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Edexcel S2 2017 June Q6
16 marks Standard +0.3
6. The continuous random variable \(X\) has a probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ k & 3 < x < 5 \\ k ( 6 - x ) & 5 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\).
  2. Show that the value of \(k\) is \(\frac { 1 } { 3 }\)
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Hence find the 90th percentile of the distribution.
  5. Find \(\mathrm { P } [ \mathrm { E } ( X ) < X < 5.5 ]\)
    Leave
    blank
    END
Edexcel S2 2018 June Q1
12 marks Standard +0.3
In a call centre, the number of telephone calls, \(X\), received during any 10 -minute period follows a Poisson distribution with mean 9
  1. Find
    1. \(\mathrm { P } ( X > 5 )\)
    2. \(\mathrm { P } ( 4 \leqslant X < 10 )\) The length of a working day is 7 hours.
  2. Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.
  3. Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.
Edexcel S2 2018 June Q2
9 marks Moderate -0.8
2. A fair coin is spun 6 times and the random variable \(T\) represents the number of tails obtained.
  1. Give two reasons why a binomial model would be a suitable distribution for modelling \(T\).
  2. Find \(\mathrm { P } ( T = 5 )\)
  3. Find the probability of obtaining more tails than heads. A second coin is biased such that the probability of obtaining a head is \(\frac { 1 } { 4 }\) This second coin is spun 6 times.
  4. Find the probability that, for the second coin, the number of heads obtained is greater than or equal to the number of tails obtained.
Edexcel S2 2018 June Q3
18 marks Standard +0.3
  1. The length of time, \(T\), minutes, spent completing a particular task has probability density function
$$f ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } ( t - 1 ) & 1 < t \leqslant 2 \\ \frac { 1 } { 16 } \left( 14 t - 3 t ^ { 2 } - 8 \right) & 2 < t \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Use algebraic integration to find \(\mathrm { E } ( T )\) Given that \(\mathrm { E } \left( T ^ { 2 } \right) = \frac { 267 } { 40 }\)
  2. find \(\operatorname { Var } ( T )\)
  3. Find the cumulative distribution function \(\mathrm { F } ( t )\)
  4. Find the 20th percentile of the time taken to complete the task.
  5. Find the probability that the time spent completing the task is more than 1.5 minutes. Given that a person has already spent 1.5 minutes on the task,
  6. find the probability that this person takes more than 3 minutes to complete the task.
Edexcel S2 2018 June Q4
10 marks Moderate -0.3
  1. David aims to catch the train to work each morning. The scheduled departure time of the train is 0830
The number of minutes after 0830 that the train departs may be modelled by the random variable \(X\). Given that \(X\) has a continuous uniform distribution over \([ \alpha , \beta ]\) and that \(\mathrm { E } ( X ) = 4\) and \(\operatorname { Var } ( X ) = 12\)
  1. find the value of \(\alpha\) and the value of \(\beta\). Each morning, the probability that David oversleeps is 0.05 If David oversleeps he will be late for work. If he does not oversleep he will be in time to catch the train, but will be late for work if the train departs after 0835
  2. Find the probability that David will be late for work. Given that David is late for work,
  3. find the probability that he overslept.
Edexcel S2 2018 June Q5
16 marks Standard +0.3
5. Past records show that the proportion of customers buying organic vegetables from Tesson supermarket is 0.35 During a particular day, a random sample of 40 customers from Tesson supermarket was taken and 18 of them bought organic vegetables.
  1. Test, at the \(5 \%\) level of significance, whether or not this provides evidence that the proportion of customers who bought organic vegetables has increased. State your hypotheses clearly. The manager of Tesson supermarket claims that the proportion of customers buying organic eggs is different from the proportion of those buying organic vegetables. To test this claim the manager decides to take a random sample of 50 customers.
  2. Using a \(5 \%\) level of significance, find the critical region to enable the Tesson supermarket manager to test her claim. The probability for each tail of the region should be as close as possible to \(2.5 \%\) During a particular day, a random sample of 50 customers from Tesson supermarket is taken and 8 of them bought organic eggs.
