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 \([ 0,1 ]\).

1. Write down \(\mathrm { P } ( \mathrm { T } < 0.2 )\).
2. Write down E(T).
4. Use integration to find \(\operatorname { Var } ( T )\). A group of 20 children each play this game once.
5. 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 The children are allowed to practise this game so that this continuous uniform model is no longer applicable.
6. 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 $$\mathrm { g } ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5
   4 - 4 t , & 0.5 \leq t \leq 1
   0 , & \text { otherwise } \end{cases}$$
7. Using this model show that \(\mathrm { P } ( T < 0.2 ) = 0.08\). A group of 75 experienced players each played this game once.
8. 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\%.
9. Identify the population and the statistic in this situation.
10. 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.
11. 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
12. more than 196 passengers turn up for this flight,
13. 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.
14. Suggest a suitable model for the distribution of \(X\) and specify it fully.
15. Calculate the mean time of arrival of the bus.
16. Find the cumulative distribution function of \(X\). Jean will be late for work if the bus arrives after 8.05 a.m.
17. 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.
18. Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour.
19. 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.
20. Write down the distribution of the number of users failing to connect at their first attempt in an 8 -hour period.
21. 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.
22. 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.
23. Find \(\mathrm { P } ( 5 < X < 15 )\).
24. 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.
25. 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 $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 0 ,
    k x ^ { 2 } + 2 k x , & 0 \leq x \leq 2 ,
    8 k , & x > 2 . \end{array} \right.$$
26. Show that \(k = \frac { 1 } { 8 }\).
27. Find the median of \(X\).
28. Find the probability density function \(\mathrm { f } ( x )\).
29. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
30. Write down the mode of \(X\).
31. Find \(\mathrm { E } ( X )\).
32. 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.
33. 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 \(2.5 \%\).
34. 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 \(L \sim \mathrm {~N} \left( \mu , 0.3 ^ { 2 } \right)\).
35. 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.
36. Find the probability that at most 2 of the canes have length less than 150 cm . Another customer buys 500 canes.
37. 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 .
38. 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.
39. 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.
40. 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 $$f ( x ) = \begin{cases} \frac { x } { 15 } , & 0 \leq x \leq 2
    \frac { 2 } { 15 } , & 2 < x < 7
    \frac { 4 } { 9 } - \frac { 2 x } { 45 } , & 7 \leq x \leq 10
    0 , & \text { otherwise } \end{cases}$$
41. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
42. 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 \(x > 10\).
43. Find \(\mathrm { P } ( X \leq 8.2 )\).
44. 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.
45. 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 \(\mathrm { H } _ { 1 } : \lambda \neq 7\).
46. 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 \(2.5 \%\).
47. Write down the significance level of this test. The actual value of \(x\) obtained was 5 .
48. 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.
49. 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
50. the probability that in a randomly chosen plot of size \(4 \mathrm {~m} ^ { 2 }\) there will be fewer than 3 of these weeds.
51. 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 $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0
    \frac { 1 } { 3 } x ^ { 2 } \left( 4 - x ^ { 2 } \right) , & 0 \leq x \leq 1
    1 & x > 1 \end{cases}$$
52. Find \(\mathrm { P } ( X > 0.7 )\).
53. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
54. Calculate \(\mathrm { E } ( X )\) and show that, to 3 decimal places, \(\operatorname { Var } ( X ) = 0.057\). One measure of skewness is $$\frac { \text { Mean - Mode } } { \text { Standard deviation } }$$
55. 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
56. exactly one,
57. more than three. A customer bought three boxes.
58. 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.
59. 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 .
60. 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.
61. Define the population associated with the magazine.
62. Suggest a suitable sampling frame for the survey.
63. Identify the sampling units.
64. 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.
65. 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.
66. 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.
67. 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 \(Y \sim \operatorname { Po } ( 30 )\).
68. Estimate \(\mathrm { P } ( Y > 28 )\).
    3. In a town, \(30 \%\) of residents listen to the local radio station. Four residents are chosen at random.
69. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio.
70. On graph paper, draw the probability distribution of \(X\).
71. Write down the most likely number of these four residents that listen to the local radio station.
72. 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
73. 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.
74. Specify the probability density function of \(X\) and sketch its graph.
75. Find the probability that the machine dispenses
    1. less than 183 ml ,
    2. exactly 183 ml .
76. Calculate the inter-quartile range of \(X\).
77. Determine the value of \(x\) such that \(\mathrm { P } ( X \geq x ) = 2 \mathrm { P } ( X \leq x )\).
78. Interpret in words your value of \(x\).
    6. A doctor expects to see, on average, 1 patient per week with a particular disease.
79. 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.
80. 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.
81. 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.
82. 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.
