Questions - Edexcel S1 - A-Level Maths

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
$\mathrm { E } ( 3 X - 2 )$,
$\operatorname { Var } ( 2 X + 6 )$.

Edexcel S1 2002 November Q7

7. The following stem and leaf diagram shows the aptitude scores $x$ obtained by all the applicants for a particular job.

	Aptitude score	31 means 31
3	129	(3)
4	24689	(5)
5	1335679	(7)
6	0133356889	(10)
7	1222455568889	(14)
8	01235889	(8)
9	012	(3)

Write down the modal aptitude score.
Find the three quartiles for these data. Outliers can be defined to be outside the limits $\mathrm { Q } _ { 1 } - 1.0 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)$ and $\mathrm { Q } _ { 3 } + 1.0 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)$.
On a graph paper, draw a box plot to represent these data. For these data, $\Sigma x = 3363$ and $\Sigma x ^ { 2 } = 238305$.
Calculate, to 2 decimal places, the mean and the standard deviation for these data.
Use two different methods to show that these data are negatively skewed.

Edexcel S1 2003 November Q1

A company wants to pay its employees according to their performance at work. The performance score $x$ and the annual salary, $y$ in $\pounds 100$ s, for a random sample of 10 of its employees for last year were recorded. The results are shown in the table below.

$x$	15	40	27	39	27	15	20	30	19	24
$y$	216	384	234	399	226	132	175	316	187	196

$$\text { [You may assume } \left. \Sigma x y = 69798 , \Sigma x ^ { 2 } = 7266 \right]$$

Draw a scatter diagram to represent these data.
Calculate exact values of $S _ { x y }$ and $S _ { x x }$.
1. Calculate the equation of the regression line of $y$ on $x$, in the form $y = a + b x$. Give the values of $a$ and $b$ to 3 significant figures.
2. Draw this line on your scatter diagram.
Interpret the gradient of the regression line. The company decides to use this regression model to determine future salaries.
Find the proposed annual salary for an employee who has a performance score of 35 .

Edexcel S1 2003 November Q2

2. A fairground game involves trying to hit a moving target with a gunshot. A round consists of up to 3 shots. Ten points are scored if a player hits the target, but the round is over if the player misses. Linda has a constant probability of 0.6 of hitting the target and shots are independent of one another.

Find the probability that Linda scores 30 points in a round. The random variable $X$ is the number of points Linda scores in a round.
Find the probability distribution of $X$.
Find the mean and the standard deviation of $X$. A game consists of 2 rounds.
Find the probability that Linda scores more points in round 2 than in round 1.

Edexcel S1 2003 November Q3

3. Cooking sauces are sold in jars containing a stated weight of 500 g of sauce The jars are filled by a machine. The actual weight of sauce in each jar is normally distributed with mean 505 g and standard deviation 10 g .

1. Find the probability of a jar containing less than the stated weight.
2. In a box of 30 jars, find the expected number of jars containing less than the stated weight. The mean weight of sauce is changed so that $1 \%$ of the jars contain less than the stated weight. The standard deviation stays the same.
Find the new mean weight of sauce.

Edexcel S1 2003 November Q4

4. Explain what you understand by

a sample space,
an event. Two events $A$ and $B$ are independent, such that $\mathrm { P } ( A ) = \frac { 1 } { 3 }$ and $\mathrm { P } ( B ) = \frac { 1 } { 4 }$.
Find
$\mathrm { P } ( A \cap B )$,
$\mathrm { P } ( A B )$,
$\mathrm { P } ( A \cup B )$.

Edexcel S1 2003 November Q5

5. The random variable $X$ has the discrete uniform distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { n } , \quad x = 1,2 , \ldots , n$$ Given that $\mathrm { E } ( X ) = 5$,

show that $n = 9$. Find
$\mathrm { P } ( X < 7 )$,
$\operatorname { Var } ( X )$.

Edexcel S1 2003 November Q6

6. A travel agent sells holidays from his shop. The price, in $\pounds$, of 15 holidays sold on a particular day are shown below.

