Questions - Edexcel S2 - A-Level Maths

7. In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key $T \mathrm {~s}$ after the star first appears, a simple model of the game assumes that $T$ is a continuous uniform random variable defined over the interval $[ 0,1 ]$.

Write down $\mathrm { P } ( \mathrm { T } < 0.2 )$.
Write down E(T).
Use integration to find $\operatorname { Var } ( T )$. A group of 20 children each play this game once.
Find the probability that no more than 4 children stop the star in less than 0.2 s . The children are allowed to practise this game so that this continuous uniform model is no longer applicable.
Explain how you would expect the mean and variance of T to change. It is found that a more appropriate model of the game when played by experienced players assumes that $T$ has a probability density function $\mathrm { g } ( t )$ given by $$\mathrm { g } ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5
4 - 4 t , & 0.5 \leq t \leq 1
0 , & \text { otherwise } \end{cases}$$
Using this model show that $\mathrm { P } ( T < 0.2 ) = 0.08$. A group of 75 experienced players each played this game once.
Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s . \section*{END} Items included with question papers Nil Materials required for examination
Answer Book (AB16)
Graph Paper (ASG2)
Mathematical Formulae (Lilac) Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. \section*{Edexcel GCE
Statistics S2
(New Syllabus)
Advanced/Advanced Subsidiary} Wednesday 23 January 2002 - Afternoon
Time: 1 hour 30 minutes In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. Explain what you understand by
2. a population,
3. a statistic.
A questionnaire concerning attitudes to classes in a college was completed by a random sample of 50 students. The students gave the college a mean approval rating of 75\%.
Identify the population and the statistic in this situation.
Explain what you understand by the sampling distribution of this statistic.
2. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2.5. The firm appoints a new salesman and wants to find out whether or not house sales increase as a result. After the appointment of the salesman, the number of house sales in a randomly chosen 4-week period is 14 . Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not the new salesman has increased house sales.
3. An airline knows that overall $3 \%$ of passengers do not turn up for flights. The airline decides to adopt a policy of selling more tickets than there are seats on a flight. For an aircraft with 196 seats, the airline sold 200 tickets for a particular flight.
Write down a suitable model for the number of passengers who do not turn up for this flight after buying a ticket. By using a suitable approximation, find the probability that
more than 196 passengers turn up for this flight,
there is at least one empty seat on this flight.
4. Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable $X$ represents the time, in minutes, after 7.55 a.m. when the bus arrives.
Suggest a suitable model for the distribution of $X$ and specify it fully.
Calculate the mean time of arrival of the bus.
Find the cumulative distribution function of $X$. Jean will be late for work if the bus arrives after 8.05 a.m.
Find the probability that Jean is late for work.
5. An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.
Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour.
Find the probability that in a randomly chosen hour
1. all Internet users connect at their first attempt,
2. more than 4 users fail to connect at their first attempt.
Write down the distribution of the number of users failing to connect at their first attempt in an 8 -hour period.
Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period.
6. The owner of a small restaurant decides to change the menu. A trade magazine claims that $40 \%$ of all diners choose organic foods when eating away from home. On a randomly chosen day there are 20 diners eating in the restaurant.
Assuming the claim made by the trade magazine to be correct, suggest a suitable model to describe the number of diners $X$ who choose organic foods.
Find $\mathrm { P } ( 5 < X < 15 )$.
Find the mean and standard deviation of $X$. The owner decides to survey her customers before finalising the new menu. She surveys 10 randomly chosen diners and finds 8 who prefer eating organic foods.
Test, at the $5 \%$ level of significance, whether or not there is reason to believe that the proportion of diners in her restaurant who prefer to eat organic foods is higher than the trade magazine's claim. State your hypotheses clearly.
7. A continuous random variable $X$ has cumulative distribution function $\mathrm { F } ( x )$ given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 0 ,
k x ^ { 2 } + 2 k x , & 0 \leq x \leq 2 ,
8 k , & x > 2 . \end{array} \right.$$
Show that $k = \frac { 1 } { 8 }$.
Find the median of $X$.
Find the probability density function $\mathrm { f } ( x )$.
Sketch $\mathrm { f } ( x )$ for all values of $x$.
Write down the mode of $X$.
Find $\mathrm { E } ( X )$.
Comment on the skewness of this distribution. Items included with question papers
Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided. Full marks may be obtained for answers to ALL questions.
This paper has seven questions. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. The manager of a leisure club is considering a change to the club rules. The club has a large membership and the manager wants to take the views of the members into consideration before deciding whether or not to make the change.
2. Explain briefly why the manager might prefer to use a sample survey rather than a census to obtain the views.
3. Suggest a suitable sampling frame.
4. Identify the sampling units.
5. A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ is taken from a finite population. A statistic $Y$ is based on this sample.
6. Explain what you understand by the statistic $Y$.
7. Give an example of a statistic.
8. Explain what you understand by the sampling distribution of $Y$.
9. The continuous random variable $R$ is uniformly distributed on the interval $\alpha \leq R \leq \beta$. Given that $\mathrm { E } ( R ) = 3$ and $\operatorname { Var } ( R ) = \frac { 25 } { 3 }$, find
10. the value of $\alpha$ and the value of $\beta$,
11. $\mathrm { P } ( R < 6.6 )$.
12. Past records show that $20 \%$ of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps was taken and 2 of them had bought them in single packets.
13. Use these data to test, at the $5 \%$ level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly.
  (6)
At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03 . To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.
Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03 . The probability for each tail of the region should be as close as possible to $2.5 \%$.
Write down the significance level of this test.
5. A garden centre sells canes of nominal length 150 cm . The canes are bought from a supplier who uses a machine to cut canes of length $L$ where $L \sim \mathrm {~N} \left( \mu , 0.3 ^ { 2 } \right)$.
Find the value of $\mu$, to the nearest 0.1 cm , such that there is only a $5 \%$ chance that a cane supplied to the garden centre will have length less than 150 cm . A customer buys 10 of these canes from the garden centre.
Find the probability that at most 2 of the canes have length less than 150 cm . Another customer buys 500 canes.
Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm .
6. From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m .
Find the probability that in a randomly chosen 25 m length of twine there will be exactly 4 faults. The twine is usually sold in balls of length 100 m . A customer buys three balls of twine.
Find the probability that only one of them will have fewer than 6 faults. As a special order a ball of twine containing 500 m is produced.
Using a suitable approximation, find the probability that it will contain between 23 and 33 faults inclusive.
7. The continuous random variable $X$ has probability density function $$f ( x ) = \begin{cases} \frac { x } { 15 } , & 0 \leq x \leq 2
\frac { 2 } { 15 } , & 2 < x < 7
\frac { 4 } { 9 } - \frac { 2 x } { 45 } , & 7 \leq x \leq 10
0 , & \text { otherwise } \end{cases}$$
Sketch $\mathrm { f } ( x )$ for all values of $x$.
1. Find expressions for the cumulative distribution function, $\mathrm { F } ( x )$, for $0 \leq x \leq 2$ and for $7 \leq x \leq 10$.
2. Show that for $2 < x < 7 , \mathrm {~F} ( x ) = \frac { 2 x } { 15 } - \frac { 2 } { 15 }$.
3. Specify $\mathrm { F } ( x )$ for $x < 0$ and for $x > 10$.
Find $\mathrm { P } ( X \leq 8.2 )$.
Find, to 3 significant figures, $\mathrm { E } ( X )$. Items included with question papers Nil Answer Book (AB16)
Graph Paper (ASG2)
Mathematical Formulae (Lilac) Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Paper Reference(s)
6684 \section*{Edexcel GCE
Statistics S2
Advanced/Advanced Subsidiary Friday 24 January 2003 - Morning Time: 1 hour 30 minutes} In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has six questions. You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. An engineer measures, to the nearest cm , the lengths of metal rods.
2. Suggest a suitable model to represent the difference between the true lengths and the measured lengths.
3. Find the probability that for a randomly chosen rod the measured length will be within 0.2 cm of the true length.
Two rods are chosen at random.
Find the probability that for both rods the measured lengths will be within 0.2 cm of their true lengths.
2. A single observation $x$ is to be taken from a Poisson distribution with parameter $\lambda$. This observation is to be used to test $\mathrm { H } _ { 0 } : \lambda = 7$ against $\mathrm { H } _ { 1 } : \lambda \neq 7$.
