Questions - Edexcel S1 - A-Level Maths

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

Edexcel S1 Q4

4. The owner of a mobile burger-bar believes that hot weather reduces his sales. To investigate the effect on his business he collected data on his daily sales, $\pounds P$, and the maximum temperature, $T ^ { \circ } \mathrm { C }$, on each of 20 days. He then coded the data, using $x = T - 20$ and $y = P - 300$, and calculated the summary statistics given below. $$\Sigma x = 57 , \quad \Sigma y = 2222 , \quad \Sigma x ^ { 2 } = 401 , \quad \Sigma y ^ { 2 } = 305576 , \quad \Sigma x y = 3871 .$$

Find an equation of the regression line of $P$ on $T$. The owner of the bar doesn't believe it is profitable for him to run the bar if he takes less than $\pounds 460$ in a day.
According to your regression line at what maximum daily temperature, to the nearest degree Celsius, does it become unprofitable for him to run the bar?
(3 marks)

Edexcel S1 Q5

5. The discrete random variable $X$ has the probability function shown below. $$P ( X = x ) = \left\{ \begin{array} { c c } k x , & x = 2,3,4,5,6
0 , & \text { otherwise } \end{array} \right.$$

Find the value of $k$.
Show that $\mathrm { E } ( X ) = \frac { 9 } { 2 }$. Find
$\mathrm { P } [ X > \mathrm { E } ( X ) ]$,
$\mathrm { E } ( 2 X - 5 )$,
$\operatorname { Var } ( X )$.

Edexcel S1 Q6

6. A geologist is analysing the size of quartz crystals in a sample of granite. She estimates that the longest diameter of $75 \%$ of the crystals is greater than 2 mm , but only $10 \%$ of the crystals have a longest diameter of more than 6 mm . The geologist believes that the distribution of the longest diameters of the quartz crystals can be modelled by a normal distribution.

Find the mean and variance of this normal distribution. The geologist also estimated that only $2 \%$ of the longest diameters were smaller than 1 mm .
Calculate the corresponding percentage that would be predicted by a normal distribution with the parameters you calculated in part (a).
Hence, comment on the suitability of the normal distribution as a model in this situation.
(2 marks)

Edexcel S1 Q7

7. Jane and Tahira play together in a basketball team. The list below shows the number of points that Jane scored in each of 30 games.

39	19	28	30	18	21	23	15	34	24
29	17	43	12	24	25	41	19	26	40
45	23	21	32	37	24	18	15	24	36

Construct a stem and leaf diagram for these data.
Find the median and quartiles for these data.
Represent these data with a boxplot. Tahira played in the same 30 games and her lowest and highest points total in a game were 19 and 41 respectively. The quartiles for Tahira were 27, 31 and 35 respectively.
Using the same scale draw a boxplot for Tahira's points totals.
Compare and contrast the number of points scored per game by Jane and Tahira. \section*{END}

Edexcel S1 Q1

Joel buys a box of second-hand Jazz and Blues CDs at a car boot sale.

In the box there are 30 CDs, 8 of which were recorded live. 16 of the CDs are predominantly Jazz and 13 of these were recorded in the studio. This information is shown in the following table.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Studio	Live	Total
Jazz	13		16
Blues
Total		8	30

Copy and complete the table above. Joel picks a CD at random to play first.
Find the probability that it is
a Blues CD that was recorded live,
a Jazz CD, given that it was recorded in the studio.

Edexcel S1 Q2

2. The discrete random variable $Q$ has the following probability distribution.

$q$	1	2	3	4	5
$\mathrm { P } ( Q = q )$	$\frac { 1 } { 5 }$	$\frac { 1 } { 5 }$	$\frac { 1 } { 5 }$	$\frac { 1 } { 5 }$	$\frac { 1 } { 5 }$

Write down the name of this distribution. The discrete random variable $R$ has the following probability distribution.
$r$ 14 24 34 44 54
$\mathrm { P } ( R = r )$ $\frac { 1 } { 5 }$ $\frac { 1 } { 5 }$ $\frac { 1 } { 5 }$ $\frac { 1 } { 5 }$ $\frac { 1 } { 5 }$
State the relationship between $R$ and $Q$ in the form $R = a Q + b$. Given that $\mathrm { E } ( Q ) = 3$ and $\operatorname { Var } ( Q ) = 2$,
find $\mathrm { E } ( R )$ and $\operatorname { Var } ( R )$.

