Questions S1 - A-Level Maths

Find the proportion of men that take longer than 300 minutes to run a marathon.
(3) Nathaniel is preparing to run a marathon. He aims to finish in the first 20\% of male runners.
Using the above model estimate the longest time that Nathaniel can take to run the marathon and achieve his aim.
(3) The time, $W$ minutes, taken by women to run a marathon is modelled by a normal distribution with mean $\mu$ minutes. Given that $\mathrm { P } ( W < \mu + 30 ) = 0.82$
find $\mathrm { P } ( W < \mu - 30 \mid W < \mu )$

Edexcel S1 2017 June Q1

A clothes shop manager records the weekly sales figures, $\pounds s$, and the average weekly temperature, $t ^ { \circ } \mathrm { C }$, for 6 weeks during the summer. The sales figures were coded so that $w = \frac { s } { 1000 }$

The data are summarised as follows $$\mathrm { S } _ { w w } = 50 \quad \sum w t = 784 \quad \sum t ^ { 2 } = 2435 \quad \sum t = 119 \quad \sum w = 42$$

Find $\mathrm { S } _ { w t }$ and $\mathrm { S } _ { t t }$
Write down the value of $\mathrm { S } _ { s s }$ and the value of $\mathrm { S } _ { s t }$
Find the product moment correlation coefficient between $s$ and $t$. The manager of the clothes shop believes that a linear regression model may be appropriate to describe these data.
State, giving a reason, whether or not your value of the correlation coefficient supports the manager's belief.
Find the equation of the regression line of $w$ on $t$, giving your answer in the form $w = a + b t$
Hence find the equation of the regression line of $s$ on $t$, giving your answer in the form $s = c + d t$, where $c$ and $d$ are correct to 3 significant figures.
Using your equation in part (f), interpret the effect of a $1 ^ { \circ } \mathrm { C }$ increase in average weekly temperature on weekly sales during the summer.

Edexcel S1 2017 June Q2

2. An estate agent is studying the cost of office space in London. He takes a random sample of 90 offices and calculates the cost, $\pounds x$ per square foot. His results are given in the table below.

Cost (£ $\boldsymbol { x }$ )	Frequency (f)	Midpoint (£y)
$20 \leqslant x < 40$	12	30
$40 \leqslant x < 45$	13	42.5
$45 \leqslant x < 50$	25	47.5
$50 \leqslant x < 60$	32	55
$60 \leqslant x < 80$	8	70

A histogram is drawn for these data and the bar representing $50 \leqslant x < 60$ is 2 cm wide and 8 cm high.

Calculate the width and height of the bar representing $20 \leqslant x < 40$
Use linear interpolation to estimate the median cost.
Estimate the mean cost of office space for these data.
Estimate the standard deviation for these data.
Describe, giving a reason, the skewness. Rika suggests that the cost of office space in London can be modelled by a normal distribution with mean $\pounds 50$ and standard deviation $\pounds 10$
With reference to your answer to part (e), comment on Rika's suggestion.
Use Rika's model to estimate the 80th percentile of the cost of office space in London.

Edexcel S1 2017 June Q3

The Venn diagram shows three events $A , B$ and $C$, where $p , q , r , s$ and $t$ are probabilities.
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(b) Find the value of $r$.
(c) Hence write down the value of $s$ and the value of $t$.
(d) State, giving a reason, whether or not the events $A$ and $B$ are independent.
(e) Find $\mathrm { P } ( B \mid A \cup C )$.
$\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6$ and $\mathrm { P } ( C ) = 0.25$ and the events $B$ and $C$ are independent.
(a) Find the value of $p$ and the value of $q$.

Edexcel S1 2017 June Q4

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

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

The cumulative distribution function of $X$ is given by

$x$	- 1	0	1	2
$\mathrm {~F} ( x )$	$\frac { 1 } { 3 }$	$d$	$\frac { 5 } { 6 }$	$e$

Find the values of $a , b , c , d$ and $e$.
Write down the value of $\mathrm { P } \left( X ^ { 2 } = 1 \right)$.
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} $T$

Edexcel S1 2017 June Q5

5. Yuto works in the quality control department of a large company. The time, $T$ minutes, it takes Yuto to analyse a sample is normally distributed with mean 18 minutes and standard deviation 5 minutes.