Using a $5 \%$ significance level, find the critical region for this test assuming that the probability of rejection in either tail is as close as possible to $2.5 \%$.
Write down the significance level of this test. The actual value of $x$ obtained was 5 .
State a conclusion that can be drawn based on this value.
3. A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.
Write down two conditions that must apply for this model to be applicable. Assuming this model and a mean of 0.7 weeds per $\mathrm { m } ^ { 2 }$, find
the probability that in a randomly chosen plot of size $4 \mathrm {~m} ^ { 2 }$ there will be fewer than 3 of these weeds.
Using a suitable approximation, find the probability that in a plot of $100 \mathrm {~m} ^ { 2 }$ there will be more than 66 of these weeds.
4. The continuous random variable $X$ has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0
\frac { 1 } { 3 } x ^ { 2 } \left( 4 - x ^ { 2 } \right) , & 0 \leq x \leq 1
1 & x > 1 \end{cases}$$
Find $\mathrm { P } ( X > 0.7 )$.
Find the probability density function $\mathrm { f } ( x )$ of $X$.
Calculate $\mathrm { E } ( X )$ and show that, to 3 decimal places, $\operatorname { Var } ( X ) = 0.057$. One measure of skewness is $$\frac { \text { Mean - Mode } } { \text { Standard deviation } }$$
Evaluate the skewness of the distribution of $X$.
5. A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05 . Eggs are packed in boxes of 12 . Find the probability that in a box, the number of eggs with double yolks will be
exactly one,
more than three. A customer bought three boxes.
Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. The farmer delivered 10 boxes to a local shop.
Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g .
Find the probability that a randomly chosen egg weighs more than 68 g .
(3)
6. A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that $40 \%$ of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.
Define the population associated with the magazine.
Suggest a suitable sampling frame for the survey.
Identify the sampling units.
Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. As a pilot study the editor took a random sample of 25 subscribers.
Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. In fact only 6 subscribers agreed to the name being changed.
Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not the percentage agreeing to the change is less that the editor believes. The full survey is to be carried out using 200 randomly chosen subscribers.
Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. \section*{END} Answer Book (AB16)
Graph Paper (ASG2)
Mathematical Formulae (Lilac) Nil Paper Reference(s)
6684 \section*{Edexcel GCE
Statistics S2} Advanced/Advanced Subsidiary
Tuesday 17 June 2003 - Afternoon
Time: $\mathbf { 1 }$ hour $\mathbf { 3 0 }$ minutes Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. Explain briefly what you understand by
2. a statistic,
3. a sampling distribution.
4. (a) Write down the condition needed to approximate a Poisson distribution by a Normal distribution.
The random variable $Y \sim \operatorname { Po } ( 30 )$.
Estimate $\mathrm { P } ( Y > 28 )$.
3. In a town, $30 \%$ of residents listen to the local radio station. Four residents are chosen at random.
State the distribution of the random variable $X$, the number of these four residents that listen to local radio.
On graph paper, draw the probability distribution of $X$.
Write down the most likely number of these four residents that listen to the local radio station.
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
4. (a) Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that
1. the first 5 will occur on the sixth throw,
2. in the first eight throws there will be exactly three 5 s .
  5. A drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml . The random variable $X$ is the volume of lemonade dispensed into a cup.
Specify the probability density function of $X$ and sketch its graph.
Find the probability that the machine dispenses
1. less than 183 ml ,
2. exactly 183 ml .
Calculate the inter-quartile range of $X$.
Determine the value of $x$ such that $\mathrm { P } ( X \geq x ) = 2 \mathrm { P } ( X \leq x )$.
Interpret in words your value of $x$.
6. A doctor expects to see, on average, 1 patient per week with a particular disease.
Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer.
Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.
Test, at the $5 \%$ level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. Medical research into the nature of the disease discovers that it can be passed from one patient to another.
Explain whether or not this research supports your choice of model. Give a reason for your answer.
7. A continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \begin{cases} k \left( x ^ { 2 } + 2 x + 1 \right) & - 1 \leq x \leq 0
0 , & \text { otherwise } \end{cases}$$ where $k$ is a positive integer.
Show that $k = 3$. Find
$\mathrm { E } ( X )$,
the cumulative distribution function $\mathrm { F } ( x )$,
$\mathrm { P } ( - 0.3 < X < 0.3 )$. \section*{END} Answer Book (AB16)
Graph Paper (ASG2)
Mathematical Formulae (Lilac) Items included with question papers Nil Paper Reference(s)
6684 Statistics S2
Advanced/Advanced Subsidiary
Friday 23 January 2004 - Morning
Time: 1 hour 30 minutes Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. A large dental practice wishes to investigate the level of satisfaction of its patients.
2. Suggest a suitable sampling frame for the investigation.
3. Identify the sampling units.
4. State one advantage and one disadvantage of using a sample survey rather than a census.
5. Suggest a problem that might arise with the sampling frame when selecting patients.
6. The random variable $R$ has the binomial distribution $\mathrm { B } ( 12,0.35 )$.
7. Find $\mathrm { P } ( R \geq 4 )$.
The random variable $S$ has the Poisson distribution with mean 2.71.
Find $\mathrm { P } ( S \leq 1 )$. The random variable $T$ has the normal distribution $\mathrm { N } \left( 25,5 ^ { 2 } \right)$.
Find $\mathrm { P } ( T \leq 18 )$.
3. The discrete random variable $X$ is distributed $\mathrm { B } ( n , p )$.
Write down the value of $p$ that will give the most accurate estimate when approximating the binomial distribution by a normal distribution.
Give a reason to support your value.
Given that $n = 200$ and $p = 0.48$, find $\mathrm { P } ( 90 \leq X < 105 )$.
4. (a) Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution.
(2) A researcher has suggested that 1 in 150 people is likely to catch a particular virus.
Assuming that a person catching the virus is independent of any other person catching it,
find the probability that in a random sample of 12 people, exactly 2 of them catch the virus.
Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus.
5. Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.
Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. Find the probability that in any randomly selected 10 minute interval
exactly 6 cars pass this point,
at least 9 cars pass this point. After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.
Test, at the $5 \%$ level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly.
(6)
6. From past records a manufacturer of ceramic plant pots knows that $20 \%$ of them will have defects. To monitor the production process, a random sample of 25 pots is checked each day and the number of pots with defects is recorded.
Find the critical regions for a two-tailed test of the hypothesis that the probability that a plant pot has defects is 0.20 . The probability of rejection in either tail should be as close as possible to $2.5 \%$.
Write down the significance level of the above test. A garden centre sells these plant pots at a rate of 10 per week. In an attempt to increase sales, the price was reduced over a six-week period. During this period a total of 74 pots was sold.
Using a $5 \%$ level of significance, test whether or not there is evidence that the rate of sales per week has increased during this six-week period.
7. The continuous random variable $X$ has probability density function $$f ( x ) = \begin{cases} k x ( 5 - x ) , & 0 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$ where $k$ is a constant.
Show that $k = \frac { 3 } { 56 }$.
Find the cumulative distribution function $\mathrm { F } ( x )$ for all values of $x$.
Evaluate $\mathrm { E } ( X )$.
Find the modal value of $X$.
Verify that the median value of $X$ lies between 2.3 and 2.5.
Comment on the skewness of $X$. Justify your answer. \section*{END} Answer Book (AB16)
Graph Paper (ASG2)
Mathematical Formulae (Lilac) Items included with question papers
Nil Paper Reference(s)
6684 Statistics S2
Advanced/Advanced Subsidiary
Wednesday 23 June 2004 - Morning Time: $\mathbf { 1 }$ hour $\mathbf { 3 0 }$ minutes Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. Explain briefly what you understand by
2. a sampling frame,
3. a statistic.
4. The continuous random variable $X$ is uniformly distributed over the interval $[ - 1,4 ]$.
Find
$\mathrm { P } ( X < 2.7 )$,
$\mathrm { E } ( X )$,
$\operatorname { Var } ( X )$.
3. Brad planted 25 seeds in his greenhouse. He has read in a gardening book that the probability of one of these seeds germinating is 0.25 . Ten of Brad's seeds germinated. He claimed that the gardening book had underestimated this probability. Test, at the $5 \%$ level of significance, Brad's claim. State your hypotheses clearly.
(7)
4. (a) State two conditions under which a random variable can be modelled by a binomial distribution.
(2) In the production of a certain electronic component it is found that $10 \%$ are defective.
The component is produced in batches of 20.
