Questions - Edexcel S2 - A-Level Maths

find the value of $\sigma$ Graham selects 15 people at random.
Find the probability that fewer than 2 of these people will take less than 12 minutes to complete the test. Jovanna takes a random sample of $n$ people. Using a normal approximation, the probability that fewer than 9 of these $n$ people will take less than 12 minutes to complete the test is 0.3085 to 4 decimal places.
Find the value of $n$.

Edexcel S2 2017 June Q6

6. The continuous random variable $X$ has a probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( x - 2 ) & 2 \leqslant x \leqslant 3
k & 3 < x < 5
k ( 6 - x ) & 5 \leqslant x \leqslant 6
0 & \text { otherwise } \end{array} \right.$$ where $k$ is a positive constant.

Sketch the graph of $\mathrm { f } ( x )$.
Show that the value of $k$ is $\frac { 1 } { 3 }$
Define fully the cumulative distribution function $\mathrm { F } ( x )$.
Hence find the 90th percentile of the distribution.
Find $\mathrm { P } [ \mathrm { E } ( X ) < X < 5.5 ]$
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Edexcel S2 2018 June Q1

In a call centre, the number of telephone calls, $X$, received during any 10 -minute period follows a Poisson distribution with mean 9
1. Find
  1. $\mathrm { P } ( X > 5 )$
  2. $\mathrm { P } ( 4 \leqslant X < 10 )$
The length of a working day is 7 hours.
Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.
Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.

Edexcel S2 2018 June Q2

2. A fair coin is spun 6 times and the random variable $T$ represents the number of tails obtained.

Give two reasons why a binomial model would be a suitable distribution for modelling $T$.
Find $\mathrm { P } ( T = 5 )$
Find the probability of obtaining more tails than heads. A second coin is biased such that the probability of obtaining a head is $\frac { 1 } { 4 }$ This second coin is spun 6 times.
Find the probability that, for the second coin, the number of heads obtained is greater than or equal to the number of tails obtained.

Edexcel S2 2018 June Q3

The length of time, $T$, minutes, spent completing a particular task has probability density function

$$f ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } ( t - 1 ) & 1 < t \leqslant 2
\frac { 1 } { 16 } \left( 14 t - 3 t ^ { 2 } - 8 \right) & 2 < t \leqslant 4
0 & \text { otherwise } \end{array} \right.$$

Use algebraic integration to find $\mathrm { E } ( T )$ Given that $\mathrm { E } \left( T ^ { 2 } \right) = \frac { 267 } { 40 }$
find $\operatorname { Var } ( T )$
Find the cumulative distribution function $\mathrm { F } ( t )$
Find the 20th percentile of the time taken to complete the task.
Find the probability that the time spent completing the task is more than 1.5 minutes. Given that a person has already spent 1.5 minutes on the task,
find the probability that this person takes more than 3 minutes to complete the task.

Edexcel S2 2018 June Q4

David aims to catch the train to work each morning. The scheduled departure time of the train is 0830

The number of minutes after 0830 that the train departs may be modelled by the random variable $X$. Given that $X$ has a continuous uniform distribution over $[ \alpha , \beta ]$ and that $\mathrm { E } ( X ) = 4$ and $\operatorname { Var } ( X ) = 12$

find the value of $\alpha$ and the value of $\beta$. Each morning, the probability that David oversleeps is 0.05 If David oversleeps he will be late for work. If he does not oversleep he will be in time to catch the train, but will be late for work if the train departs after 0835
Find the probability that David will be late for work. Given that David is late for work,
find the probability that he overslept.

Edexcel S2 2018 June Q5

5. Past records show that the proportion of customers buying organic vegetables from Tesson supermarket is 0.35 During a particular day, a random sample of 40 customers from Tesson supermarket was taken and 18 of them bought organic vegetables.

Test, at the $5 \%$ level of significance, whether or not this provides evidence that the proportion of customers who bought organic vegetables has increased. State your hypotheses clearly. The manager of Tesson supermarket claims that the proportion of customers buying organic eggs is different from the proportion of those buying organic vegetables. To test this claim the manager decides to take a random sample of 50 customers.
Using a $5 \%$ level of significance, find the critical region to enable the Tesson supermarket manager to test her claim. The probability for each tail of the region should be as close as possible to $2.5 \%$ During a particular day, a random sample of 50 customers from Tesson supermarket is taken and 8 of them bought organic eggs.
Using your answer to part (b), state whether or not this sample supports the manager's claim. Use a $5 \%$ level of significance.
State the actual significance level of this test. The proportion of customers who buy organic fruit from Tesson supermarket is 0.2 During a particular day, a random sample of 200 customers from Tesson supermarket is taken. Using a suitable approximation, the probability that fewer than $n$ of these customers bought organic fruit is 0.0465 correct to 4 decimal places.
Find the value of $n$.