Find the probability that Yuto takes longer than 20 minutes to analyse the next sample. (3) The company has a large store of samples analysed by Yuto with the time taken for each analysis recorded. Serena is investigating the samples that took Yuto longer than 15 minutes to analyse. She selects, at random, one of the samples that took Yuto longer than 15 minutes to analyse.
Find the probability that this sample took Yuto more than 20 minutes to analyse. Serena can identify, in advance, the samples that Yuto can analyse in under 15 minutes and in future she will assign these to someone else.
Estimate the median time taken by Yuto to analyse samples in future.

Edexcel S1 2017 June Q6

6. The score, $X$, for a biased spinner is given by the probability distribution

$x$	0	3	6
$\mathrm { P } ( X = x )$	$\frac { 1 } { 12 }$	$\frac { 2 } { 3 }$	$\frac { 1 } { 4 }$

Find

$\mathrm { E } ( X )$
$\operatorname { Var } ( X )$ A biased coin has one face labelled 2 and the other face labelled 5 The score, $Y$, when the coin is spun has $$\mathrm { P } ( Y = 5 ) = p \quad \text { and } \quad \mathrm { E } ( Y ) = 3$$
Form a linear equation in $p$ and show that $p = \frac { 1 } { 3 }$
Write down the probability distribution of $Y$. Sam plays a game with the spinner and the coin.
Each is spun once and Sam calculates his score, $S$, as follows $$\begin{aligned} & \text { if } X = 0 \text { then } S = Y ^ { 2 }
& \text { if } X \neq 0 \text { then } S = X Y \end{aligned}$$
Show that $\mathrm { P } ( S = 30 ) = \frac { 1 } { 12 }$
Find the probability distribution of $S$.
Find $\mathrm { E } ( S )$. Charlotte also plays the game with the spinner and the coin.
Each is spun once and Charlotte ignores the score on the coin and just uses $X ^ { 2 }$ as her score. Sam and Charlotte each play the game a large number of times.
State, giving a reason, which of Sam and Charlotte should achieve the higher total score.
END

Edexcel S1 2018 June Q1

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

$x$	2	4	7	10
$\mathrm { P } ( X = x )$	$a$	$b$	0.1	$c$

where $a , b$ and $c$ are probabilities.
The cumulative distribution function of $X$ is $\mathrm { F } ( x )$ and $\mathrm { F } ( 3 ) = 0.2$ and $\mathrm { F } ( 6 ) = 0.8$

Find the value of $a$, the value of $b$ and the value of $c$.
Write down the value of $\mathrm { F } ( 7 )$.

Edexcel S1 2018 June Q2

2. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 100 motorists were delayed by roadworks on a stretch of motorway one Monday.

Delay (minutes)	Number of motorists (f)	Delay midpoint (x)
3-6	38	4.5
7-8	25	7.5
9-10	18	9.5
11-15	12	13
16-20	7	18

(You may use $\sum \mathrm { f } x ^ { 2 } = 8096.25$ ) A histogram has been drawn to represent these data. The bar representing a delay of (3-6) minutes has a width of 2 cm and a height of 9.5 cm .

Calculate the width and the height of the bar representing a delay of (11-15) minutes.
Use linear interpolation to estimate the median delay.
Calculate an estimate of the mean delay.
Calculate an estimate of the standard deviation of the delays. One coefficient of skewness is given by $\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$
Evaluate this coefficient for the above data, giving your answer to 2 significant figures. On the following Friday, the coefficient of skewness for the delays on this stretch of motorway was - 0.22
State, giving a reason, how the delays on this stretch of motorway on Friday are different from the delays on Monday.