Write down a suitable model for the distribution of defective components in a batch. Find the probability that a batch contains
no defective components,
more than 6 defective components.
Find the mean and the variance of the defective components in a batch. A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.
Using a suitable approximation, find the probability that the supplier will receive a refund.
(4)
5. (a) Explain what you understand by a critical region of a test statistic. The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean $\frac { 1 } { 7 }$.
Find the probability that on a particular day there are fewer than 2 breakdowns.
Find the probability that during a 14-day period there are at most 4 breakdowns. The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28 -day period after the checks are introduced, only 1 hire car broke down.
Test, at the $5 \%$ level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly.
(7)
6. Minor defects occur in a particular make of carpet at a mean rate of 0.05 per $\mathrm { m } ^ { 2 }$.
Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer. A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires $30 \mathrm {~m} ^ { 2 }$ of this carpet. Find the probability that the foyer carpet contains
exactly 2 defects,
more than 5 defects. The carpet fitter orders a total of $355 \mathrm {~m} ^ { 2 }$ of the carpet for the whole hotel.
Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects.
7. A random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 3 } , & 0 \leq x \leq 1
\frac { 8 x ^ { 3 } } { 45 } , & 1 \leq x \leq 2
0 , & \text { otherwise } \end{cases}$$
Calculate the mean of $X$.
Specify fully the cumulative distribution function $\mathrm { F } ( x )$.
Find the median of $X$.
Comment on the skewness of the distribution of $X$. \section*{END} Paper Reference(s)
6684 \section*{Edexcel GCE
Statistics S2
Advanced/Advanced Subsidiary Tuesday 25 January 2005 - Morning Time: $\mathbf { 1 }$ hour $\mathbf { 3 0 }$ minutes } Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. The random variables $R , S$ and $T$ are distributed as follows
$$R \sim \mathrm {~B} ( 15,0.3 ) , \quad S \sim \mathrm { Po } ( 7.5 ) , \quad T \sim \mathrm {~N} \left( 8,2 ^ { 2 } \right) .$$ Find
$\mathrm { P } ( R = 5 )$,
$\mathrm { P } ( S = 5 )$,
$\mathrm { P } ( T = 5 )$.
2. (a) Explain what you understand by (i) a population and (ii) a sampling frame. The population and the sampling frame may not be the same.
Explain why this might be the case.
Give an example, justifying your choices, to illustrate when you might use
1. a census,
2. a sample.
  3. A rod of length $2 l$ was broken into 2 parts. The point at which the rod broke is equally likely to be anywhere along the rod. The length of the shorter piece of rod is represented by the random variable $X$.
Write down the name of the probability density function of $X$, and specify it fully.
Find $\mathrm { P } \left( X < \frac { 1 } { 3 } l \right)$.
Write down the value of $\mathrm { E } ( X )$. Two identical rods of length $2 l$ are broken.
Find the probability that both of the shorter pieces are of length less than $\frac { 1 } { 3 } l$.
4. In an experiment, there are 250 trials and each trial results in a success or a failure.
Write down two other conditions needed to make this into a binomial experiment. It is claimed that $10 \%$ of students can tell the difference between two brands of baked beans. In a random sample of 250 students, 40 of them were able to distinguish the difference between the two brands.
Using a normal approximation, test at the $1 \%$ level of significance whether or not the claim is justified. Use a one-tailed test.
Comment on the acceptability of the assumptions you needed to carry out the test.
5. From company records, a manager knows that the probability that a defective article is produced by a particular production line is 0.032 . A random sample of 10 articles is selected from the production line.
Find the probability that exactly 2 of them are defective. On another occasion, a random sample of 100 articles is taken.
Using a suitable approximation, find the probability that fewer than 4 of them are defective. At a later date, a random sample of 1000 is taken.
Using a suitable approximation, find the probability that more than 42 are defective.
6. Over a long period of time, accidents happened on a stretch of road at random at a rate of 3 per month. Find the probability that
in a randomly chosen month, more than 4 accidents occurred,
in a three-month period, more than 4 accidents occurred. At a later date, a speed restriction was introduced on this stretch of road. During a randomly chosen month only one accident occurred.
Test, at the $5 \%$ level of significance, whether or not there is evidence to support the claim that this speed restriction reduced the mean number of road accidents occurring per month. The speed restriction was kept on this road. Over a two-year period, 55 accidents occurred.
Test, at the $5 \%$ level of significance, whether or not there is now evidence that this speed restriction reduced the mean number of road accidents occurring per month.
7. The random variable $X$ has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { l c } k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4
0 , & \text { otherwise } \end{array} \right.$$
Show that $k = \frac { 2 } { 9 }$. Find
$\mathrm { E } ( X )$,
the mode of $X$.
the cumulative distribution function $\mathrm { F } ( x )$ for all $x$.
Evaluate $\mathrm { P } ( X \leq 2.5 )$,
Deduce the value of the median and comment on the shape of the distribution. Materials required for examination
Mathematical Formulae (Lilac or Green) Items included with question papers Nil Paper Reference(s)
6684/01 \section*{Advanced/Advanced Subsidiary} \section*{Wednesday 22 June 2005 - Afternoon} Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. It is estimated that $4 \%$ of people have green eyes. In a random sample of size $n$, the expected number of people with green eyes is 5 .
2. Calculate the value of $n$.
The expected number of people with green eyes in a second random sample is 3 .
Find the standard deviation of the number of people with green eyes in this second sample.
2. The continuous random variable X is uniformly distributed over the interval $[ 2,6 ]$.
Write down the probability density function $\mathrm { f } ( x )$. Find
$\mathrm { E } ( X )$,
$\operatorname { Var } ( X )$,
the cumulative distribution function of $X$, for all $x$,
$\mathrm { P } ( 2.3 < X < 3.4 )$.
3. The random variable $X$ is the number of misprints per page in the first draft of a novel.
State two conditions under which a Poisson distribution is a suitable model for $X$. The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
a randomly chosen page has no misprints,
the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.
4. Explain what you understand by
a sampling unit,
a sampling frame,
a sampling distribution.
5. In a manufacturing process, $2 \%$ of the articles produced are defective. A batch of 200 articles is selected.
Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
Estimate the probability that there are less than 5 defective articles.
6. A continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \begin{cases} k \left( 4 x - x ^ { 3 } \right) , & 0 \leq x \leq 2
0 , & \text { otherwise } \end{cases}$$ where $k$ is a positive constant.
Show that $k = \frac { 1 } { 4 }$. Find
$\mathrm { E } ( X )$,
the mode of $X$,
the median of $X$.
Comment on the skewness of the distribution.
Sketch $\mathrm { f } ( x )$.
7. A drugs company claims that $75 \%$ of patients suffering from depression recover when treated with a new drug. A random sample of 10 patients with depression is taken from a doctor's records.
Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug. Given that the claim is correct,
find the probability that the treatment will be successful for exactly 6 patients. The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.
Stating your hypotheses clearly, test, at the $5 \%$ level of significance, the doctor's belief.
From a sample of size 20, find the greatest number of patients who need to recover from the test in part (c) to be significant at the $1 \%$ level.

8. The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by $$f ( x ) = \left\{ \begin{array} { l c } 2 ( x - 2 ) & 2 \leq x \leq 3
0 & \text { otherwise } \end{array} \right.$$

Sketch $\mathrm { f } ( x )$ for all values of $x$.
Write down the mode of $X$. Find
$\mathrm { E } ( X )$,
the median of $X$.
Comment on the skewness of this distribution. Give a reason for your answer. \section*{Advanced Level} \section*{Friday 23 May 2008 - Morning} $\frac { \text { Materials required for examination } } { \text { Mathematical Formulae (Lilac or Green) } } \quad \frac { \text { Items included with question papers } } { \text { Nil } }$ Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit. \section*{H32581A}
1. Jean regularly takes a break from work to go to the post office. The amount of time Jean waits in the queue to be served at the post office has a continuous uniform distribution between 0 and 10 minutes.
2. Find the mean and variance of the time Jean spends in the post office queue.
3. Find the probability that Jean does not have to wait more than 2 minutes.
Jean visits the post office 5 times.
Find the probability that she never has to wait more than 2 minutes. Jean is in the queue when she receives a message that she must return to work for an urgent meeting. She can only wait in the queue for a further 3 minutes. Given that Jean has already been queuing for 5 minutes,
find the probability that she must leave the post office queue without being served.