  3. Using your answer to part (b), state whether or not this sample supports the manager's claim. Use a \(5 \%\) level of significance.
  4. State the actual significance level of this test. The proportion of customers who buy organic fruit from Tesson supermarket is 0.2 During a particular day, a random sample of 200 customers from Tesson supermarket is taken. Using a suitable approximation, the probability that fewer than \(n\) of these customers bought organic fruit is 0.0465 correct to 4 decimal places.
  5. Find the value of \(n\).
Edexcel S2 2018 June Q6
10 marks Challenging +1.2
  1. The continuous random variable \(X\) has the following cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 1 \\ \frac { 4 } { 15 } ( x - 1 ) & 1 < x \leqslant 2 \\ k \left( \frac { a x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } \right) + b & 2 < x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$ where \(k , a\) and \(b\) are constants.
Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)
  1. show that \(a = 4\)
  2. Find \(\mathrm { P } ( X < 2.5 )\) giving your answer to 3 significant figures.
Edexcel S2 Q1
9 marks Standard +0.3
  1. The lifetime, in tens of hours, of a certain delicate electrical component can be modelled by the random variable \(X\) with probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 42 } x , & 0 \leq x < 6 \\ \frac { 1 } { 7 } & 6 \leq x \leq 10 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Find the probability that a component lasts at least 50 hours. A particular device requires two of these components and it will not operate if one or more of the components fail. The device has just been fitted with two new components and the lifetimes of these two components are independent.
  3. Find the probability that the device breaks down within the next 50 hours.
Edexcel S2 Q2
11 marks Standard +0.3
2. The continuous random variable \(X\) represents the error, in mm, made when a machine cuts piping to a target length. The distribution of \(X\) is rectangular over the interval \([ - 5.0,5.0 ]\). Find
  1. \(\mathrm { P } ( X < - 4.2 )\),
  2. \(\mathrm { P } ( | X | < 1.5 )\). A supervisor checks a random sample of 10 lengths of piping cut by the machine.
  3. Find the probability that more than half of them are within 1.5 cm of the target length.
    (3 marks)
    If \(X < - 4.2\), the length of piping cannot be used. At the end of each day the supervisor checks a random sample of 60 lengths of piping.
  4. Use a suitable approximation to estimate the probability that no more than 2 of these lengths of piping cannot be used.
    (5 marks)
Edexcel S2 Q3
11 marks Moderate -0.8
3. An athletics teacher has kept careful records over the past 20 years of results from school sports days. There are always 10 competitors in the javelin competition. Each competitor is allowed 3 attempts and the teacher has a record of the distances thrown by each competitor at each attempt. The random variable \(D\) represents the greatest distance thrown by each competitor and the random variable \(A\) represents the number of the attempt in which the competitor achieved their greatest distance.
  1. State which of the two random variables \(D\) or \(A\) is continuous. A new athletics coach wishes to take a random sample of the records of 36 javelin competitors.
  2. Specify a suitable sampling frame and explain how such a sample could be taken.
    (2 marks)
    The coach assumes that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\), and is therefore surprised to find that 20 of the 36 competitors in the sample achieved their greatest distance on their second attempt. Using a suitable approximation, and assuming that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\),
  3. find the probability that at least 20 of the competitors achieved their greatest distance on their second attempt.
    (6 marks)
  4. Comment on the assumption that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\).
Edexcel S2 Q4
12 marks Standard +0.3
4. From past records a manufacturer of glass vases knows that \(15 \%\) of the production have slight defects. To monitor the production, a random sample of 20 vases is checked each day and the number of vases with slight defects is recorded.
  1. Using a 5\% significance level, find the critical regions for a two-tailed test of the hypothesis that the probability of a vase with slight defects is 0.15 . The probability of rejecting, in either tail, should be as close as possible to \(2.5 \%\).