Edexcel S2 2018 June Q6

The continuous random variable $X$ has the following cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 1
\frac { 4 } { 15 } ( x - 1 ) & 1 < x \leqslant 2
k \left( \frac { a x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } \right) + b & 2 < x \leqslant 4
1 & x > 4 \end{array} \right.$$ where $k , a$ and $b$ are constants.
Given that the mode of $X$ is $\frac { 8 } { 3 }$

show that $a = 4$
Find $\mathrm { P } ( X < 2.5 )$ giving your answer to 3 significant figures.

Edexcel S2 Q1

The lifetime, in tens of hours, of a certain delicate electrical component can be modelled by the random variable $X$ with probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 42 } x , & 0 \leq x < 6
\frac { 1 } { 7 } & 6 \leq x \leq 10
0 , & \text { otherwise } \end{cases}$$

Sketch $\mathrm { f } ( x )$ for all values of $x$.
Find the probability that a component lasts at least 50 hours. A particular device requires two of these components and it will not operate if one or more of the components fail. The device has just been fitted with two new components and the lifetimes of these two components are independent.
Find the probability that the device breaks down within the next 50 hours.

Edexcel S2 Q2

2. The continuous random variable $X$ represents the error, in mm, made when a machine cuts piping to a target length. The distribution of $X$ is rectangular over the interval $[ - 5.0,5.0 ]$. Find

$\mathrm { P } ( X < - 4.2 )$,
$\mathrm { P } ( | X | < 1.5 )$. A supervisor checks a random sample of 10 lengths of piping cut by the machine.
Find the probability that more than half of them are within 1.5 cm of the target length.
(3 marks)
If $X < - 4.2$, the length of piping cannot be used. At the end of each day the supervisor checks a random sample of 60 lengths of piping.
Use a suitable approximation to estimate the probability that no more than 2 of these lengths of piping cannot be used.
(5 marks)

Edexcel S2 Q3

3. An athletics teacher has kept careful records over the past 20 years of results from school sports days. There are always 10 competitors in the javelin competition. Each competitor is allowed 3 attempts and the teacher has a record of the distances thrown by each competitor at each attempt. The random variable $D$ represents the greatest distance thrown by each competitor and the random variable $A$ represents the number of the attempt in which the competitor achieved their greatest distance.

State which of the two random variables $D$ or $A$ is continuous. A new athletics coach wishes to take a random sample of the records of 36 javelin competitors.
Specify a suitable sampling frame and explain how such a sample could be taken.
(2 marks)
The coach assumes that $\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }$, and is therefore surprised to find that 20 of the 36 competitors in the sample achieved their greatest distance on their second attempt. Using a suitable approximation, and assuming that $\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }$,
find the probability that at least 20 of the competitors achieved their greatest distance on their second attempt.
(6 marks)
Comment on the assumption that $\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }$.

Edexcel S2 Q4

4. From past records a manufacturer of glass vases knows that $15 \%$ of the production have slight defects. To monitor the production, a random sample of 20 vases is checked each day and the number of vases with slight defects is recorded.

Using a 5\% significance level, find the critical regions for a two-tailed test of the hypothesis that the probability of a vase with slight defects is 0.15 . The probability of rejecting, in either tail, should be as close as possible to $2.5 \%$.
State the actual significance level of the test described in part (a). A shop sells these vases at a rate of 2.5 per week. In the 4 weeks of December the shop sold 15 vases.
Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there is evidence that the rate of sales per week had increased in December.
(6 marks)

Edexcel S2 Q5

5. The continuous random variable $T$ represents the time in hours that students spend on homework. The cumulative distribution function of $T$ is $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0
k \left( 2 t ^ { 3 } - t ^ { 4 } \right) & 0 \leq t \leq 1.5
1 , & t > 1.5 \end{cases}$$ where $k$ is a positive constant.

Show that $k = \frac { 16 } { 27 }$.
Find the proportion of students who spend more than 1 hour on homework.
Find the probability density function $\mathrm { f } ( t )$ of $T$.
Show that $\mathrm { E } ( T ) = 0.9$.
Show that $\mathrm { F } ( \mathrm { E } ( T ) ) = 0.4752$. A student is selected at random. Given that the student spent more than the mean amount of time on homework,
find the probability that this student spent more than 1 hour on homework.

Edexcel S2 Q6

6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that

at least 4 customers arrive in the next 10 minutes,
no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
Find the probability that no customers arrive in at most one of these periods. The post office is open for $3 \frac { 1 } { 2 }$ hours on Wednesday mornings.
Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END

Edexcel S2 Specimen Q1

A school held a disco for years 9,10 and 11 which was attended by 500 pupils. The pupils were registered as they entered the disco. The disco organisers were keen to assess the success of the event. They designed a questionnaire to obtain information from those who attended.
1. State one advantage and one disadvantage of using a sample survey rather than a census.
2. Suggest a suitable sampling frame.
3. Identify the sampling units.
4. A piece of string $A B$ has length 12 cm . A child cuts the string at a randomly chosen point $P$, into two pieces. The random variable $X$ represents the length, in cm, of the piece $A P$.
5. Suggest a suitable model for the distribution of $X$ and specify it fully
6. Find the cumulative distribution function of $X$.
7. Write down $\mathrm { P } ( X < 4 )$.
8. A manufacturer of chocolates produces 3 times as many soft centred chocolates as hard centred ones.
Assuming that chocolates are randomly distributed within boxes of chocolates, find the probability that in a box containing 20 chocolates there are
equal numbers of soft centred and hard centred chocolates,
fewer than 5 hard centred chocolates. A large box of chocolates contains 100 chocolates.
Write down the expected number of hard centred chocolates in a large box.