83. Show that \(k = 3\). Find
84. \(\mathrm { E } ( X )\),
85. the cumulative distribution function \(\mathrm { F } ( x )\),
86. \(\mathrm { P } ( - 0.3 < X < 0.3 )\). \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.
87. Find \(\mathrm { P } ( S \leq 1 )\). The random variable \(T\) has the normal distribution \(\mathrm { N } \left( 25,5 ^ { 2 } \right)\).
88. Find \(\mathrm { P } ( T \leq 18 )\).
    3. The discrete random variable \(X\) is distributed \(\mathrm { B } ( n , p )\).
89. Write down the value of \(p\) that will give the most accurate estimate when approximating the binomial distribution by a normal distribution.
90. Give a reason to support your value.
91. 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,
92. find the probability that in a random sample of 12 people, exactly 2 of them catch the virus.
93. 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.
94. 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
95. exactly 6 cars pass this point,
96. 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.
97. 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.
98. 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 \(2.5 \%\).
99. 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.
100. 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.
101. Show that \(k = \frac { 3 } { 56 }\).
102. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
103. Evaluate \(\mathrm { E } ( X )\).
104. Find the modal value of \(X\).
105. Verify that the median value of \(X\) lies between 2.3 and 2.5.
106. 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
107. \(\mathrm { P } ( X < 2.7 )\),
108. \(\mathrm { E } ( X )\),
109. \(\operatorname { Var } ( X )\).
     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.
110. Write down a suitable model for the distribution of defective components in a batch. Find the probability that a batch contains
111. no defective components,
112. more than 6 defective components.
113. 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.
114. 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 \(\frac { 1 } { 7 }\).
115. Find the probability that on a particular day there are fewer than 2 breakdowns.
116. 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.
117. 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 \(\mathrm { m } ^ { 2 }\).
118. 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
119. exactly 2 defects,
120. more than 5 defects. The carpet fitter orders a total of \(355 \mathrm {~m} ^ { 2 }\) of the carpet for the whole hotel.
121. 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 $$f ( x ) = \begin{cases} \frac { 1 } { 3 } , & 0 \leq x \leq 1
     \frac { 8 x ^ { 3 } } { 45 } , & 1 \leq x \leq 2
     0 , & \text { otherwise } \end{cases}$$
122. Calculate the mean of \(X\).
123. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\).
124. Find the median of \(X\).
125. 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
126. \(\mathrm { P } ( R = 5 )\),
127. \(\mathrm { P } ( S = 5 )\),
128. \(\mathrm { P } ( T = 5 )\).
     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.
129. Explain why this might be the case.
130. 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 \(X\).
131. Write down the name of the probability density function of \(X\), and specify it fully.
132. Find \(\mathrm { P } \left( X < \frac { 1 } { 3 } l \right)\).
133. Write down the value of \(\mathrm { E } ( X )\). Two identical rods of length \(2 l\) are broken.
134. 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.
135. 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.
136. Using a normal approximation, test at the \(1 \%\) level of significance whether or not the claim is justified. Use a one-tailed test.
137. 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.
138. Find the probability that exactly 2 of them are defective. On another occasion, a random sample of 100 articles is taken.
139. 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.
140. 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
141. in a randomly chosen month, more than 4 accidents occurred,
142. 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.
143. 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.
144. 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 $$\mathrm { f } ( x ) = \left\{ \begin{array} { l c } k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4
     0 , & \text { otherwise } \end{array} \right.$$
145. Show that \(k = \frac { 2 } { 9 }\). Find
146. \(\mathrm { E } ( X )\),
147. the mode of \(X\).
148. the cumulative distribution function \(\mathrm { F } ( x )\) for all \(x\).
149. Evaluate \(\mathrm { P } ( X \leq 2.5 )\),
150. 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 .
151. 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 \([ 2,6 ]\).
152. Write down the probability density function \(\mathrm { f } ( x )\). Find
153. \(\mathrm { E } ( X )\),
154. \(\operatorname { Var } ( X )\),
155. the cumulative distribution function of \(X\), for all \(x\),
156. \(\mathrm { P } ( 2.3 < X < 3.4 )\).
     3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.
157. 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
158. a randomly chosen page has no misprints,
159. the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
160. 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
161. a sampling unit,
162. a sampling frame,
163. a sampling distribution.
     5. In a manufacturing process, \(2 \%\) of the articles produced are defective. A batch of 200 articles is selected.
164. Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
165. 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.
166. Show that \(k = \frac { 1 } { 4 }\). Find
167. \(\mathrm { E } ( X )\),
168. the mode of \(X\),
169. the median of \(X\).
170. Comment on the skewness of the distribution.
171. 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.
172. 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,
173. 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.
174. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, the doctor's belief.
175. 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.