  2. State the actual significance level of the test described in part (a). A shop sells these vases at a rate of 2.5 per week. In the 4 weeks of December the shop sold 15 vases.
  3. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales per week had increased in December.
    (6 marks)
Edexcel S2 Q5
14 marks Standard +0.3
5. The continuous random variable \(T\) represents the time in hours that students spend on homework. The cumulative distribution function of \(T\) is $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0 \\ k \left( 2 t ^ { 3 } - t ^ { 4 } \right) & 0 \leq t \leq 1.5 \\ 1 , & t > 1.5 \end{cases}$$ where \(k\) is a positive constant.
  1. Show that \(k = \frac { 16 } { 27 }\).
  2. Find the proportion of students who spend more than 1 hour on homework.
  3. Find the probability density function \(\mathrm { f } ( t )\) of \(T\).
  4. Show that \(\mathrm { E } ( T ) = 0.9\).
  5. Show that \(\mathrm { F } ( \mathrm { E } ( T ) ) = 0.4752\). A student is selected at random. Given that the student spent more than the mean amount of time on homework,
  6. find the probability that this student spent more than 1 hour on homework.
Edexcel S2 Q6
18 marks Standard +0.8
6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that
  1. at least 4 customers arrive in the next 10 minutes,
  2. no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
  3. Find the probability that no customers arrive in at most one of these periods. The post office is open for \(3 \frac { 1 } { 2 }\) hours on Wednesday mornings.
  4. Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END
Edexcel S3 2021 January Q1
6 marks Easy -1.8
  1. A journalist is going to interview a sample of 10 players from the 60 players in a local football club. The journalist uses the random numbers on page 27 of the formula booklet and starts at the top of the 10th column, where the first number is 96
The journalist worked down the 10th column to select 10 numbers. The first 3 numbers selected were: 33, 15 and 23
  1. Find the other 7 numbers to complete the sample of ten. There are 24 girls and 36 boys who play football for the club.
    The journalist labels the girls from 1 to 24 and the boys from 25 to 60
  2. Show how the journalist can use her 10 random numbers to select a stratified sample of 10 players from the club to interview. The club provided the journalist with a list of the players in ascending order of ages, numbered 1 to 60. The journalist uses the 10 random numbers to select a simple random sample of the players.
  3. State, giving a reason, a group of players who may not be represented in this sample.
Edexcel S3 2021 January Q2
9 marks Standard +0.3
2. A teacher believes that those of her students with strong mathematical ability may also have enhanced short-term memory. She shows a random sample of 11 students a tray of different objects for eight seconds and then asks them to write down as many of the objects as they can remember. The results, along with their percentage score in a recent mathematics test, are given in the table below.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)
No. of objects811915176101412135
\% in maths test3062578075436551485532
  1. Calculate Spearman's rank correlation coefficient for these data. Show your working clearly.
  2. Stating your hypotheses clearly, carry out a suitable test to assess the teacher's belief. Use a \(5 \%\) level of significance and state your critical value. The teacher shows these results to her class and argues that spending more time trying to improve their short-term memory would improve their mathematical ability.
  3. Explain whether or not you agree with the teacher's argument.
Edexcel S3 2021 January Q3
10 marks Standard +0.3
3. The students in a group of schools can choose a club to join. There are 4 clubs available: Music, Art, Sports and Computers. The director collected information about the number of students in each club, using a random sample of 88 students from across the schools. The results are given in Table 1 below. \begin{table}[h]
\cline { 2 - 5 } \multicolumn{1}{c|}{}MusicArtSportsComputers
No. of students14282719
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} The director uses a chi-squared test to determine whether or not the students are uniformly distributed across the 4 clubs.