2. In a large college $58 \%$ of students are female and $42 \%$ are male. A random sample of 100 students is chosen from the college. Using a suitable approximation find the probability that more than half the sample are female.
3. A test statistic has a Poisson distribution with parameter $\lambda$. Given that $$\mathrm { H } _ { 0 } : \lambda = 9 , \mathrm { H } _ { 1 } : \lambda \neq 9$$
find the critical region for the test statistic such that the probability in each tail is as close as possible to $2.5 \%$.
State the probability of incorrectly rejecting $\mathrm { H } _ { 0 }$ using this critical region.
4. Each cell of a certain animal contains 11000 genes. It is known that each gene has a probability 0.0005 of being damaged. A cell is chosen at random.
Suggest a suitable model for the distribution of the number of damaged genes in the cell.
Find the mean and variance of the number of damaged genes in the cell.
Using a suitable approximation, find the probability that there are at most 2 damaged genes in the cell.
(4)
5. Sue throws a fair coin 15 times and records the number of times it shows a head.
State the distribution to model the number of times the coin shows a head. Find the probability that Sue records
exactly 8 heads,
at least 4 heads. Sue has a different coin which she believes is biased in favour of heads. She throws the coin 15 times and obtains 13 heads.
Test Sue's belief at the $1 \%$ level of significance. State your hypotheses clearly.
6. A call centre agent handles telephone calls at a rate of 18 per hour.
Give two reasons to support the use of a Poisson distribution as a suitable model for the number of calls per hour handled by the agent.
Find the probability that in any randomly selected 15 minute interval the agent handles
1. exactly 5 calls,
2. more than 8 calls. The agent received some training to increase the number of calls handled per hour. During a randomly selected 30 minute interval after the training the agent handles 14 calls.
Test, at the $5 \%$ level of significance, whether or not there is evidence to support the suggestion that the rate at which the agent handles calls has increased. State your hypotheses clearly.
7. A random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } x & 0 \leq x < 1
k x ^ { 3 } & 1 \leq x \leq 2
0 & \text { otherwise } \end{cases}$$ where $k$ is a constant.
Show that $k = \frac { 1 } { 5 }$.
Calculate the mean of $X$.
Specify fully the cumulative distribution function $\mathrm { F } ( x )$.
Find the median of $X$.
Comment on the skewness of the distribution of $X$. \section*{Materials required for examination
Mathematical Formulae (Green)} Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. \section*{Advanced} \section*{Wednesday 21 January 2009 - Afternoon} In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 7 questions on this paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3 . The daisies are distributed randomly throughout the field.
Find the probability that, in a randomly chosen square there will be
more than 2 daisies,
either 5 or 6 daisies. The botanist decides to count the number of daisies, $x$, in each of 80 randomly selected squares within the field. The results are summarised below $$\sum x = 295 \quad \sum x ^ { 2 } = 1386$$
Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places.
Explain how the answers from part (c) support the choice of a Poisson distribution as a model.
Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square.
2. The continuous random variable $X$ is uniformly distributed over the interval [-2, 7].
Write down fully the probability density function $\mathrm { f } ( x )$ of $X$.
Sketch the probability density function $\mathrm { f } ( x )$ of $X$. Find
$\mathrm { E } \left( X ^ { 2 } \right)$,
$\mathrm { P } ( - 0.2 < X < 0.6 )$.
3. A single observation $x$ is to be taken from a Binomial distribution $\mathrm { B } ( 20 , p )$. This observation is used to test $\mathrm { H } _ { 0 } : p = 0.3$ against $\mathrm { H } _ { 1 } : p \neq 0.3$.
Using a $5 \%$ level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to $2.5 \%$.
State the actual significance level of this test. The actual value of $x$ obtained is 3 .
State a conclusion that can be drawn based on this value, giving a reason for your answer.
4. The length of a telephone call made to a company is denoted by the continuous random variable $T$. It is modelled by the probability density function $$\mathrm { f } ( t ) = \begin{cases} k t , & 0 \leq t \leq 10
0 , & \text { otherwise } \end{cases}$$
Show that the value of $k$ is $\frac { 1 } { 50 }$.
Find $\mathrm { P } ( T > 6 )$.
Calculate an exact value for $\mathrm { E } ( T )$ and for $\operatorname { Var } ( T )$.
Write down the mode of the distribution of $T$. It is suggested that the probability density function, $\mathrm { f } ( t )$, is not a good model for $T$.
Sketch the graph of a more suitable probability density function for $T$.
5. A factory produces components of which $1 \%$ are defective. The components are packed in boxes of 10 . A box is selected at random.
Find the probability that the box contains exactly one defective component.
Find the probability that there are at least 2 defective components in the box.
Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components.
6. A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.
1. Test, at the $10 \%$ level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
2. State the minimum number of visits required to obtain a significant result.
State an assumption that has been made about the visits to the server. In a random two minute period on a Saturday the web server is visited 20 times.
Using a suitable approximation, test at the $10 \%$ level of significance, whether or not the rate of visits is greater on a Saturday.
7. A random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} - \frac { 2 } { 9 } x + \frac { 8 } { 9 } , & 1 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$
Show that the cumulative distribution function $\mathrm { F } ( x )$ can be written in the form $a x ^ { 2 } + b x + c$, for $1 \leq x \leq 4$ where $a , b$ and $c$ are constants.
Define fully the cumulative distribution function $\mathrm { F } ( x )$.
Show that the upper quartile of $X$ is 2.5 and find the lower quartile. Given that the median of $X$ is 1.88 ,
describe the skewness of the distribution. Give a reason for your answer. \section*{TOTAL FOR PAPER: 75 MARKS} Materials required for examination
Mathematical Formulae (Orange or Green) Items included with question papers
Nil \section*{Advanced Level} \section*{Monday 1 June 2009 - Morning} Candidates may use any calculator allowed by the regulations of the Joint
Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 8 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.
1. A bag contains a large number of counters of which $15 \%$ are coloured red. A random sample of 30 counters is selected and the number of red counters is recorded.
2. Find the probability of no more than 6 red counters in this sample.
A second random sample of 30 counters is selected and the number of red counters is recorded.
Using a Poisson approximation, estimate the probability that the total number of red counters in the combined sample of size 60 is less than 13 .
2. An effect of a certain disease is that a small number of the red blood cells are deformed. Emily has this disease and the deformed blood cells occur randomly at a rate of 2.5 per ml of her blood. Following a course of treatment, a random sample of 2 ml of Emily's blood is found to contain only 1 deformed red blood cell. Stating your hypotheses clearly and using a $5 \%$ level of significance, test whether or not there has been a decrease in the number of deformed red blood cells in Emily's blood.
3. A random sample $X _ { 1 } , X _ { 2 } , \ldots X _ { n }$ is taken from a population with unknown mean $\mu$ and unknown variance $\sigma ^ { 2 }$. A statistic $Y$ is based on this sample.
Explain what you understand by the statistic $Y$.
Explain what you understand by the sampling distribution of $Y$.
State, giving a reason which of the following is not a statistic based on this sample.
1. $\sum _ { i = 1 } ^ { n } \frac { \left( X _ { i } - \bar { X } \right) ^ { 2 } } { n }$
2. $\sum _ { i = 1 } ^ { n } \left( \frac { X _ { i } - \mu } { \sigma } \right) ^ { 2 }$
3. $\sum _ { i = 1 } ^ { n } X _ { i } { } ^ { 2 }$
  4. Past records suggest that $30 \%$ of customers who buy baked beans from a large supermarket buy them in single tins. A new manager questions whether or not there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
Using a $10 \%$ level of significance, find the critical region for a two-tailed test to answer the manager's question. You should state the probability of rejection in each tail which should be less than 0.05 .
Write down the actual significance level of a test based on your critical region from part (a). The manager found that 11 customers from the sample of 20 had bought baked beans in single tins.
Comment on this finding in the light of your critical region found in part (a).
5. An administrator makes errors in her typing randomly at a rate of 3 errors every 1000 words.
In a document of 2000 words find the probability that the administrator makes 4 or more errors. The administrator is given an 8000 word report to type and she is told that the report will only be accepted if there are 20 or fewer errors.
Use a suitable approximation to calculate the probability that the report is accepted.
(7) ⟶
6. The three independent random variables $A , B$ and $C$ each has a continuous uniform distribution over the interval $[ 0,5 ]$.
Find $\mathrm { P } ( A > 3 )$.
