Questions S2 (1597 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel S2 Q3
3. The random variable \(X\) is modelled by a binomial distribution \(\mathrm { B } ( n , p )\), with \(n = 20\) and \(p\) unknown. It is suspected that \(p = 0 \cdot 4\).
  1. Find the critical region for the test of \(\mathrm { H } _ { 0 } : p = 0.4\) against \(\mathrm { H } _ { 1 } : p \neq 0.4\), at the \(5 \%\) significance level.
  2. Find the critical region if, instead, the alternative hypothesis is \(\mathrm { H } _ { 1 } : p < 0.4\).
Edexcel S2 Q4
4. A random variable \(X\) has the distribution \(\mathrm { B } ( 80,0.375 )\).
  1. Write down the mean and variance of \(X\).
  2. Use the Normal approximation to the binomial distribution to estimate \(\mathrm { P } ( X > 40 )\).
Edexcel S2 Q5
5. A traffic analyst is interested in the number of heavy lorries passing a certain junction. He counts the numbers of lorries in 100 five-minute intervals, and gets the following results:
Number of lorries in
five-minute interval, \(X\)
01234567
Number of intervals7132519151074
Q. 5 continued on next page ... \section*{STATISTICS 2 (A) TEST PAPER 9 Page 2}
  1. continued ...
    1. Show that the mean of \(X\) is 3 , and find the variance of \(X\).
    2. Give two reasons for thinking that \(X\) can be modelled by a Poisson distribution. (2 marks)
    After a new landfill site has been established nearby, a member of an environmental group notices that 18 lorries pass the junction in a period of 15 minutes. The group claims that this is evidence that the mean number of lorries per five-minute interval has increased.
  2. Test whether the group's claim is valid. Work at the \(5 \%\) significance level, and state your hypotheses clearly.
Edexcel S2 Q6
6. In a particular parliamentary constituency, the percentage of Conservative voters at the last election was \(35 \%\), and the percentage who voted for the Monster Raving Loony party was \(2 \%\).
  1. Find the probability that a random sample of 10 electors includes at least two Conservative voters. Use suitable approximations to find
  2. the probability that a random sample of 500 electors will include at least 200 who voted either Conservative or Monster Raving Loony,
  3. the probability that a random sample of 200 electors will have at least 5 Monster Raving Loony voters in it.
  4. One of (b) or (c) requires an adjustment to be made before a calculation is done. Explain what this adjustment is, and why it is necessary.
Edexcel S2 Q7
7. The fraction of sky covered by cloud is modelled by the random variable \(X\) with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 0
\mathrm { f } ( x ) = k x ^ { 2 } ( 1 - x ) & 0 \leq x \leq 1 ,
\mathrm { f } ( x ) = 0 & x > 1 . \end{array}$$
  1. Find \(k\) and sketch the graph of \(\mathrm { f } ( x )\).
  2. Find the mean and the variance of \(X\).
  3. Find the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Given that flying is prohibited when \(85 \%\) of the sky is covered by cloud, show that cloud conditions allow flying nearly \(90 \%\) of the time.
Edexcel S2 Q1
  1. Briefly explain what is meant by
    1. a statistical model,
      (2 marks)
    2. a sampling frame,
    3. a sampling unit.
    4. (a) Explain what is meant by the critical region of a statistical test.
    5. Under a hypothesis \(\mathrm { H } _ { 0 }\), an event \(A\) can happen with probability \(4 \cdot 2 \%\). The event \(A\) does then happen. State, with justification, whether \(\mathrm { H } _ { 0 }\) should be accepted or rejected at the \(5 \%\) significance level.
Edexcel S2 Q3
3
  1. Briefly describe the main features of a binomial distribution. I conduct an experiment by randomly selecting 10 cards, without replacement, from a normal pack of 52 .
  2. Explain why the distribution of \(X\), the number of hearts obtained, is not \(\mathrm { B } \left( 10 , \frac { 1 } { 4 } \right)\).
    (2 marks)
    After making the appropriate adjustment to the experiment, which should be stated, so that the distribution is \(\mathrm { B } \left( 10 , \frac { 1 } { 4 } \right)\), find
  3. the probability of getting no hearts,
  4. the probability of getting 4 or more hearts.
  5. If the modified experiment is repeated 50 times, find the total number of hearts that you would you expect to have drawn.
Edexcel S2 Q4
4. A Geiger counter is observed in the presence of a radioactive source. In 100 one-minute intervals, the number of counts recorded are as follows:
No of counts, \(X\)0123456
Frequency102429161263
  1. Find the mean and variance of this data, and show that it supports the idea that the random variable \(X\) is following a Poisson distribution.
  2. Use a Poisson distribution with the mean found in part (a) to calculate, to 3 decimal places, the probability that more than 6 counts will be recorded in any particular minute.