299	1050	2315	999	485
350	169	1015	650	830
99	2100	689	550	475

For these data, find

the mean and the standard deviation,
the median and the inter-quartile range. An outlier is an observation that falls either more than $1.5 \times$ (inter-quartile range) above the upper quartile or more than $1.5 \times$ (inter-quartile range) below the lower quartile.
Determine if any of the prices are outliers. The travel agent also sells holidays from a website on the Internet. On the same day, he recorded the price, $\pounds x$, of each of 20 holidays sold on the website. The cheapest holiday sold was $\pounds 98$, the most expensive was $\pounds 2400$ and the quartiles of these data were $\pounds 305 , \pounds 1379$ and $\pounds 1805$. There were no outliers.
On graph paper, and using the same scale, draw box plots for the holidays sold in the shop and the holidays sold on the website.
Compare and contrast sales from the shop and sales from the website. \section*{END}

Edexcel S1 2004 November Q1

As part of their job, taxi drivers record the number of miles they travel each day. A random sample of the mileages recorded by taxi drivers Keith and Asif are summarised in the back-toback stem and leaf diagram below.

Totals													Asif						Totals
(9)		8	7	4	3	2	1	1	0	18	4	4	5	7					(4)
(11)	9	8	6	5	4	3	3	1	1	19	5	7	8	9	9				(5)
(6)				8	7	4	2	2	0	20	0	2	2	4	4	8			(6)
(6)				9	4	3	1	0	0	21	2	3	5	6	6	7	9		(7)
(4)						6	4	1	1	22	1	1	2	4	5	5	8		(7)
(2)								2	0	23	1	1	3	4	6	6	7	8	(8)
(2)								7	1	24	2	4	8	9					(4)
(1)									9	25	4								(1)
(2)								9	3	26									(0)

Key: 0184 means 180 for Keith and 184 for Asif
The quartiles for these two distributions are summarised in the table below.

	Keith	Asif
Lower quartile	191	$a$
Median	$b$	218
Upper quartile	221	$c$

Find the values of $a , b$ and $c$. Outliers are values that lie outside the limits $$Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \text { and } Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) .$$
On graph paper, and showing your scale clearly, draw a box plot to represent Keith's data.
Comment on the skewness of the two distributions.

Edexcel S1 2004 November Q2

2. An experiment carried out by a student yielded pairs of $( x , y )$ observations such that $$\bar { x } = 36 , \quad \bar { y } = 28.6 , \quad S _ { x x } = 4402 , \quad S _ { x y } = 3477.6$$

Calculate the equation of the regression line of $y$ on $x$ in the form $y = a + b x$. Give your values of $a$ and $b$ to 2 decimal places.
Find the value of $y$ when $x = 45$.

Edexcel S1 2004 November Q3

3. The random variable $X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$. It is known that $$\mathrm { P } ( X \leq 66 ) = 0.0359 \text { and } \mathrm { P } ( X \geq 81 ) = 0.1151 .$$

In the space below, give a clearly labelled sketch to represent these probabilities on a Normal curve.
1. Show that the value of $\sigma$ is 5 .
2. Find the value of $\mu$.
Find $\mathrm { P } ( 69 \leq X \leq 83 )$.

Edexcel S1 2004 November Q4

4. The discrete random variable $X$ has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } 0.2 , & x = - 3 , - 2
\alpha , & x = - 1,0
0.1 , & x = 1,2 . \end{array}$$ Find

$\alpha$,
$\mathrm { P } ( - 1 \leq X < 2 )$,
$\mathrm { F } ( 0.6 )$,
the value of $a$ such that $\mathrm { E } ( a X + 3 ) = 1.2$,
$\operatorname { Var } ( X )$,
$\operatorname { Var } ( 3 X - 2 )$.

Edexcel S1 2004 November Q5

5. The events $A$ and $B$ are such that $\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( B ) = \frac { 1 } { 3 }$ and $\mathrm { P } ( A \cap B ) = \frac { 1 } { 4 }$.

Using the space below, represent these probabilities in a Venn diagram. Hence, or otherwise, find
$\mathrm { P } ( A \cup B )$,
$\mathrm { P } \left( \begin{array} { l l } A & B ^ { \prime } \end{array} \right)$

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	\(\beta\)

\(x\)	15	40	27	39	27	15	20	30	19	24
\(y\)	216	384	234	399	226	132	175	316	187	196

Questions — Edexcel S1 (574 questions)

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