Find the probability that $A , B$ and $C$ are all greater than 3 . The random variable $Y$ represents the maximum value of $A , B$ and $C$.
The cumulative distribution function of $Y$ is $$\mathrm { F } ( y ) = \begin{cases} 0 , & y < 0
\frac { y ^ { 3 } } { 125 } , & 0 \leq y \leq 5
1 , & y > 5 \end{cases}$$
Find the probability density function of $Y$.
Sketch the probability density function of $Y$.
Write down the mode of $Y$.
(1)
Find $\mathrm { E } ( Y )$.
Find $\mathrm { P } ( Y > 3 )$.
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d6f7790-b966-4104-97cc-796d28a8f169-33_327_559_310_1885} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the probability density function $\mathrm { f } ( x )$ of the random variable $X$. The part of the sketch from $x = 0$ to $x = 4$ consists of an isosceles triangle with maximum at ( $2,0.5$ ).
Write down $\mathrm { E } ( X )$. The probability density function $\mathrm { f } ( x )$ can be written in the following form. $$f ( x ) = \begin{cases} a x & 0 \leq x < 2
b - a x & 2 \leq x \leq 4
0 & \text { otherwise } \end{cases}$$
Find the values of the constants $a$ and $b$.
Show that $\sigma$, the standard deviation of $X$, is 0.816 to 3 decimal places.
Find the lower quartile of $X$.
State, giving a reason, whether $\mathrm { P } ( 2 - \sigma < X < 2 + \sigma )$ is more or less than 0.5
8. A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
Find the probability of exactly 4 faults in a 15 metre length of cloth.
Find the probability of more than 10 faults in 60 metres of cloth. A retailer buys a large amount of this cloth and sells it in pieces of length $x$ metres. He chooses $x$ so that the probability of no faults in a piece is 0.80 .
Write down an equation for $x$ and show that $x = 1.7$ to 2 significant figures. The retailer sells 1200 of these pieces of cloth. He makes a profit of 60 p on each piece of cloth that does not contain a fault but a loss of $\pounds 1.50$ on any pieces that do contain faults.
Find the retailer's expected profit. \section*{TOTAL FOR PAPER: 75 MARKS} \section*{END} Materials required for examination
Mathematical Formulae (Pink or Green) Items included with question papers
Nil \section*{Advanced Level} \section*{Tuesday 19 January 2010 - Morning} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. \section*{M35712A} This publication may only be reproduced in accordance with Edexcel Limited copyright policy.
©2010 Edexcel Limited
1. A manufacturer supplies DVD players to retailers in batches of 20 . It has $5 \%$ of the players returned because they are faulty.
2. Write down a suitable model for the distribution of the number of faulty DVD players in a batch.
Find the probability that a batch contains
no faulty DVD players,
more than 4 faulty DVD players.
Find the mean and variance of the number of faulty DVD players in a batch.
2. A continuous random variable $X$ has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < - 2
\frac { x + 2 } { 6 } , & - 2 \leq x \leq 4
1 , & x > 4 \end{array} \right.$$
Find $\mathrm { P } ( X < 0 )$.
Find the probability density function $\mathrm { f } ( x )$ of $X$.
Write down the name of the distribution of $X$.
Find the mean and the variance of $X$.
Write down the value of $\mathrm { P } ( X = 1 )$.
3. A robot is programmed to build cars on a production line. The robot breaks down at random at a rate of once every 20 hours.
Find the probability that it will work continuously for 5 hours without a breakdown. Find the probability that, in an 8 hour period,
the robot will break down at least once,
there are exactly 2 breakdowns. In a particular 8 hour period, the robot broke down twice.
Write down the probability that the robot will break down in the following 8 hour period. Give a reason for your answer.
4. The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by $$\mathrm { f } ( x ) = \begin{cases} k \left( x ^ { 2 } - 2 x + 2 \right) , & 0 < x \leq 3
3 k , & 3 < x \leq 4
0 , & \text { otherwise } \end{cases}$$ where $k$ is a constant.
Show that $k = \frac { 1 } { 9 }$.
Find the cumulative distribution function $\mathrm { F } ( x )$.
Find the mean of $X$.
Show that the median of $X$ lies between $x = 2.6$ and $x = 2.7$.
5. A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes. Find the probability that
fewer than 9 customers arrive for breakfast on a Monday morning between 10 a.m. and 11 a.m. The café serves breakfast every day between 8 a.m. and 12 noon.
Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.
6. (a) Define the critical region of a test statistic. A discrete random variable $X$ has a Binomial distribution $\mathrm { B } ( 30 , p )$. A single observation is used to test $\mathrm { H } _ { 0 } : p = 0.3$ against $\mathrm { H } _ { 1 } : p \neq 0.3$
Using a $1 \%$ level of significance find the critical region of this test. You should state the probability of rejection in each tail which should be as close as possible to 0.005 .
Write down the actual significance level of the test. The value of the observation was found to be 15 .
Comment on this finding in light of your critical region.
7. A bag contains a large number of coins. It contains only 1 p and 2 p coins in the ratio $1 : 3$.
Find the mean $\mu$ and the variance $\sigma ^ { 2 }$ of the values of this population of coins. A random sample of size 3 is taken from the bag.
List all the possible samples.
Find the sampling distribution of the mean value of the samples. Nil \section*{Advanced Level} \section*{Wednesday 9 June 2010 - Afternoon} \section*{Time: $\mathbf { 1 }$ hour $\mathbf { 3 0 }$ minutes} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. \section*{H35396A} \footnotetext{This publication may only be reproduced in accordance with Edexcel Limited copyright policy.
©2010 Edexcel Limited }
1. Explain what you understand by
2. a population,
3. a statistic.
A researcher took a sample of 100 voters from a certain town and asked them who they would vote for in an election. The proportion who said they would vote for Dr Smith was $35 \%$.
State the population and the statistic in this case.
Explain what you understand by the sampling distribution of this statistic.
2. Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 . Find the probability that, in 9 games, Bhim loses
exactly 3 of the games,
fewer than half of the games. Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 . Bhim and Joe agree to play a further 60 games.
Calculate the mean and variance for the number of these 60 games that Bhim loses.
Using a suitable approximation calculate the probability that Bhim loses more than 4 games.
3. A rectangle has a perimeter of 20 cm . The length, $X \mathrm {~cm}$, of one side of this rectangle is uniformly distributed between 1 cm and 7 cm . Find the probability that the length of the longer side of the rectangle is more than 6 cm long.
4. The lifetime, $X$, in tens of hours, of a battery has a cumulative distribution function $\mathrm { F } ( x )$ given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1
\frac { 4 } { 9 } \left( x ^ { 2 } + 2 x - 3 \right) & 1 \leq x \leq 1.5
1 & x > 1.5 \end{array} \right.$$
Find the median of $X$, giving your answer to 3 significant figures.
Find, in full, the probability density function of the random variable $X$.
Find $\mathrm { P } ( X \geq 1.2 )$ A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.
Find the probability that the lantern will still be working after 12 hours.
5. A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company's website at their first attempt.
Explain why the Poisson distribution may be a suitable model in this case. Find the probability that, in a randomly chosen 2 hour period,
1. all users connect at their first attempt,
2. at least 4 users fail to connect at their first attempt. The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60 .
Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a $5 \%$ level of significance and state your hypotheses clearly.
(9)
6. A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.
Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.
Using a $5 \%$ significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is $\frac { 1 } { 4 }$. The probability of rejection in either tail should be as close as possible to 0.025 .
Find the actual significance level of this test. In the sample of 50 the actual number of faulty bolts was 8 .
Comment on the company's claim in the light of this value. Justify your answer. The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.
Test at the $1 \%$ level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly.
7. The random variable $Y$ has probability density function $f ( y )$ given by $$\mathrm { f } ( y ) = \begin{cases} k y ( a - y ) & 0 \leq y \leq 3
0 & \text { otherwise } \end{cases}$$ where $k$ and $a$ are positive constants.
1. Explain why $a \geq 3$.
2. Show that $k = \frac { 2 } { 9 ( a - 2 ) }$. Given that $\mathrm { E } ( Y ) = 1.75$,
show that $a = 4$ and write down the value of $k$. For these values of $a$ and $k$,
sketch the probability density function,
write down the mode of $Y$. \section*{Advanced Level} \section*{Friday 14 January 2011 - Morning} Mathematical Formulae (Pink) Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. \section*{H35411A}
1. A disease occurs in $3 \%$ of a population.
2. State any assumptions that are required to model the number of people with the disease in a random sample of size n as a binomial distribution.
3. Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people.
4. Find the mean and variance of the number of people with the disease in a random sample of 100 people.
A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.
Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination.
2. A student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the $5 \%$ level of significance, test whether or not there is evidence to reject the teacher's claim. State your hypotheses clearly.
3. The continuous random variable $X$ is uniformly distributed over the interval $[ - 1,3 ]$. Find
$\mathrm { E } ( X )$
$\operatorname { Var } ( X )$
$\mathrm { E } \left( X ^ { 2 } \right)$
$\mathrm { P } ( X < 1.4 )$ A total of 40 observations of $X$ are made.
Find the probability that at least 10 of these observations are negative.
4. Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4 -week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the $5 \%$ level of significance. State your hypotheses clearly.
5. A continuous random variable $X$ has the probability density function $\mathrm { f } ( x )$ shown in Figure 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d6f7790-b966-4104-97cc-796d28a8f169-40_396_465_612_520} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
Show that $\mathrm { f } ( x ) = 4 - 8 x$ for $0 \leq x \leq 0.5$ and specify $\mathrm { f } ( x )$ for all real values of $x$.
Find the cumulative distribution function $\mathrm { F } ( x )$.
Find the median of $X$.
Write down the mode of $X$.
State, with a reason, the skewness of $X$.
6. Cars arrive at a motorway toll booth at an average rate of 150 per hour.
Suggest a suitable distribution to model the number of cars arriving at the toll booth, $X$, per minute.
State clearly any assumptions you have made by suggesting this model. Using your model,
find the probability that in any given minute
1. no cars arrive,
2. more than 3 cars arrive.
In any given 4 minute period, find $m$ such that $\mathrm { P } ( \mathrm { X } > m ) = 0.0487$
Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period.
7. The queuing time in minutes, $X$, of a customer at a post office is modelled by the probability density function $$f ( x ) = \begin{cases} k x \left( 81 - x ^ { 2 } \right) & 0 \leq x \leq 9
0 & \text { otherwise } \end{cases}$$
Show that $k = \frac { 4 } { 6561 }$. Using integration, find
the mean queuing time of a customer,
the probability that a customer will queue for more than 5 minutes. Three independent customers shop at the post office.
Find the probability that at least 2 of the customers queue for more than 5 minutes. \section*{Advanced Level} \section*{Thursday 26 May 2011 - Morning} Materials required for examination
Mathematical Formulae (Pink) Items included with question papers
Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. \section*{P38165A}
1. A factory produces components. Each component has a unique identity number and it is assumed that $2 \%$ of the components are faulty. On a particular day, a quality control manager wishes to take a random sample of 50 components.
2. Identify a sampling frame.
The statistic $F$ represents the number of faulty components in the random sample of size 50 .
Specify the sampling distribution of $F$.
2. A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.
Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval. The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.
Stating your hypotheses clearly, and using a $5 \%$ level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.
Using a 5\% level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.
(3)
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d6f7790-b966-4104-97cc-796d28a8f169-42_287_627_301_404} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the probability density function $\mathrm { f } ( x )$ of the random variable $X$.
For $0 \leq x \leq 3 , \mathrm { f } ( x )$ is represented by a curve $O B$ with equation $\mathrm { f } ( x ) = k x ^ { 2 }$, where $k$ is a constant.
For $3 \leq x \leq a$, where $a$ is a constant, $\mathrm { f } ( x )$ is represented by a straight line passing through $B$ and the point $( a , 0 )$. For all other values of $x , \mathrm { f } ( x ) = 0$.
Given that the mode of $X =$ the median of $X$, find
the mode,
the value of $k$,
the value of $a$. Without calculating $\mathrm { E } ( X )$ and with reference to the skewness of the distribution
state, giving your reason, whether $\mathrm { E } ( X ) < 3 , \mathrm { E } ( X ) = 3$ or $\mathrm { E } ( X ) > 3$.
(2)
[0pt] 4. In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm , of a randomly chosen stick has a continuous uniform distribution over the interval [7,10]. A stick is selected at random from the box.
Find the probability that the stick is shorter than 9.5 cm . To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm .
Find the probability of winning a bag of sweets. To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm .
Find the probability of winning a soft toy.
5. Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm .
Find the probability that Jim's plank contains at most 3 defects. Shivani buys 6 planks each of length 100 cm .
Find the probability that fewer than 2 of Shivani's planks contain at most 3 defects.
Using a suitable approximation, estimate the probability that the total number of defects on Shivani's 6 planks is less than 18.
(6)
6. A shopkeeper knows, from past records, that $15 \%$ of customers buy an item from the display next to the till. After a refurbishment of the shop, he takes a random sample of 30 customers and finds that only 1 customer has bought an item from the display next to the till.
Stating your hypotheses clearly, and using a $5 \%$ level of significance, test whether or not there has been a change in the proportion of customers buying an item from the display next to the till. During the refurbishment a new sandwich display was installed. Before the refurbishment $20 \%$ of customers bought sandwiches. The shopkeeper claims that the proportion of customers buying sandwiches has now increased. He selects a random sample of 120 customers and finds that 31 of them have bought sandwiches.
Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper's claim. Use a $10 \%$ level of significance.
7. The continuous random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 32 } ( x - 1 ) ( 5 - x ) & 1 \leq x \leq 5
0 & \text { otherwise } \end{cases}$$
Sketch $\mathrm { f } ( x )$ showing clearly the points where it meets the $x$-axis.
Write down the value of the mean, $\mu$, of $X$.
Show that $\mathrm { E } \left( X ^ { 2 } \right) = 9.8$.
Find the standard deviation, $\sigma$, of $X$. The cumulative distribution function of $X$ is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1
\frac { 1 } { 32 } \left( a - 15 x + 9 x ^ { 2 } - x ^ { 3 } \right) & 1 \leq x \leq 5
1 & x > 5 \end{cases}$$ where $a$ is a constant.
Find the value of $a$.
Show that the lower quartile of $X , q _ { 1 }$, lies between 2.29 and 2.31.
Hence find the upper quartile of $X$, giving your answer to 1 decimal place.
Find, to 2 decimal places, the value of $k$ so that $$\mathrm { P } ( \mu - k \sigma < X < \mu + k \sigma ) = 0.5$$ END \section*{Advanced Level} \section*{Tuesday 17 January 2012 - Morning} Mathematical Formulae (Pink) Items included with question papers
Nil Candidates may use any calculator allowed by the regulations of the Joint
Council for Qualifications. Calculators must not have the facility for symbolic
algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.
1. The time in minutes that Elaine takes to checkout at her local supermarket follows a continuous uniform distribution defined over the interval [ 3,9 ].
Find
Elaine's expected checkout time,
the variance of the time taken to checkout at the supermarket,
the probability that Elaine will take more than 7 minutes to checkout. Given that Elaine has already spent 4 minutes at the checkout,
find the probability that she will take a total of less than 6 minutes to checkout.
2. David claims that the weather forecasts produced by local radio are no better than those achieved by tossing a fair coin and predicting rain if a head is obtained or no rain if a tail is obtained. He records the weather for 30 randomly selected days. The local radio forecast is correct on 21 of these days. Test David's claim at the $5 \%$ level of significance.
State your hypotheses clearly.
3. The probability of a telesales representative making a sale on a customer call is 0.15 . Find the probability that
no sales are made in 10 calls,
more than 3 sales are made in 20 calls. Representatives are required to achieve a mean of at least 5 sales each day.
Find the least number of calls each day a representative should make to achieve this requirement.
Calculate the least number of calls that need to be made by a representative for the probability of at least 1 sale to exceed 0.95 .
4. A website receives hits at a rate of 300 per hour.
State a distribution that is suitable to model the number of hits obtained during a 1 minute interval.
State two reasons for your answer to part (a). Find the probability of
10 hits in a given minute,
at least 15 hits in 2 minutes. The website will go down if there are more than 70 hits in 10 minutes.
Using a suitable approximation, find the probability that the website will go down in a particular 10 minute interval.
5. The probability of an electrical component being defective is 0.075 . The component is supplied in boxes of 120 .
Using a suitable approximation, estimate the probability that there are more than 3 defective components in a box. A retailer buys 2 boxes of components.
Estimate the probability that there are at least 4 defective components in each box.