Edexcel S1 Q3

3. The random variable $X$ is normally distributed with a mean of 42 and a variance of 18 . Find

$\mathrm { P } ( X \leq 45 )$,
$\mathrm { P } ( 32 \leq X \leq 38 )$,
the value of $x$ such that $\mathrm { P } ( X \leq x ) = 0.95$

Edexcel S1 Q4

4. The ages of 300 houses in a village are recorded giving the following table of results.

Age (years)	Number of houses
0 -	36
20 -	92
40 -	74
60 -	39
100 -	14
200 -	27
300-500	18

Use linear interpolation to estimate for these data

the median,
the limits between which the middle $80 \%$ of the ages lie. An estimate of the mean of these data is calculated to be 86.6 years.
Explain why the mean and median are so different and hence say which you consider best represents the data.

Edexcel S1 Q5

5. The discrete random variable $Y$ has the following cumulative distribution function.

$y$	0	1	2	3	4
$\mathrm {~F} ( Y )$	0.05	0.15	0.35	0.75	1

Write down the probability distribution of $Y$.
Find $\mathrm { P } ( 1 \leq Y < 3 )$.
Show that $\mathrm { E } ( Y ) = 2.7$
Find $\mathrm { E } ( 2 Y + 4 )$.
Find $\operatorname { Var } ( Y )$.

Edexcel S1 Q6

6. A software company sets exams for programmers who wish to qualify to use their packages. Past records show that $55 \%$ of candidates taking the exam for the first time will pass, $60 \%$ of those taking it for the second time will pass, but only $40 \%$ of those taking the exam for the third time will pass. Candidates are not allowed to sit the exam more than three times. A programmer decides to keep taking the exam until he passes or is allowed no further attempts. Find the probability that he will

pass the exam on his second attempt,
pass the exam. Another programmer already has the qualification.
Find, correct to 3 significant figures, the probability that she passed first time. At a particular sitting of the exam there are 400 candidates.
The ratio of those sitting the exam for the first time to those sitting it for the second time to those sitting it for the third time is $5 : 3 : 2$
How many of the 400 candidates would be expected to pass?

Edexcel S1 Q7

7. A doctor wished to investigate the effects of staying awake for long periods on a person's ability to complete simple tasks. She recorded the number of times, $n$, that a subject could clinch his or her fist in 30 seconds after being awake for $h$ hours. The results for one subject were as follows.

$h$ (hours)	16	17	18	19	20	21	22	23	24
$n$	116	114	109	101	94	94	86	81	80

Plot a scatter diagram of $n$ against $h$ for these results. You may use $$\Sigma h = 180 , \quad \Sigma n = 875 , \quad \Sigma h ^ { 2 } = 3660 , \quad \Sigma h n = 17204 .$$
Obtain the equation of the regression line of $n$ on $h$ in the form $n = a + b h$.
Give a practical interpretation of the constant b.
Explain why this regression line would be unlikely to be appropriate for values of $h$ between 0 and 16 .
(2 marks)
Another subject underwent the same tests giving rise to a regression line of $n = 213.4 - 5.87$ h
After how many hours of being awake together would you expect these two subjects to be able to clench their fists the same number of times in 30 seconds?

Edexcel S1 Q1

The weight in kilograms, $w$, of the 15 players in a rugby team was recorded and the results summarised as follows.

$$\Sigma w = 1145.3 , \quad \Sigma w ^ { 2 } = 88042.14$$

Calculate the mean and variance of the weight of the players. Due to injury, one of the players who weighed 79.2 kg was replaced with another player who weighed 63.5 kg .
Without further calculation state the effect of this change on the mean and variance of the weight of the players in the team. Explain your answers.
(4 marks)

Edexcel S1 Q2

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

$x$	1	2	3	4	5
$\mathrm { P } ( X = x )$	$a$	$b$	$\frac { 1 } { 4 }$	$2 a$	$\frac { 1 } { 8 }$

Find an expression for $b$ in terms of $a$.
Find an expression for $\mathrm { E } ( X )$ in terms of $a$. Given that $\mathrm { E } ( X ) = \frac { 45 } { 16 }$,
find the values of $a$ and $b$,

Edexcel S1 Q3

3. The time it takes girls aged 15 to complete an obstacle course is found to be normally distributed with a mean of 21.5 minutes and a standard deviation of 2.2 minutes.