Edexcel S2 Specimen Q4

4. A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:

No. of errors	0	1	2	3	4	5
No. of pages	37	65	60	49	27	12

Show that the mean number of errors per page in this sample of pages is 2 .
Find the variance of the number of errors per page in this sample.
Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution.
(1) Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2 .
Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a $5 \%$ level of significance.
(6)

Edexcel S2 Specimen Q5

5. In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30 . During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.

Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10 . The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.
Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to $2.5 \%$.
State the significance level of this test giving your answer to 2 significant figures.

Edexcel S2 Specimen Q6

6. A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25 . The sheep are randomly spread throughout the field.

Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. Calculate the probability that a randomly selected sample square contains
no sheep,
more than 2 sheep. A sheepdog has been sent into the field to round up the sheep.
Explain why the model may no longer be applicable. In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.
Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep.

Edexcel S2 Specimen Q7

7. The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by $$f ( x ) = \left\{ \begin{aligned} \frac { 1 } { 20 } x ^ { 3 } , & 1 \leq x \leq 3
0 , & \text { otherwise } \end{aligned} \right.$$

Sketch $\mathrm { f } ( x )$ for all values of $x$.
Calculate $\mathrm { E } ( X )$.
Show that the standard deviation of $X$ is 0.459 to 3 decimal places.
Show that for $1 \leq x \leq 3 , \mathrm { P } ( X \leq x )$ is given by $\frac { 1 } { 80 } \left( x ^ { 4 } - 1 \right)$ and specify fully the cumulative distribution function of $X$.
Find the interquartile range for the random variable $X$. Some statisticians use the following formula to estimate the interquartile range: $$\text { interquartile range } = \frac { 4 } { 3 } \times \text { standard deviation. }$$
Use this formula to estimate the interquartile range in this case, and comment.

Edexcel S2 Q1

(a) Briefly describe the difference between a census and a sample survey.
(b) Illustrate the difference by considering the case of a village council which has to decide whether or not to build a new village hall.

Given that the council decides to use a sample survey,
(c) suggest suitable sampling units.

Edexcel S2 Q2

2. The number of copies of The Statistician that a newsagent sells each week is modelled by a Poisson distribution. On average, he sells 1.5 copies per week.

Find the probability that he sells no copies in a particular week.
If he stocks 5 copies each week, find the probability he will not have enough copies to meet that week's demand.
Find the minimum number of copies that he should stock in order to have at least a $95 \%$ probability of being able to satisfy the week's demand.

Edexcel S2 Q3

3. A die is rolled 60 times, and results in 16 sixes.

Use a suitable approximation to test, at the $5 \%$ significance level, whether the probability of scoring a six is $\frac { 1 } { 6 }$ or not. State your hypotheses clearly.
Describe how you would change the test if you wished to investigate whether the probability of scoring a six is greater than $\frac { 1 } { 6 }$. Carry out this modified test.

Edexcel S2 Q4

4. A continuous random variable $X$ has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l } 0
\frac { 1 } { 84 } \left( x ^ { 2 } - 16 \right)
1 \end{array} \right.$$ $$\begin{aligned} & x < 4 ,
& 4 \leq x \leq 10 ,
& x > 10 . \end{aligned}$$

Find the median value of $X$.
Find the interquartile range for $X$.
Find the probability density function $\mathrm { f } ( x )$ of $X$.
Sketch the graph of $\mathrm { f } ( x )$ and hence write down the mode of $X$, explaining how you obtain your answer from the graph. \section*{STATISTICS 2 (A) TEST PAPER 1 Page 2}

Edexcel S2 Q5

Lupin seeds are sold in packets of 15 . On average, 9 seeds in a packet are green and 6 are red. Find, to 2 decimal places, the probability that in any particular packet there are
1. less than 2 red seeds,
2. more red than green seeds.
The seeds from 10 packets are then combined together.
Use a suitable approximation to find the probability that the total number of green seeds is more than 100 .

Edexcel S2 Q6

6. Patients suffering from 'flu are treated with a drug. The number of days, $t$, that it then takes for them to recover is modelled by the continuous random variable $T$ with the probability density function $$\begin{array} { l l } \mathrm { f } ( t ) = \frac { 3 t ^ { 2 } ( 4 - t ) } { 64 } & 0 \leq t \leq 4
\mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$

Find the mean and standard deviation of $T$.
Find the probability that a patient takes more than 3 days to recover.
Two patients are selected at random. Find the probability that they both recover within three days.
Comment on the suitability of the model.

Questions — Edexcel S2 (494 questions)

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Edexcel S2 2017 June Q5

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