6. A random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 2 } , & 0 \leq x < 1
x - \frac { 1 } { 2 } , & 1 \leq x \leq k
0 & \text { otherwise } \end{cases}$$ where $k$ is a positive constant.
Sketch the graph of $\mathrm { f } ( x )$.
Show that $k = \frac { 1 } { 2 } ( 1 + \sqrt { } 5 )$.
Define fully the cumulative distribution function $\mathrm { F } ( x )$.
Find $\mathrm { P } ( 0.5 < X < 1.5 )$.
Write down the median of $X$ and the mode of $X$.
Describe the skewness of the distribution of $X$. Give a reason for your answer.
7. (a) Explain briefly what you understand by
1. a critical region of a test statistic,
2. the level of significance of a hypothesis test.
An estate agent has been selling houses at a rate of 8 per month. She believes that the rate of sales will decrease in the next month.
1. Using a 5\% level of significance, find the critical region for a one tailed test of the hypothesis that the rate of sales will decrease from 8 per month.
2. Write down the actual significance level of the test in part (b)(i). The estate agent is surprised to find that she actually sold 13 houses in the next month. She now claims that this is evidence of an increase in the rate of sales per month.
Test the estate agent's claim at the $5 \%$ level of significance. State your hypotheses clearly. Mathematical Formulae (Pink) Items included with question papers
Nil \section*{Advanced Level} \section*{Thursday 24 May 2012 - Morning} \section*{Time: $\mathbf { 1 }$ hour $\mathbf { 3 0 }$ minutes} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 8 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. \section*{P40106A}
1. A manufacturer produces sweets of length $L \mathrm {~mm}$ where $L$ has a continuous uniform distribution with range [15,30].
2. Find the probability that a randomly selected sweet has length greater than 24 mm .
These sweets are randomly packed in bags of 20 sweets.
Find the probability that a randomly selected bag will contain at least 8 sweets with length greater than 24 mm .
Find the probability that 2 randomly selected bags will both contain at least 8 sweets with length greater than 24 mm .
2. A test statistic has a distribution $\mathrm { B } ( 25 , p )$. Given that $$\mathrm { H } _ { 0 } : p = 0.5 , \quad \mathrm { H } _ { 1 } : p \neq 0.5$$
find the critical region for the test statistic such that the probability in each tail is as close as possible to $2.5 \%$.
State the probability of incorrectly rejecting $\mathrm { H } _ { 0 }$ using this critical region.
3. (a) Write down the two conditions needed to approximate the binomial distribution by the Poisson distribution. A machine which manufactures bolts is known to produce $3 \%$ defective bolts. The machine breaks down and a new machine is installed. A random sample of 200 bolts is taken from those produced by the new machine and 12 bolts are defective.
Using a suitable approximation, test at the $5 \%$ level of significance whether or not the proportion of defective bolts is higher with the new machine than with the old machine. State your hypotheses clearly.
4. The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week.
Find the probability that in the next four weeks the estate agent sells
1. exactly 3 houses,
2. more than 5 houses. The estate agent monitors sales in periods of 4 weeks.
Find the probability that in the next twelve of those 4 week periods there are exactly nine periods in which more than 5 houses are sold. The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks.
Use a suitable approximation to estimate the probability that the estate agent receives a bonus.
5. The queuing time, $X$ minutes, of a customer at a till of a supermarket has probability density function $$f ( x ) = \begin{cases} \frac { 3 } { 32 } x ( k - x ) & 0 \leq x \leq k
0 & \text { otherwise } \end{cases}$$
Show that the value of $k$ is 4 .
Write down the value of $\mathrm { E } ( X )$.
Calculate Var ( $X$ ).
Find the probability that a randomly chosen customer's queuing time will differ from the mean by at least half a minute.
6. A bag contains a large number of balls. 65\% are numbered 1
35\% are numbered 2
A random sample of 3 balls is taken from the bag.
Find the sampling distribution for the range of the numbers on the 3 selected balls.
7. The continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ given by $$\mathrm { f } ( x ) = \begin{cases} \frac { x ^ { 2 } } { 45 } & 0 \leq x \leq 3
\frac { 1 } { 5 } & 3 < x < 4
\frac { 1 } { 3 } - \frac { x } { 30 } & 4 \leq x \leq 10
0 & \text { otherwise } \end{cases}$$
Sketch $\mathrm { f } ( x )$ for $0 \leq x \leq 10$.
Find the cumulative distribution function $\mathrm { F } ( x )$ for all values of $x$.
Find $\mathrm { P } ( X \leq 8 )$.
8. In a large restaurant an average of 3 out of every 5 customers ask for water with their meal. A random sample of 10 customers is selected.
Find the probability that
1. exactly 6 ask for water with their meal,
2. less than 9 ask for water with their meal. A second random sample of 50 customers is selected.
Find the smallest value of $n$ such that $$\mathrm { P } ( X < n ) \geq 0.9$$ where the random variable $X$ represents the number of these customers who ask for water. END \section*{Advanced Level} \section*{Friday 18 January 2013 - Afternoon} \section*{Materials required for examination
Mathematical Formulae (Pink)} \section*{Items included with question papers
Nil} Candidates may use any calculator allowed by the regulations of the Joint
Council for Qualifications. Calculators must not have the facility for symbolic
algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. \section*{P41482A}
1. (a) Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution.
The probability of any one letter being delivered to the wrong house is 0.01 .
On a randomly selected day Peter delivers 1000 letters.
Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house. Give your answer to 4 decimal places.
2. In a village, power cuts occur randomly at a rate of 3 per year.
Find the probability that in any given year there will be
1. exactly 7 power cuts,
2. at least 4 power cuts.
Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20 .
3. A random variable $X$ has the distribution $\mathrm { B } ( 12 , p )$.
Given that $p = 0.25$, find
1. $\mathrm { P } ( X < 5 )$,
2. $\mathrm { P } ( X \geq 7 )$.
Given that $\mathrm { P } ( X = 0 ) = 0.05$, find the value of $p$ to 3 decimal places.
Given that the variance of $X$ is 1.92 , find the possible values of $p$.
4. The continuous random variable $X$ is uniformly distributed over the interval $[ - 4,6 ]$.
Write down the mean of $X$.
Find $\mathrm { P } ( X \leq 2.4 )$.
Find $\mathrm { P } ( - 3 < X - 5 < 3 )$. The continuous random variable $Y$ is uniformly distributed over the interval $[ a , 4 a ]$.
Use integration to show that $\mathrm { E } \left( Y ^ { 2 } \right) = 7 a ^ { 2 }$.
Find $\operatorname { Var } ( Y )$.
Given that $\mathrm { P } \left( X < \frac { 8 } { 3 } \right) = \mathrm { P } \left( Y < \frac { 8 } { 3 } \right)$, find the value of $a$.
5. The continuous random variable $T$ is used to model the number of days, $t$, a mosquito survives after hatching. The probability that the mosquito survives for more than $t$ days is $$\frac { 225 } { ( t + 15 ) ^ { 2 } } , \quad t \geq 0$$
Show that the cumulative distribution function of $T$ is given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \frac { 225 } { ( t + 15 ) ^ { 2 } } , & t \geq 0
0 , & \text { otherwise } \end{cases}$$
Find the probability that a randomly selected mosquito will die within 3 days of hatching.
Given that a mosquito survives for 3 days, find the probability that it will survive for at least 5 more days. A large number of mosquitoes hatch on the same day.
Find the number of days after which only $10 \%$ of these mosquitoes are expected to survive.
6. (a) Explain what you understand by a hypothesis.
Explain what you understand by a critical region. Mrs George claims that $45 \%$ of voters would vote for her.
In an opinion poll of 20 randomly selected voters it was found that 5 would vote for her.
Test at the $5 \%$ level of significance whether or not the opinion poll provides evidence to support Mrs George's claim. In a second opinion poll of $n$ randomly selected people it was found that no one would vote for Mrs George.
Using a $1 \%$ level of significance, find the smallest value of $n$ for which the hypothesis $\mathrm { H } _ { 0 } : p = 0.45$ will be rejected in favour of $\mathrm { H } _ { 1 } : p < 0.45$.
7. The continuous random variable $X$ has the following probability density function $$f ( x ) = \begin{cases} a + b x , & 0 \leq x \leq 5
0 , & \text { otherwise } \end{cases}$$ where $a$ and $b$ are constants.
Show that $10 a + 25 b = 2$. Given that $\mathrm { E } ( X ) = \frac { 35 } { 12 }$,
find a second equation in $a$ and $b$,
hence find the value of $a$ and the value of $b$.
