Questions S2 - A-Level Maths

$\boldsymbol { x }$	0	1	2	3	4	5	6
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0	0.19	0.26	0.20	0.13	0.07	0.15

Find the probability that a member borrows more than 3 books.
Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books.
Show that the mean of $X$ is 3.08, and calculate the variance of $X$.
One of the library staff notices that the values of the mean and the variance of $X$ are similar and suggests that a Poisson distribution could be used to model $X$. Without further calculations, give two reasons why a Poisson distribution would not be suitable to model $X$.
The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
1. the mean amount that will be paid by a member;
2. the standard deviation of the amount that will be paid by a member.

AQA S2 2016 June Q4

4 A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2 . The error, $X ^ { \circ } \mathrm { C }$, resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f ( x ) = \left\{ \begin{array} { l c } k & 0 \leqslant x \leqslant 0.1
0 & \text { otherwise } \end{array} \right.$$

State the value of $k$.
Find the probability that the error resulting from this rounding down is greater than $0.03 ^ { \circ } \mathrm { C }$.
1. State the value for $\mathrm { E } ( X )$.
2. Use integration to find the value for $\mathrm { E } \left( X ^ { 2 } \right)$.
3. Hence find the value for the standard deviation of $X$.
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AQA S2 2016 June Q5

2 marks

5 A car manufacturer keeps a record of how many of the new cars that it has sold experience mechanical problems during the first year. The manufacturer also records whether the cars have a petrol engine or a diesel engine. Data for a random sample of 250 cars are shown in the table.

	Problems during first 3 months	Problems during first year but after first 3 months	No problems during first year	Total
Petrol engine	10	35	170	215
Diesel engine	4	8	23	35
Total	14	43	193	250

Use a $\chi ^ { 2 }$-test to investigate, at the $10 \%$ significance level, whether there is an association between the mechanical problems experienced by a new car from this manufacturer and the type of engine.
Arisa is planning to buy a new car from this manufacturer. She would prefer to buy a car with a diesel engine, but a friend has told her that cars with diesel engines experience more mechanical problems. Based on your answer to part (a), state, with a reason, the advice that you would give to Arisa.
[0pt] [2 marks]

AQA S2 2016 June Q6

2 marks

6 Gerald is a scientist who studies sand lizards. He believes that sand lizards on islands are, on average, shorter than those on the mainland. The population of sand lizards on the mainland has a mean length of 18.2 cm and a standard deviation of 1.8 cm . Gerald visited three islands, $\mathrm { A } , \mathrm { B }$ and C , and measured the length, $X$ centimetres, of each of a sample of $n$ sand lizards on each island. The samples may be regarded as random. The data are shown in the table.

AQA S2 2016 June Q7

7 The continuous random variable $X$ has a cumulative distribution function $\mathrm { F } ( x )$, where $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1
\frac { 1 } { 4 } ( x - 1 ) & 1 \leqslant x < 4
\frac { 1 } { 16 } \left( 12 x - x ^ { 2 } - 20 \right) & 4 \leqslant x \leqslant 6
1 & x > 6 \end{array} \right.$$

Sketch the probability density function, $\mathrm { f } ( x )$, on the grid below.
Find the mean value of $X$.

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.

Edexcel S2 Q7

7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per $\mathrm { cm } ^ { 2 }$. One particular patch, of area $1 \mathrm {~cm} ^ { 2 }$, is selected at random. Assuming that the number of daisies per $\mathrm { cm } ^ { 2 }$ has a Poisson distribution,

find the probability that the chosen patch contains
1. no daisies,
2. one daisy. Ten such patches are chosen. Using your answers to part (a),
find the probability that the total number of daisies is less than two.
By considering the distribution of daisies over patches of $10 \mathrm {~cm} ^ { 2 }$, use the Poisson distribution to find the probability that a particular area of $10 \mathrm {~cm} ^ { 2 }$ contains no more than one daisy.
Compare your answers to parts (b) and (c).
Use a suitable approximation to find the probability that a patch of area $1 \mathrm {~m} ^ { 2 }$ contains more than 8100 daisies.

Edexcel S2 Q1

(a) Explain why it is often useful to take samples as a means of obtaining information.
(b) Briefly define the term sampling frame.
(c) Suggest a suitable sampling frame for a sample survey on a proposal to install speed humps on a road.
An insurance company conducts its business by using a Call Centre. The average number of calls per minute is $3 \cdot 5$. In the first minute after a TV advertisement is shown, the number of calls received is 7 .
(a) Stating your hypotheses carefully, and working at the $5 \%$ significance level, test whether the advertisement has had an effect.
(b) Find the number of calls that would be required in the first minute for the null hypothesis to be rejected at the $0.1 \%$ significance level.
On average, $35 \%$ of the candidates in a certain subject get an A or B grade in their exam. In a class of 20 students, find the probability that
(a) less than 5 get A or B grades,
(b) exactly 8 get A or B grades.

Five such classes of 20 students are combined to sit the exam.
(c) Use a suitable approximation to find the probability that less than a quarter of the total get A or B grades.