Find the probability that a randomly chosen 15 year-old girl completes the course in less than 25 minutes. A 13 year-old girl completes the course in exactly 19 minutes.
What percentage of 15 year-old girls would she beat over the course? Anyone completing the course in less than 20 minutes is presented with a certificate of achievement. Three friends all complete the course one afternoon.
What is the probability that exactly two of them get certificates?

Edexcel S1 Q4

4. The events $A$ and $B$ are such that $$\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.42 \text { and } \mathrm { P } ( A \cup B ) = 0.76$$ Find

$\mathrm { P } ( A \cap B )$,
$\quad \mathrm { P } \left( A ^ { \prime } \cup B \right)$,
$\mathrm { P } \left( B \mid A ^ { \prime } \right)$.
Show that events $A$ and $B$ are not independent.

Edexcel S1 Q5

5. Each child in class 3A was given a packet of seeds to plant. The stem and leaf diagram below shows how many seedlings were visible in each child's tray one week after planting.

Number of seedlings	(2 \| 1 means 21)	Totals
0	02	(2)
0		(0)
1	1	(1)
1	57	(2)
2	01334	(5)
2	5777899	(7)
3	0001224	(7)
3	5688	(4)
4	134	(3)

Find the median and interquartile range for these data.
Use the quartiles to describe the skewness of the data. Show your method clearly. The mean and standard deviation for these data were 27.2 and 10.3 respectively.
Explaining your answer, state whether you would recommend using these values or your answers to part (a) to summarise these data. Outliers are defined to be values outside of the limits $\mathrm { Q } _ { 1 } - 2 s$ and $\mathrm { Q } _ { 3 } + 2 s$ where $s$ is the standard deviation given above.
Represent these data with a boxplot identifying clearly any outliers.

Edexcel S1 Q6

6. A school introduced a new programme of support lessons in 1994 with a view to improving grades in GCSE English. The table below shows the number of years since 1994, n, and the corresponding percentage of students achieving A to C grades in GCSE English, $p$, for each year.

$n$	1	2	3	4	5	6
$p ( \% )$	35.2	37.1	40.6	39.0	43.4	44.8

Represent these data on a scatter diagram. You may use the following values. $$\Sigma n = 21 , \quad \Sigma p = 240.1 , \quad \Sigma n ^ { 2 } = 91 , \quad \Sigma p ^ { 2 } = 9675.41 , \quad \Sigma n p = 873 .$$
Find an equation of the regression line of $p$ on $n$ and draw it on your graph.
Calculate the product moment correlation coefficient for these data and comment on the suitability of a linear model for the relationship between $n$ and $p$ during this period.

Edexcel S1 Q1

The discrete random variable $Y$ has the following probability distribution.

$y$	${ } ^ { - } 2$	${ } ^ { - } 1$	0	1	2
$\mathrm { P } ( Y = y )$	0.1	0.15	0.2	0.3	0.25

Find

$\mathrm { F } ( 0.5 )$,
$\mathrm { P } \left( { } ^ { - } 1 < Y < 1.9 \right)$,
$\mathrm { E } ( Y )$,
$\mathrm { E } ( 3 Y - 1 )$.

Edexcel S1 Q2

2. A supermarket manager believes that those of her staff on lower rates of pay tend to work more hours of overtime.

Suggest why this might be the case. To investigate her theory the manager recorded the number of hours of overtime, $h$, worked by each of the store's 18 full-time staff during one week. She also recorded each employee's hourly rate of pay, $\pounds p$, and summarised her results as follows: $$\Sigma p = 86 , \quad \Sigma h = 104.5 , \quad \Sigma p ^ { 2 } = 420.58 , \quad \Sigma h ^ { 2 } = 830.25 , \quad \Sigma p h = 487.3$$
Calculate the product moment correlation coefficient for these data.
Comment on the manager's hypothesis.

Edexcel S1 Q3

3. A magazine collected data on the total cost of the reception at each of a random sample of 80 weddings. The data is grouped and coded using $y = \frac { C - 3250 } { 250 }$, where $C$ is the mid-point in pounds of each class, giving $\sum f y = 37$ and $\sum f y ^ { 2 } = 2317$.

