Questions S2 - A-Level Maths

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.

AQA S2 2006 January Q1

1 A study undertaken by Goodhealth Hospital found that the number of patients each month, $X$, contracting a particular superbug can be modelled by a Poisson distribution with a mean of 1.5 .

1. Calculate $\mathrm { P } ( X = 2 )$.
2. Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.
1. Write down the distribution of $Y$, the number of patients contracting this superbug in a given 6-month period.
2. Find the probability that at least 12 patients will contract this superbug during a given 6-month period.
State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.

AQA S2 2006 January Q2

2 Year 12 students at Newstatus School choose to participate in one of four sports during the Spring term. The students' choices are summarised in the table.

	Squash	Badminton	Archery	Hockey	Total
Male	5	16	30	19	70
Female	4	20	33	53	110
Total	9	36	63	72	180

Use a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to determine whether the choice of sport is independent of gender.
Interpret your result in part (a) as it relates to students choosing hockey.

AQA S2 2006 January Q3

3 The time, $T$ minutes, that parents have to wait before seeing a mathematics teacher at a school parents' evening can be modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$. At a recent parents' evening, a random sample of 9 parents was asked to record the times that they waited before seeing a mathematics teacher. The times, in minutes, are $$\begin{array} { l l l l l l l l l } 5 & 12 & 10 & 8 & 7 & 6 & 9 & 7 & 8 \end{array}$$

Construct a $90 \%$ confidence interval for $\mu$.
Comment on the headteacher's claim that the mean time that parents have to wait before seeing a mathematics teacher is 5 minutes.

AQA S2 2006 January Q4

A random variable $X$ has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} k & a < x < b
0 & \text { otherwise } \end{cases}$$
1. Show that $k = \frac { 1 } { b - a }$.
2. Prove, using integration, that $\mathrm { E } ( X ) = \frac { 1 } { 2 } ( a + b )$.
The error, $X$ grams, made when a shopkeeper weighs out loose sweets can be modelled by a rectangular distribution with the following probability density function: $$f ( x ) = \begin{cases} k & - 2 < x < 4
0 & \text { otherwise } \end{cases}$$
1. Write down the value of the mean, $\mu$, of $X$.
2. Evaluate the standard deviation, $\sigma$, of $X$.
3. Hence find $\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)$.

AQA S2 2006 January Q5

5 The Globe Express agency organises trips to the theatre. The cost, $\pounds X$, of these trips can be modelled by the following probability distribution:

$\boldsymbol { x }$	40	45	55	74
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.30	0.24	0.36	0.10

Calculate the mean and standard deviation of $X$.
For special celebrity charity performances, Globe Express increases the cost of the trips to $\pounds Y$, where $$Y = 10 X + 250$$ Determine the mean and standard deviation of $Y$.

AQA S2 2006 January Q6

6 In previous years, the marks obtained in a French test by students attending Topnotch College have been modelled satisfactorily by a normal distribution with a mean of 65 and a standard deviation of 9 . Teachers in the French department at Topnotch College suspect that this year their students are, on average, underachieving. In order to investigate this suspicion, the teachers selected a random sample of 35 students to take the French test and found that their mean score was 61.5.

Investigate, at the $5 \%$ level of significance, the teachers' suspicion.
Explain, in the context of this question, the meaning of a Type I error.

AQA S2 2006 January Q7

7 Engineering work on the railway network causes an increase in the journey time of commuters travelling into work each morning. The increase in journey time, $T$ hours, is modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} 4 t \left( 1 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 1
0 & \text { otherwise } \end{cases}$$

Show that $\mathrm { E } ( T ) = \frac { 8 } { 15 }$.
1. Find the cumulative distribution function, $\mathrm { F } ( t )$, for $0 \leqslant t \leqslant 1$.
2. Hence, or otherwise, for a commuter selected at random, find $$\mathrm { P } ( \text { mean } < T < \text { median } )$$

AQA S2 2006 January Q8

8 Bottles of sherry nominally contain 1000 millilitres. After the introduction of a new method of filling the bottles, there is a suspicion that the mean volume of sherry in a bottle has changed. In order to investigate this suspicion, a random sample of 12 bottles of sherry is taken and the volume of sherry in each bottle is measured. The volumes, in millilitres, of sherry in these bottles are found to be

996	1006	1009	999	1007	1003
998	1010	997	996	1008	1007

Assuming that the volume of sherry in a bottle is normally distributed, investigate, at the $5 \%$ level of significance, whether the mean volume of sherry in a bottle differs from 1000 millilitres.

AQA S2 2007 January Q1

1 Alan's journey time, in minutes, to travel home from work each day is known to be normally distributed with mean $\mu$. Alan records his journey time, in minutes, on a random sample of 8 days as being $$\begin{array} { l l l l l l l l } 36 & 38 & 39 & 40 & 50 & 35 & 36 & 42 \end{array}$$ Construct a $95 \%$ confidence interval for $\mu$.

AQA S2 2007 January Q2

2 The number of computers, $A$, bought during one day from the Amplebuy computer store can be modelled by a Poisson distribution with a mean of 3.5. The number of computers, $B$, bought during one day from the Bestbuy computer store can be modelled by a Poisson distribution with a mean of 5.0 .

1. Calculate $\mathrm { P } ( A = 4 )$.
2. Determine $\mathrm { P } ( B \leqslant 6 )$.
3. Find the probability that a total of fewer than 10 computers is bought from these two stores on one particular day.
Calculate the probability that a total of fewer than 10 computers is bought from these two stores on at least 4 out of 5 consecutive days.
The numbers of computers bought from the Choicebuy computer store over a 10-day period are recorded as $$\begin{array} { l l l l l l l l l l } 8 & 12 & 6 & 6 & 9 & 15 & 10 & 8 & 6 & 12 \end{array}$$
1. Calculate the mean and variance of these data.
2. State, giving a reason based on your results in part (c)(i), whether or not a Poisson distribution provides a suitable model for these data.

AQA S2 2007 January Q3

3 The handicap committee of a golf club has indicated that the mean score achieved by the club's members in the past was 85.9 . A group of members believes that recent changes to the golf course have led to a change in the mean score achieved by the club's members and decides to investigate this belief. A random sample of the scores, $x$, of 100 club members was taken and is summarised by $$\sum x = 8350 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 15321$$ where $\bar { x }$ denotes the sample mean.
Test, at the $5 \%$ level of significance, the group's belief that the mean score of 85.9 has changed.

AQA S2 2007 January Q4

4 The number of fish, $X$, caught by Pearl when she goes fishing can be modelled by the following discrete probability distribution:

$\boldsymbol { x }$	1	2	3	4	5	6	$\geqslant 7$
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.01	0.05	0.14	0.30	$k$	0.12	0

Find the value of $k$.
Find:
1. $\mathrm { E } ( X )$;
2. $\operatorname { Var } ( X )$.
When Pearl sells her fish, she earns a profit, in pounds, given by $$Y = 5 X + 2$$ Find:
1. $\mathrm { E } ( Y )$;
2. the standard deviation of $Y$.

\(\boldsymbol { x }\)	40	45	55	74
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.30	0.24	0.36	0.10

\(\boldsymbol { x }\)	1	2	3	4	5	6	\(\geqslant 7\)
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.01	0.05	0.14	0.30	\(k\)	0.12	0

Questions S2 (1597 questions)

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Edexcel S2 2018 June Q5

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Edexcel S2 Specimen Q1

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AQA S2 2006 January Q1

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