Edexcel S2 Q4

4. Light bulbs produced in a certain factory have lifetimes, in 100 s of hours, whose distribution is modelled by the random variable $X$ with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x ( 3 - x ) } { 9 } , & 0 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$

Sketch $\mathrm { f } ( x )$.
Write down the mean lifetime of a bulb.
Show that ten times as many bulbs fail before 200 hours as survive beyond 250 hours.
Given that a bulb lasts for 200 hours, find the probability that it will then last for at least another 50 hours.
State, with a reason, whether you consider that the density function $f$ is a realistic model for the lifetimes of light bulbs. \section*{STATISTICS 2 (A) TEST PAPER 2 Page 2}

Edexcel S2 Q5

In a packet of 40 biscuits, the number of currants in each biscuit is as follows

Number of currants, $x$	0	1	2	3	4	5	6
Number of biscuits	4	9	11	8	4	3	1

Find the mean and variance of the random variable $X$ representing the number of currants per biscuit.
State an appropriate model for the distribution of $X$, giving two reasons for your answer. Another machine produces biscuits with a mean of 1.9 currants per biscuit.
Determine which machine is more likely to produce a biscuit with at least two currants.

Edexcel S2 Q6

6. A greengrocer sells apples from a barrel in his shop. He claims that no more than $5 \%$ of the apples are of poor quality. When he takes 10 apples out for a customer, 2 of them are bad.

Stating your hypotheses clearly, test his claim at the $1 \%$ significance level.
State an assumption that has been made about the selection of the apples.
When five other customers also buy 10 apples each, the numbers of bad apples they get are $1,3,1,2$ and 1 respectively. By combining all six customers' results, and using a suitable approximation, test at the $1 \%$ significance level whether the combined results provide evidence that the proportion of bad apples in the barrel is greater than $5 \%$.
Comment briefly on your results in parts (a) and (c).

Edexcel S2 Q7

7. Some children are asked to mark the centre of a scale 10 cm long. The position they choose is indicated by the variable $X$, where $0 \leq X \leq 10$. Initially, $X$ is modelled as a random variable with a continuous uniform distribution.

Find the mean and the standard deviation of $X$. It is suggested that a better model would be the distribution with probability density function $$f ( x ) = c x , 0 \leq x \leq 5 , \quad f ( x ) = c ( 10 - x ) , 5 < x \leq 10 , \quad f ( x ) = 0 \text { otherwise. }$$
Write down the mean of $X$.
Find $c$, and hence find the standard deviation of $X$ in this model.
Find $\mathrm { P } ( 4 < X < 6 )$. It is then proposed that an even better model for $X$ would be a Normal distribution with the mean and standard deviation found in parts (b) and (c).
Use these results to find $\mathrm { P } ( 4 < X < 6 )$ in the third model.
Compare your answer with (d). Which model do you think is most appropriate? (1 mark)

Edexcel S2 Q1

Explain what is meant by
1. a population,
2. a sampling unit.
Suggest suitable sampling frames for surveys of
families who have holidays in Greece,
mothers with children under two years old.

Edexcel S2 Q2

2. A continuous random variable $X$ has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k & 5 \leq x \leq 15 ,
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

Find $k$ and specify the cumulative density function $\mathrm { F } ( x )$.
Write down the value of $\mathrm { P } ( X < 8 )$.

Edexcel S2 Q3

3. A coin is tossed 20 times, giving 16 heads.

Test at the $1 \%$ significance level whether the coin is fair, stating your hypotheses clearly.
Find the critical region for the same test at the $0.1 \%$ significance level.

Edexcel S2 Q4

4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is $p$.

Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
Evaluate this probability when $p = 0.6$.

Edexcel S2 Q5

5. In a certain school, $32 \%$ of Year 9 pupils are left-handed. A random sample of 10 Year 9 pupils is chosen.

Find the probability that none are left-handed.
Find the probability that at least two are left-handed.
Use a suitable approximation to find the probability of getting more than 5 but less than 15 left-handed pupils in a group of 35 randomly selected Year 9 pupils.
Explain what adjustment is necessary when using this approximation. \section*{STATISTICS 2 (A) TEST PAPER 3 Page 2}

Edexcel S2 Q6

A sample of radioactive material decays randomly, with an approximate mean of 1.5 counts per minute.
1. Name a distribution that would be suitable for modelling the number of counts per minute.
Give any parameters required for the model.
Find the probability of at least 4 counts in a randomly chosen minute.
Find the probability of 3 counts or fewer in a random interval lasting 5 minutes. More careful measurements, over 50 one-minute intervals, give the following data for $x$, the number of counts per minute: $$\sum x = 84 , \quad \sum x ^ { 2 } = 226$$
Decide whether these data support your answer to part (a).
Use the improved data to find probability of exactly two counts in a given one-minute interval.

Edexcel S2 Q7

7. Each day on the way to work, a commuter encounters a similar traffic jam. The length of time, in 10-minute units, spent waiting in the traffic jam is modelled by the random variable $T$ with the cumulative distribution function: $$\begin{array} { l l } \mathrm { F } ( t ) = 0 & t < 0 ,
\mathrm {~F} ( t ) = \frac { t ^ { 2 } \left( 3 t ^ { 2 } - 16 t + 24 \right) } { 16 } & 0 \leq t \leq 2 ,
\mathrm {~F} ( t ) = 1 & t > 2 . \end{array}$$

Show that 0.77 is approximately the median value of $T$.
Given that he has already waited for 12 minutes, find the probability that he will have to wait another 3 minutes.
Find, and sketch, the probability density function of $T$.
Hence find the modal value of $T$.
Comment on the validity of this model.

Edexcel S2 Q1

A random sample is to be taken from the A-level results obtained by the final-year students in a Sixth Form College. Suggest
1. suitable sampling units,
2. a suitable sampling frame.
3. Would it be advisable simply to use the results of all those doing A-level Maths?
Explain your answer.

\(\boldsymbol { x }\)	0	1	2	3	4	5	6
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0	0.19	0.26	0.20	0.13	0.07	0.15

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