Edexcel S1 2018 June Q3

3. The random variable $Y$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$ The $\mathrm { P } ( Y > 17 ) = 0.4$ Find

$\mathrm { P } ( \mu < Y < 17 )$
$\mathrm { P } ( \mu - \sigma < Y < 17 )$

Edexcel S1 2018 June Q4

4．A bag contains 64 coloured beads．There are $r$ red beads，$y$ yellow beads and 1 green bead and $r + y + 1 = 64$ Two beads are selected at random，one at a time without replacement．
（a）Find the probability that the green bead is one of the beads selected． The probability that both of the beads are red is $\frac { 5 } { 84 }$
（b）Show that $r$ satisfies the equation $r ^ { 2 } - r - 240 = 0$
（c）Hence show that the only possible value of $r$ is 16
（d）Given that at least one of the beads is red，find the probability that they are both red．

Edexcel S1 2018 June Q5

5. The score when a spinner is spun is given by the discrete random variable $X$ with the following probability distribution, where $a$ and $b$ are probabilities.

$x$	- 1	0	2	4	5
$\mathrm { P } ( X = x )$	$b$	$a$	$a$	$a$	$b$

Explain why $\mathrm { E } ( X ) = 2$
Find a linear equation in $a$ and $b$. Given that $\operatorname { Var } ( X ) = 7.1$
find a second equation in $a$ and $b$ and simplify your answer.
Solve your two equations to find the value of $a$ and the value of $b$. The discrete random variable $Y = 10 - 3 X$
Find
1. $\mathrm { E } ( Y )$
2. $\operatorname { Var } ( Y )$ The spinner is spun once.
Find $\mathrm { P } ( Y > X )$.

Edexcel S1 2018 June Q6

6. A group of climbers collected information about the height above sea level, $h$ metres, and the air temperature, $t ^ { \circ } \mathrm { C }$, at the same time at 8 different points on the same mountain. The data are summarised by $$\sum h = 6370 \quad \sum t = 61 \quad \sum t h = 31070 \quad \sum t ^ { 2 } = 693$$

Show that $\mathrm { S } _ { \text {th } } = - 17501.25$ and $\mathrm { S } _ { \text {tt } } = 227.875$ The product moment correlation coefficient for these data is - 0.985
State, giving a reason, whether or not this value supports the use of a regression equation to predict the air temperature at different heights on this mountain.
Find the equation of the regression line of $t$ on $h$, giving your answer in the form $t = a + b h$. Give the value of your coefficients to 3 significant figures.
Give an interpretation of your value of $a$. One of the climbers has just stopped for a short break before climbing the next 150 metres.
Estimate the drop in temperature over this 150 metre climb.

Edexcel S1 2018 June Q7

7. Farmer Adam grows potatoes. The weights of potatoes, in grams, grown by Adam are normally distributed with a mean of 140 g and a standard deviation of 40 g . Adam cannot sell potatoes with a weight of less than 92 g .

Find the percentage of potatoes that Adam grows but cannot sell. The upper quartile of the weight of potatoes sold by Adam is $q _ { 3 }$
Find the probability that the weight of a randomly selected potato grown by Adam is more than $q _ { 3 }$
Find the lower quartile, $q _ { 1 }$, of the weight of potatoes sold by Adam. Betty selects a random sample of 3 potatoes sold by Adam.
Find the probability that one weighs less than $q _ { 1 }$, one weighs more than $q _ { 3 }$ and one has a weight between $q _ { 1 }$ and $q _ { 3 }$

END

Edexcel S1 Q1

The weight of coffee in glass jars labelled 100 g is normally distributed with mean 101.80 g and standard deviation 0.72 g . The weight of an empty glass jar is normally distributed with mean 260.00 g and standard deviation 5.45 g . The weight of a glass jar is independent of the weight of the coffee it contains.

Find the probability that a randomly selected jar weighs less than 266 g and contains less than 100 g of coffee. Give your answer to 2 significant figures.
(8 marks)

Edexcel S1 Q2

2. A botany student counted the number of daisies in each of 42 randomly chosen areas of 1 m by 1 m in a large field. The results are summarised in the following stem and leaf diagram.