  3. Find the number of one-minute intervals, in the sample of 100 , in which more than 6 counts would be expected. \section*{STATISTICS 2 (A) TEST PAPER 10 Page 2}
Edexcel S2 Q5
  1. A continuous random variable \(X\) has the cumulative distribution function
$$\begin{array} { l l } \mathrm { F } ( x ) = 0 & x < 2 ,
\mathrm {~F} ( x ) = k ( x - a ) ^ { 2 } & 2 \leq x \leq 6 ,
\mathrm {~F} ( x ) = 1 & x \geq 6 . \end{array}$$
  1. Find the values of the constants \(a\) and \(k\).
  2. Show that the median of \(X\) is \(2 ( 1 + \sqrt { 2 } )\).
  3. Given that \(X > 4\), find the probability that \(X > 5\).
Edexcel S2 Q6
6. A small opinion poll shows that the Trendies have a \(10 \%\) lead over the Oldies. The poll is based on a survey of 20 voters, in which the Trendies got 11 and the Oldies 9. The Oldies spokesman says that the result is consistent with a \(10 \%\) lead for the Oldies, whilst the Trendies spokesperson says that this is impossible.
  1. At the \(5 \%\) significance level, test which is right, stating your null hypothesis carefully.
  2. If it is indeed true that the Trendies are supported by \(55 \%\) of the population, use a suitable approximation to find the probability that in a random sample of 200 voters they would obtain less than half of the votes.
Edexcel S2 Q7
7. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 6 x } { 175 } & 0 \leq x < 5
\mathrm { f } ( x ) = \frac { 6 x ( 10 - x ) } { 875 } & 5 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$
  1. Verify that f is a probability density function.
  2. Write down the probability that \(X < 1\).
  3. Find the cumulative distribution function of \(X\), carefully showing how it changes for different domains.
  4. Find the probability that \(2 < X < 7\).
Edexcel S2 Q1
  1. A golfer believes that the distance, in metres, that she hits a ball with a 5 iron, follows a continuous uniform distribution over the interval [100, 150].
    1. Find the median and interquartile range of the distance she hits a ball, that would be predicted by this model.
    2. Explain why the continuous uniform distribution may not be a suitable model.
      (2 marks)
    3. The continuous random variable \(X\) has the following cumulative distribution function:
    $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0
    \frac { 1 } { 64 } \left( 16 x - x ^ { 2 } \right) , & 0 \leq x \leq 8
    1 , & x > 8 \end{cases}$$
  2. Find \(\mathrm { P } ( X > 5 )\).
  3. Find and specify fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
  4. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
Edexcel S2 Q3
3. An electrician records the number of repairs of different types of appliances that he makes each day. His records show that over 40 working days he repaired a total of 180 CD players.
  1. Explain why a Poisson distribution may be suitable for modelling the number of CD players he repairs each day and find the parameter for this distribution.
  2. Find the probability that on one particular day he repairs
    1. no CD players,
    2. more than 6 CD players.
  3. Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days.
    (3 marks)
Edexcel S2 Q4
4. A teacher wants to investigate the sports played by students at her school in their free time. She decides to ask a random sample of 120 pupils to complete a short questionnaire.
  1. Give two reasons why the teacher might choose to use a sample survey rather than a census.
  2. Suggest a suitable sampling frame that she could use. The teacher believes that 1 in 20 of the students play tennis in their free time. She uses the data collected from her sample to test if the proportion is different from this.
  3. Using a suitable approximation and stating the hypotheses that she should use, find the critical region for this test. The probability for each tail of the region should be as close as possible to 5\%.
  4. State the significance level of this test.
Edexcel S2 Q5
5. As part of a business studies project, 8 groups of students are each randomly allocated 10 different shares from a listing of over 300 share prices in a newspaper. Each group has to follow the changes in the price of their shares over a 3-month period. At the end of the 3 months, \(35 \%\) of all the shares in the listing have increased in price and the rest have decreased.
  1. Find the probability that, for the 10 shares of one group,
    1. exactly 6 have gone up in price,
    2. more than 5 have gone down in price.
  2. Using a suitable approximation, find the probability that of the 80 shares allocated in total to the groups, more than 55 will have decreased in value.
Edexcel S2 Q6
6. A shoe shop sells on average 4 pairs of shoes per hour on a weekday morning.
  1. Suggest a suitable distribution for modelling the number of sales made per hour on a weekday morning and state the value of any parameters needed.
  2. Explain why this model might have to be modified for modelling the number of sales made per hour on a Saturday morning.
  3. Find the probability that on a weekday morning the shop sells
    1. more than 4 pairs in a one-hour period,
    2. no pairs in a half-hour period,
    3. more than 4 pairs during each hour from 9 am until noon. The area manager visits the shop on a weekday morning, the day after an advert appears in a local paper. In a one-hour period the shop sells 7 pairs of shoes, leading the manager to believe that the advert has increased the shop’s sales.