Find, to 3 significant figures, the median of $X$.
Comment on the skewness. Give a reason for your answer. \section*{Statistics S2 (R)} \section*{Advanced/Advanced Subsidiary} \section*{Friday 24 May 2013 - Morning} Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. This paper is strictly for students outside the UK. In the boxes above, write your centre number, candidate number, your surname, initials and signature.
Check that you have the correct question paper.
Answer ALL the questions.
You must write your answer for each question in the space following the question.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for the parts of questions are shown in round brackets, e.g. (2).
There are 7 questions in this question paper. The total mark for this paper is 75 .
There are 24 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.
1. A bag contains a large number of counters. A third of the counters have a number 5 on them and the remainder have a number 1 .
A random sample of 3 counters is selected.
List all possible samples.
Find the sampling distribution for the range.
2. The continuous random variable $Y$ has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0
\frac { 1 } { 4 } \left( y ^ { 3 } - 4 y ^ { 2 } + k y \right) & 0 \leq y \leq 2
1 & y > 2 \end{array} \right.$$ where $k$ is a constant.
Find the value of $k$.
Find the probability density function of $Y$, specifying it for all values of $y$.
Find $\mathrm { P } ( Y > 1 )$.
3. The random variable $X$ has a continuous uniform distribution on $[ a , b ]$ where $a$ and $b$ are positive numbers. Given that $\mathrm { E } ( X ) = 23$ and $\operatorname { Var } ( X ) = 75$,
find the value of $a$ and the value of $b$. Given that $\mathrm { P } ( X > c ) = 0.32$,
find $\mathrm { P } ( 23 < X < c )$.
4. The random variable $X$ has probability density function $\mathrm { f } ( x )$ given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( 3 + 2 x - x ^ { 2 } \right) & 0 \leq x \leq 3
0 & \text { otherwise } \end{array} \right.$$ where $k$ is a constant.
Show that $k = \frac { 1 } { 9 }$.
Find the mode of $X$.
Use algebraic integration to find $\mathrm { E } ( X )$. By comparing your answers to parts (b) and (c),
describe the skewness of $X$, giving a reason for your answer.
5. In a village shop the customers must join a queue to pay. The number of customers joining the queue in a 10 minute interval is modelled by a Poisson distribution with mean 3. Find the probability that
exactly 4 customers join the queue in the next 10 minutes,
more than 10 customers join the queue in the next 20 minutes. When a customer reaches the front of the queue the customer pays the assistant. The time each customer takes paying the assistant, $T$ minutes, has a continuous uniform distribution over the interval $[ 0,5 ]$. The random variable $T$ is independent of the number of people joining the queue.
Find $\mathrm { P } ( T > 3.5 )$. In a random sample of 5 customers, the random variable $C$ represents the number of customers who took more than 3.5 minutes paying the assistant.
Find $\mathrm { P } ( C \geq 3 )$. Bethan has just reached the front of the queue and starts paying the assistant.
Find the probability that in the next 4 minutes Bethan finishes paying the assistant and no other customers join the queue.
6. Frugal bakery claims that their packs of 10 muffins contain on average 80 raisins per pack. A Poisson distribution is used to describe the number of raisins per muffin. A muffin is selected at random to test whether or not the mean number of raisins per muffin has changed.
Find the critical region for a two-tailed test using a $10 \%$ level of significance. The probability of rejection in each tail should be less than 0.05 .
Find the actual significance level of this test. The bakery has a special promotion claiming that their muffins now contain even more raisins. A random sample of 10 muffins is selected and is found to contain a total of 95 raisins.
Use a suitable approximation to test the bakery's claim. You should state your hypotheses clearly and use a $5 \%$ level of significance.
7. As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0.2 and the number of correct answers is represented by the random variable $X$.
Suggest a suitable distribution for $X$. Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable $S$ represents the final score, in points, for an applicant who chooses answers to this test at random.
Show that $S = 5 X - 20$.
Find $\mathrm { E } ( S )$ and $\operatorname { Var } ( S )$. An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process.
Find $\mathrm { P } ( S \geq 20 )$. Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0.4 .
Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly. 6684/01
Edexcel GCE \section*{Advanced Level} \section*{Friday 24 May 2013 - Morning} Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint
Council for Qualifications. Calculators must not have the facility for symbolic
algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S2), the paper reference (6684), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has 7 questions.
The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. \section*{P42035A} This publication may only be reproduced in accordance with Edexcel Limited copyright policy.
©2013 Edexcel Limited
1. A bag contains a large number of $1 \mathrm { p } , 2 \mathrm { p }$ and 5 p coins.
  $50 \%$ are 1 p coins
  $20 \%$ are 2 p coins
  $30 \%$ are 5 p coins
  A random sample of 3 coins is chosen from the bag.
2. List all the possible samples of size 3 with median 5p.
3. Find the probability that the median value of the sample is 5p.
4. Find the sampling distribution of the median of samples of size 3 .
5. The number of defects per metre in a roll of cloth has a Poisson distribution with mean 0.25 .
Find the probability that
a randomly chosen metre of cloth has 1 defect,
the total number of defects in a randomly chosen 6 metre length of cloth is more than 2 . A tailor buys 300 metres of cloth.
Using a suitable approximation find the probability that the tailor's cloth will contain less than 90 defects.
(5)
3. An online shop sells a computer game at an average rate of 1 per day.
Find the probability that the shop sells more than 10 games in a 7 day period. Once every 7 days the shop has games delivered before it opens.
Find the least number of games the shop should have in stock immediately after a delivery so that the probability of running out of the game before the next delivery is less than 0.05 . In an attempt to increase sales of the computer game, the price is reduced for six months. A random sample of 28 days is taken from these six months. In the sample of 28 days, 36 computer games are sold.
Using a suitable approximation and a $5 \%$ level of significance, test whether or not the average rate of sales per day has increased during these six months. State your hypotheses clearly.
4. A continuous random variable $X$ is uniformly distributed over the interval $[ b , 4 b ]$ where $b$ is a constant.
Write down $\mathrm { E } ( X )$.
Use integration to show that $\operatorname { Var } ( X ) = \frac { 3 b ^ { 2 } } { 4 }$.
Find $\operatorname { Var } ( 3 - 2 X )$. Given that $b = 1$, find
the cumulative distribution function of $X , \mathrm {~F} ( x )$, for all values of $x$,
the median of $X$.
5. The continuous random variable $X$ has a cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 1
\frac { x ^ { 3 } } { 10 } + \frac { 3 x ^ { 2 } } { 10 } + a x + b , & 1 \leq x \leq 2
1 , & x > 2 \end{cases}$$ where $a$ and $b$ are constants.
Find the value of $a$ and the value of $b$.
Show that $\mathrm { f } ( x ) = \frac { 3 } { 10 } \left( x ^ { 2 } + 2 x - 2 \right) , \quad 1 \leq x \leq 2$.
Use integration to find $\mathrm { E } ( X )$.
Show that the lower quartile of $X$ lies between 1.425 and 1.435.
6. In a manufacturing process $25 \%$ of articles are thought to be defective. Articles are produced in batches of 20.
A batch is selected at random. Using a $5 \%$ significance level, find the critical region for a two tailed test that the probability of an article chosen at random being defective is 0.25 . You should state the probability in each tail, which should be as close as possible to 0.025 . The manufacturer changes the production process to try to reduce the number of defective articles. She then chooses a batch at random and discovers there are 3 defective articles.
Test at the $5 \%$ level of significance whether or not there is evidence that the changes to the process have reduced the percentage of defective articles. State your hypotheses clearly.
7. A telesales operator is selling a magazine. Each day he chooses a number of people to telephone. The probability that each person he telephones buys the magazine is 0.1 .
Suggest a suitable distribution to model the number of people who buy the magazine from the telesales operator each day.
On Monday, the telesales operator telephones 10 people. Find the probability that he sells at least 4 magazines.
Calculate the least number of people he needs to telephone on Tuesday, so that the probability of selling at least 1 magazine, on that day, is greater than 0.95 . A call centre also sells the magazine. The probability that a telephone call made by the call centre sells a magazine is 0.05 . The call centre telephones 100 people every hour.
Using a suitable approximation, find the probability that more than 10 people telephoned by the call centre buy a magazine in a randomly chosen hour. \section*{\textbackslash section*\{END\}}

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