 2000$ in advertising budget. Comment on the validity of your answer.
(2)

Edexcel S1 2005 January Q1

Easy -1.3

A company assembles drills using components from two sources. Goodbuy supplies $85 \%$ of the components and Amart supplies the rest. It is known that $3 \%$ of the components supplied by Goodbuy are faulty and $6 \%$ of those supplied by Amart are faulty.
1. Represent this information on a tree diagram.
An assembled drill is selected at random.
Find the probability that it is not faulty.

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

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

Time (mins)	\(90 - 139\)	\(140 - 149\)	\(150 - 159\)	\(160 - 169\)	\(170 - 179\)	\(180 - 229\)
No. of days	8	10	10	4	4	4

Hours	\(1 - 10\)	\(11 - 20\)	\(21 - 25\)	\(26 - 30\)	\(31 - 40\)	\(41 - 59\)
Frequency	6	15	11	13	8	3
Mid-point	5.5	15.5		28		50

Destination	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)
\(d\)	2.2	4.0	6.0	2.5	8.0	5.0
\(f\)	18	20	25	23	32	28

Delay (mins)	Number of motorists
\(4 - 6\)	15
\(7 - 8\)	28
9	49
10	53
\(11 - 12\)	30
\(13 - 15\)	15
\(16 - 20\)	10

Hours of sunshine	Days
1	16
\(2 - 4\)	32
\(5 - 6\)	28
7	12
8	9
\(9 - 11\)	2
12	1

Food	\(t\)	\(s\)
Packet of biscuits	170	420
1 potato	90	160
1 apple	80	110
Crisp breads	10	70
Chocolate bar	260	360
1 slice white bread	75	135
1 slice brown bread	60	115
Portion of beef curry	270	350
Portion of rice pudding	165	390
Half a pint of milk	160	200

Year	Number of new car sale, \(c\)	Cost of advertising (£000), \(p\)
1990	4240	120
1991	4380	126
1992	4420	132
1993	4440	134
1994	4430	137
1995	4520	144
1996	4590	148
1997	4660	150
1998	4700	153
1999	4790	158

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