Number of daisies								$1 \mid 1$ means 11
1	1	2	2	3	4	4	4		(7)
1	5	5	6	7	8	9	9		(7)
2	0	0	1	3	3	3	3	4	(8)
2	5	5	6	7	9	9	9		(7)
3	0	0	1	2	4	4			(6)
3	6	6	7	8	8				(5)
4	1	3							(2)

Write down the modal value of these data.
Find the median and the quartiles of these data.
On graph paper and showing your scale clearly, draw a box plot to represent these data.
Comment on the skewness of this distribution. The student moved to another field and collected similar data from that field.
Comment on how the student might summarise both sets of raw data before drawing box plots.
(1 mark)

Edexcel S1 Q3

3. Data relating to the lifetimes (to the nearest hour) of a random sample of 200 light bulbs from the production line of a manufacturer were summarised in a group frequency table. The mid-point of each group in the table was represented by $x$ and the corresponding frequency for that group by $f$. The data were then coded using $y = \frac { ( x - 755.0 ) } { 2.5 }$ and summarised as follows: $$\Sigma f y = - 467 , \Sigma f y ^ { 2 } = 9179 .$$

Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
(9 marks)
An estimate of the interquartile range for these data was 27.7 hours.
Explain, giving a reason, whether you would recommend the manufacturer to use the interquartile range or the standard deviation to represent the spread of lifetimes of the bulbs from this production line.
(2 marks)

Edexcel S1 Q4

4. A customer wishes to withdraw money from a cash machine. To do this it is necessary to type a PIN number into the machine. The customer is unsure of this number. If the wrong number is typed in, the customer can try again up to a maximum of four attempts in total. Attempts to type in the correct number are independent and the probability of success at each attempt is 0.6 .

Show that the probability that the customer types in the correct number at the third attempt is 0.096 .
(2 marks)
The random variable $A$ represents the number of attempts made to type in the correct PIN number, regardless of whether or not the attempt is successful.
Find the probability distribution of $A$.
Calculate the probability that the customer types in the correct number in four or fewer attempts.
Calculate $\mathrm { E } ( A )$ and $\operatorname { Var } ( A )$.
Find $\mathrm { F } ( 1 + \mathrm { E } ( A ) )$.

Edexcel S1 Q5

5. A keep-fit enthusiast swims, runs or cycles each day with probabilities $0.2,0.3$ and 0.5 respectively. If he swims he then spends time in the sauna with probability 0.35 . The probabilities that he spends time in the sauna after running or cycling are 0.2 and 0.45 respectively.

Represent this information on a tree diagram.
Find the probability that on any particular day he uses the sauna.
Given that he uses the sauna one day, find the probability that he had been swimming.
Given that he did not use the sauna one day, find the probability that he had been swimming.

Edexcel S1 Q6

6. To test the heating of tyre material, tyres are run on a test rig at chosen speeds under given conditions of load, pressure and surrounding temperature. The following table gives values of $x$, the test rig speed in miles per hour (mph), and the temperature, $y ^ { \circ } \mathrm { C }$, generated in the shoulder of the tyre for a particular tyre material.

$x ( \mathrm { mph } )$	15	20	25	30	35	40	45	50
$y \left( { } ^ { \circ } \mathrm { C } \right)$	53	55	63	65	78	83	91	101

Draw a scatter diagram to represent these data.
Give a reason to support the fitting of a regression line of the form $y = a + b x$ through these points.
Find the values of $a$ and $b$.
(You may use $\Sigma x ^ { 2 } = 9500 , \Sigma y ^ { 2 } = 45483 , \Sigma x y = 20615$ )
Give an interpretation for each of $a$ and $b$.
Use your line to estimate the temperature at 50 mph and explain why this estimate differs from the value given in the table. A tyre specialist wants to estimate the temperature of this tyre material at 12 mph and 85 mph .
Explain briefly whether or not you would recommend the specialist to use this regression equation to obtain these estimates.

Edexcel S1 2002 November Q1

(a) Explain briefly why statistical models are used when attempting to solve real-world problems.
(b) Write down the name of the distribution you would recommend as a suitable model for each of the following situations.
1. The weight of marmalade in a jar.
2. The number on the uppermost face of a fair die after it has been rolled.
  (2)
3. There are 125 sixth-form students in a college, of whom 60 are studying only arts subjects, 40 only science subjects and the rest a mixture of both.
Three students are selected at random, without replacement.
Find the probability that
(a) all three students are studying only arts subjects,
(b) exactly one of the three students is studying only science subjects.

