Questions — Edexcel S2 (494 questions)

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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)
Edexcel S2 Q5
5. The continuous random variable \(X\) has the following cumulative distribution function: $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0
\frac { 1 } { 432 } x ^ { 2 } \left( x ^ { 2 } - 16 x + 72 \right) , & 0 \leq x \leq 6
1 , & x > 6 \end{cases}$$
  1. Find \(\mathrm { P } ( X < 2 )\).
  2. Find and specify fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
  3. Show that the mode of \(X\) is 2 .
  4. State, with a reason, whether the median of \(X\) is higher or lower than the mode of \(X\).
Edexcel S2 Q6
6. A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008
  1. Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery.
  2. Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken.
  3. State the conditions under which the binomial distribution can be approximated by the Poisson distribution.
  4. Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer.
Edexcel S2 Q7
7. The random variable \(X\) follows a continuous uniform distribution over the interval [2,11].
  1. Write down the mean of \(X\).
  2. Find \(\mathrm { P } ( X \geq 8.6 )\).
  3. Find \(\mathrm { P } ( | X - 5 | < 2 )\). The random variable \(Y\) follows a continuous uniform distribution over the interval \([ a , b ]\).
  4. Show by integration that $$\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 1 } { 3 } \left( b ^ { 2 } + a b + a ^ { 2 } \right)$$
  5. Hence, prove that $$\operatorname { Var } ( Y ) = \frac { 1 } { 12 } ( b - a ) ^ { 2 }$$ You may assume that \(\mathrm { E } ( Y ) = \frac { 1 } { 2 } ( a + b )\).
Edexcel S2 Q1
  1. The continuous random variable \(X\) has the following cumulative distribution function:
$$F ( x ) = \begin{cases} 0 , & x < 2
k \left( 19 x - x ^ { 2 } - 34 \right) , & 2 \leq x \leq 5
1 , & x > 5 \end{cases}$$
  1. Show that \(k = \frac { 1 } { 36 }\).
  2. Find \(\mathrm { P } ( X > 4 )\).
  3. Find and specify fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
Edexcel S2 Q2
2. Suggest, with reasons, suitable distributions for modelling each of the following:
  1. the number of times the letter J occurs on each page of a magazine,
  2. the length of string left over after cutting as many 3 metre long pieces as possible from partly used balls of string,
  3. the number of heads obtained when spinning a coin 15 times.
Edexcel S2 Q3
3. A primary school teacher finds that exactly half of his year 6 class have mobile phones. He decides to investigate whether the proportion of pupils with mobile phones is different from 0.5 in the year 5 class at his school. There are 25 pupils in the year 5 class.
  1. State the hypotheses that he should use.
  2. Find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
  3. Determine the significance level of this test. He finds that eight of the year 5 pupils have mobile phones and concludes that there is not sufficient evidence of the proportion being different from 0.5
  4. Stating the new hypotheses clearly, find if the number of year 5 pupils with mobile phones would have been significant if he had tested whether or not the proportion was less than 0.5 and used the largest critical region with a probability of less than \(5 \%\).
    (3 marks)
Edexcel S2 Q4
4. A hardware store is open on six days each week. On average the store sells 8 of a particular make of electric drill each week. Find the probability that the store sells
  1. no more than 4 of the drills in a week,
  2. more than 2 of the drills in one day. The store receives one delivery of drills at the same time each week.
  3. Find the number of drills that need to be in stock after a delivery for there to be at most a 5\% chance of the store not having sufficient drills to meet demand before the next delivery.
    (3 marks)
    [0pt]
Edexcel S2 Q5
5. In a party game, a bottle is spun and whoever it points to when it stops has to play next. The acute angle, in degrees, that the bottle makes with the side of the room is modelled by a rectangular distribution over the interval [0,90]. Find the probability that on one spin this angle is
  1. between \(25 ^ { \circ }\) and \(38 ^ { \circ }\),
  2. \(45 ^ { \circ }\) to the nearest degree. The bottle is spun ten times.
  3. Find the probability that the acute angle it makes with the side of the room is less than \(10 ^ { \circ }\) more than twice.
Edexcel S2 Q6
6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.
  1. Find the significance level of this test.
  2. State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
  3. Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
  4. Using a suitable approximation, test at the \(5 \%\) level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
    (6 marks)
Edexcel S2 Q7
7. In a competition at a funfair, participants have to stay on a log being rotated in a pool of water for as long as possible. The length of time, in tens of seconds, that the competitors stay on the log is modelled by the random variable \(T\) with the following probability density function: $$\mathrm { f } ( t ) = \begin{cases} k ( t - 3 ) ^ { 2 } , & 0 \leq t \leq 3
0 , & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 1 } { 9 }\).
  2. Sketch f \(( t )\) for all values of \(t\).
  3. Show that the mean time that competitors stay on the \(\log\) is 7.5 seconds. When the competition is next run the organisers decide to make it easier at first by spinning the log more slowly and then increasing the speed of rotation. The length of time, in tens of seconds, that the competitors now stay on the log is modelled by the random variable \(S\) with the following probability density function: $$f ( s ) = \begin{cases} \frac { 1 } { 12 } \left( 8 - s ^ { 3 } \right) , & 0 \leq s \leq 2
    0 , & \text { otherwise } \end{cases}$$
  4. Find the change in the mean time that competitors stay on the log.
Edexcel S2 Q1
  1. (a) State one advantage and one disadvantage in using a census rather than a sample survey in statistical work.
    (b) Give an example of a situation in which you would choose to use a census rather than a sample survey and explain why.
    (2 marks)
  2. An advert for Tatty's Crisps claims that 1 in 10 bags contain a free scratchcard game.
Tatty's Crisps can be bought in a Family Pack containing 10 bags. Find the probability that the bags in one of these Family Packs contain
(a) no scratchcards,
(b) more than 2 scratchcards. Tatty's Crisps can also be bought wholesale in boxes containing 50 bags. A pub Landlord notices that her customers only found 2 scratchcards in the crisps from one of these boxes.
(c) Stating your hypotheses clearly, test at the \(10 \%\) level of significance whether or not this gives evidence of there being fewer free scratchcards than is claimed by the advert.
(4 marks)