  4. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of an increase in sales following the appearance of the advert.
    (4 marks)
Edexcel S2 Q7
7. The continuous random variable \(T\) has the following probability density function: $$\mathrm { f } ( t ) = \begin{cases} k \left( t ^ { 2 } + 2 \right) , & 0 \leq t \leq 3
0 , & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 1 } { 15 }\).
  2. Sketch f \(( t )\) for all values of \(t\).
  3. State the mode of \(T\).
  4. Find \(\mathrm { E } ( T )\).
  5. Show that the standard deviation of \(T\) is 0.798 correct to 3 significant figures.
Edexcel S2 Q1
  1. (a) Explain what you understand by the term sampling frame when conducting a sample survey.
    (b) Suggest a suitable sampling frame and identify the sampling units when using a sample survey to study
    1. the frequency with which cars break down in the first 3 months after being serviced at a particular garage,
    2. the weight loss of people involved in trials of a new dieting programme.
      (4 marks)
    3. An ornithologist believes that on average 4.2 different species of bird will visit a bird table in a rural garden when 50 g of breadcrumbs are spread on it.
      (a) Suggest a suitable distribution for modelling the number of species that visit a bird table meeting these criteria.
      (b) Explain why the parameter used with this model may need to be changed if
    4. 50 g of nuts are used instead of breadcrumbs,
    5. 100 g of breadcrumbs are used.
    A bird table in a rural garden has 50 g of breadcrumbs spread on it.
    Find the probability that
    (c) exactly 6 different species visit the table,
    (d) more than 2 different species visit the table.
Edexcel S2 Q3
3. In a test studying reaction times, white dots appear at random on a black rectangular screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen. The distribution of \(X\) is rectangular over the interval [0,20].
  1. Find \(\mathrm { P } ( 2 < X < 3.6 )\).
  2. Find the mean and variance of \(X\). The continuous random variable \(Y\) represents the distance, in centimetres, of the dot from the bottom edge of the screen. The distribution of \(Y\) is rectangular over the interval [ 0,16 ]. Find the probability that a dot appears
  3. in a square of side 4 cm at the centre of the screen,
  4. within 2 cm of the edge of the screen.
Edexcel S2 Q4
4. It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3 . Find the probability that
  1. there will be no failures in a one-hour period,
  2. there will be more than 4 failures in a 30 -minute period. Using a suitable approximation, find the probability that in a 24-hour period there will be
  3. less than 60 failures,
  4. exactly 72 failures.
Edexcel S2 Q5
5. Six standard dice with faces numbered 1 to 6 are thrown together. Assuming that the dice are fair, find the probability that
  1. none of the dice show a score of 6,
  2. more than one of the dice shows a score of 6,
  3. there are equal numbers of odd and even scores showing on the dice. One of the dice is suspected of being biased such that it shows a score of 6 more often than the other numbers. This die is thrown eight times and gives a score of 6 three times.
  4. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not this die is biased towards scoring a 6.
    (7 marks)
Edexcel S2 Q6
6. The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} \frac { 1 } { 6 } x , & 0 \leq x \leq 2
\frac { 1 } { 12 } ( 6 - x ) , & 2 \leq x \leq 6
0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. State the mode of \(X\).
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\).
  4. Show that the median of \(X\) is 2.536, correct to 4 significant figures.
Edexcel S2 Q1
  1. (a) Explain briefly what you understand by the terms
    1. population,
    2. sample.
      (b) Giving a reason for each of your answers, state whether you would use a census or a sample survey to investigate
    3. the dietary requirements of people attending a 4-day residential course,
    4. the lifetime of a particular type of battery.
    5. The manager of a supermarket receives an average of 6 complaints per day from customers. Find the probability that on one day she receives
      (a) 3 complaints,
      (b) 10 or more complaints.
    The supermarket is open on six days each week.
    (c) Find the probability that the manager receives 10 or more complaints on no more than one day in a week.
    (4 marks)
Edexcel S2 Q3
3. The sales staff at an insurance company make house calls to prospective clients. Past records show that \(30 \%\) of the people visited will take out a new policy with the company. On a particular day, one salesperson visits 8 people. Find the probability that, of these,
  1. exactly 2 take out new policies,
  2. more than 4 take out new policies. The company awards a bonus to any salesperson who sells more than 50 policies in a month.
  3. Using a suitable approximation, find the probability that a salesperson gets a bonus in a month in which he visits 150 prospective clients.
    (5 marks)
Edexcel S2 Q4
4. A rugby player scores an average of 0.4 tries per match in which he plays.
  1. Find the probability that he scores 2 or more tries in a match. The team's coach moves the player to a different position in the team believing he will then score more frequently. In the next five matches he scores 6 tries.
  2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of an increase in the number of tries the player scores per match as a result of playing in a different position.
    (5 marks)