Using these values, calculate estimates of the mean and standard deviation of the cost of the receptions in the sample.
Explain why your answers to part (a) are only estimates. The median of the data was $\pounds 3050$.
Comment on the skewness of the data and suggest a reason for it.

Edexcel S1 Q4

4. The random variable $A$ is normally distributed with a mean of 32.5 and a variance of 18.6 Find

$\mathrm { P } ( A < 38.2 )$,
$\mathrm { P } ( 31 \leq A \leq 35 )$, The random variable $B$ is normally distributed with a standard deviation of 7.2
Given also that $\mathrm { P } ( B > 110 ) = 0.138$,
find the mean of $B$.

Edexcel S1 Q5

5. A group of children were each asked to try and complete a task to test hand-eye coordination. Each child repeated the task until he or she had been successful or had made four attempts. The number of attempts made by the children in the group are summarised in the table below.

Number of attempts	1	2	3	4
Number of children	43	26	13	3

Calculate the mean and standard deviation of the number of attempts made by each child. It is suggested that the number of attempts made by each child could be modelled by a discrete random variable $X$ with the probability function $$P ( X = x ) = \left\{ \begin{array} { c c } k \left( 20 - x ^ { 2 } \right) , & x = 1,2,3,4
0 , & \text { otherwise } \end{array} \right.$$
Show that $k = \frac { 1 } { 50 }$.
Find $\mathrm { E } ( X )$.
Comment on the suitability of this model.

Edexcel S1 Q6

6. Serving against his regular opponent, a tennis player has a $65 \%$ chance of getting his first serve in. If his first serve is in he then has a $70 \%$ chance of winning the point but if his first serve is not in, he only has a $45 \%$ chance of winning the point.

Represent this information on a tree diagram. For a point on which this player served to his regular opponent, find the probability that
he won the point,
his first serve went in given that he won the point,
his first serve didn't go in given that he lost the point.

Edexcel S1 Q7

7. Pipes-R-us manufacture a special lightweight aluminium tubing. The price $\pounds P$, for each length, $l$ metres, that the company sells is shown in the table.

$l$ (metres)	0.5	0.8	1.0	1.5	2	4	6
$P ( \pounds )$	2.50	3.40	4.00	5.20	6.00	10.50	15.00

Represent these data on a scatter diagram. You may use $$\Sigma l = 15.8 , \quad \Sigma P = 46.6 , \quad \Sigma l ^ { 2 } = 60.14 , \quad \Sigma l P = 159.77$$
Find the equation of the regression line of $P$ on $l$ in the form $P = a + b l$.
Give a practical interpretation of the constant b. In response to customer demand Pipes- $R$-us decide to start selling tubes cut to specific lengths. Initially the company decides to use the regression line found in part (b) as a pricing formula for this new service.
Calculate the price that Pipes- $R$-us should charge for 5.2 metres of the tubing.
Suggest a reason why Pipes- $R$-us might not offer prices based on the regression line for any length of tubing.

\(q\)	1	2	3	4	5
\(\mathrm { P } ( Q = q )\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)

\(r\)	14	24	34	44	54
\(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)

\(x\)	1	2	3	4	5
\(\mathrm { P } ( X = x )\)	\(a\)	\(b\)	\(\frac { 1 } { 4 }\)	\(2 a\)	\(\frac { 1 } { 8 }\)

39	19	28	30	18	21	23	15	34	24
29	17	43	12	24	25	41	19	26	40
45	23	21	32	37	24	18	15	24	36

\(y\)	\({ } ^ { - } 2\)	\({ } ^ { - } 1\)	0	1	2
\(\mathrm { P } ( Y = y )\)	0.1	0.15	0.2	0.3	0.25

\(l\) (metres)	0.5	0.8	1.0	1.5	2	4	6
\(P ( \pounds )\)	2.50	3.40	4.00	5.20	6.00	10.50	15.00

39	19	28	30	18	21	23	15	34	24
29	17	43	12	24	25	41	19	26	40
45	23	21	32	37	24	18	15	24	36

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39	19	28	30	18	21	23	15	34	24
29	17	43	12	24	25	41	19	26	40
45	23	21	32	37	24	18	15	24	36