Edexcel S1 2002 November Q3

3. The events $A$ and $B$ are independent such that $\mathrm { P } ( A ) = 0.25$ and $\mathrm { P } ( B ) = 0.30$. Find

$\mathrm { P } ( A \cap B )$,
$\mathrm { P } ( A \cup B )$,
$\mathrm { P } \left( A B ^ { \prime } \right)$.

Edexcel S1 2002 November Q4

4. Strips of metal are cut to length $L \mathrm {~cm}$, where $L \sim \mathrm {~N} \left( \mu , 0.5 ^ { 2 } \right)$.

Given that $2.5 \%$ of the cut lengths exceed 50.98 cm , show that $\mu = 50$.
Find $\mathrm { P } ( 49.25 < L < 50.75 )$. Those strips with length either less than 49.25 cm or greater than 50.75 cm cannot be used.
Two strips of metal are selected at random.
Find the probability that both strips cannot be used.

Edexcel S1 2002 November Q5

5. An agricultural researcher collected data, in appropriate units, on the annual rainfall $x$ and the annual yield of wheat $y$ at 8 randomly selected places. The data were coded using $s = x - 6$ and $t = y - 20$ and the following summations were obtained. $$\Sigma s = 48.5 , \quad \Sigma t = 65.0 , \quad \Sigma s ^ { 2 } = 402.11 , \quad \Sigma t ^ { 2 } = 701.80 , \quad \Sigma s t = 523.23$$

Find the equation of the regression line of $t$ on $s$ in the form $t = p + q s$.
Find the equation of the regression line of $y$ on $x$ in the form $y = a + b x$, giving $a$ and $b$ to 3 decimal places. The value of the product moment correlation coefficient between $s$ and $t$ is 0.943 , to 3 decimal places.
Write down the value of the product moment correlation coefficient between $x$ and $y$. Give a justification for your answer.

Edexcel S1 2002 November Q6

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

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

Given that $\mathrm { E } ( X ) = - 0.2$, find the value of $\alpha$ and the value of $\beta$.
Write down $\mathrm { F } ( 0.8 )$.
Evaluate $\operatorname { Var } ( X )$. Find the value of
$\mathrm { E } ( 3 X - 2 )$,
$\operatorname { Var } ( 2 X + 6 )$.

\(x\)	- 1	0	1	2
\(\mathrm { P } ( X = x )\)	\(a\)	\(b\)	\(b\)	\(c\)

\(x\)	- 1	0	1	2
\(\mathrm {~F} ( x )\)	\(\frac { 1 } { 3 }\)	\(d\)	\(\frac { 5 } { 6 }\)	\(e\)

\(x\)	0	3	6
\(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 12 }\)	\(\frac { 2 } { 3 }\)	\(\frac { 1 } { 4 }\)

\(x\)	2	4	7	10
\(\mathrm { P } ( X = x )\)	\(a\)	\(b\)	0.1	\(c\)

\(x\)	- 1	0	2	4	5
\(\mathrm { P } ( X = x )\)	\(b\)	\(a\)	\(a\)	\(a\)	\(b\)

Cost (£ \(\boldsymbol { x }\) )	Frequency (f)	Midpoint (£y)
\(20 \leqslant x < 40\)	12	30
\(40 \leqslant x < 45\)	13	42.5
\(45 \leqslant x < 50\)	25	47.5
\(50 \leqslant x < 60\)	32	55
\(60 \leqslant x < 80\)	8	70

Number of daisies								\(1 \mid 1\) means 11
1	1	2	2	3	4	4	4		(7)
1	5	5	6	7	8	9	9		(7)
2	0	0	1	3	3	3	3	4	(8)
2	5	5	6	7	9	9	9		(7)
3	0	0	1	2	4	4			(6)
3	6	6	7	8	8				(5)
4	1	3							(2)

\(x ( \mathrm { mph } )\)	15	20	25	30	35	40	45	50
\(y \left( { } ^ { \circ } \mathrm { C } \right)\)	53	55	63	65	78	83	91	101

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

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