    1. Find the expected frequencies he should use. Given that the test statistic he calculated was 6.09 (to 3 significant figures)
    2. use a \(5 \%\) level of significance to complete the test. You should state the degrees of freedom and the critical value used. The director wishes to examine the situation in more detail and takes a second random sample of 88 students. The director assumes that within each school, students select their clubs independently. The students come from 3 schools and the distribution of the students from each school amongst the clubs is given in Table 2 below. \begin{table}[h]
      School ClubMusicArtSportsComputers
      School \(\boldsymbol { A }\)31098
      School \(\boldsymbol { B }\)111135
      School \(\boldsymbol { C }\)11674
      \captionsetup{labelformat=empty} \caption{Table 2}
      \end{table} The director wishes to test for an association between a student's school and the club they choose.
  2. State hypotheses suitable for such a test.
  3. Calculate the expected frequency for School \(C\) and the Computers club. The director calculates the test statistic to be 7.29 (to 3 significant figures) with 4 degrees of freedom.
  4. Explain clearly why his test has 4 degrees of freedom.
  5. Complete the test using a \(5 \%\) level of significance and stating clearly your critical value.
Edexcel S3 2021 January Q4
14 marks Standard +0.3
4. The scores in a national test of seven-year-old children are normally distributed with a standard deviation of 18
A random sample of 25 seven-year-old children from town \(A\) had a mean score of 52.4
  1. Calculate a 98\% confidence interval for the mean score of the seven-year-old children from town \(A\).
    (4) An independent random sample of 30 seven-year-old children from town \(B\) had a mean score of 57.8
    A local newspaper claimed that the mean score of seven-year-old children from town \(B\) was greater than the mean score of seven-year-old children from town \(A\).
  2. Stating your hypotheses clearly, use a \(5 \%\) significance level to test the newspaper's claim. You should show your working clearly. The mean score for the national test of seven-year-old children is \(\mu\). Considering the two samples of seven-year-old children separately, at the \(5 \%\) level of significance, there is insufficient evidence that the mean score for town \(A\) is less than \(\mu\), and insufficient evidence that the mean score for town \(B\) is less than \(\mu\).
  3. Find the largest possible value for \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-11_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel S3 2021 January Q5
18 marks Standard +0.3
5. Chrystal is studying the lengths of pine cones that have fallen from a tree. She believes that the length, \(X \mathrm {~cm}\), of the pine cones can be modelled by a normal distribution with mean 6 cm and standard deviation 0.75 cm . She collects a random sample of 80 pine cones and their lengths are recorded in the table below.
Length, \(x\) cm\(x < 5\)\(5 \leqslant x < 5.5\)\(5.5 \leqslant x < 6\)\(6 \leqslant x < 6.5\)\(x \geqslant 6.5\)
Frequency614242610
  1. Stating your hypotheses clearly and using a \(10 \%\) level of significance, test Chrystal's belief. Show your working clearly and state the expected frequencies, the test statistic and the critical value used.
    (10) Chrystal's friend David asked for more information about the lengths of the 80 pine cones. Chrystal told him that $$\sum x = 464 \quad \text { and } \quad \sum x ^ { 2 } = 2722.59$$
  2. Calculate unbiased estimates of the mean and variance of the lengths of the pine cones. David used the calculations from part (b) to test whether or not the lengths of the pine cones are normally distributed using Chrystal's sample. His test statistic was 3.50 (to 3 significant figures) and he did not pool any classes.
  3. Using a \(10 \%\) level of significance, complete David's test stating the critical value and the degrees of freedom used.
  4. Estimate, to 2 significant figures, the proportion of pine cones from the tree that are longer than 7 cm . \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-15_2255_50_314_34}
Edexcel S3 2021 January Q6
18 marks Standard +0.8
6. A potter makes decorative tiles in two colours, red and yellow. The length, \(R \mathrm {~cm}\), of the red tiles has a normal distribution with mean 15 cm and standard deviation 1.5 cm . The length, \(Y \mathrm {~cm}\), of the yellow tiles has the normal distribution \(\mathrm { N } \left( 12,0.8 ^ { 2 } \right)\). The random variables \(R\) and \(Y\) are independent. A red tile and a yellow tile are chosen at random.
  1. Find the probability that the yellow tile is longer than the red tile. Taruni buys 3 red tiles and 1 yellow tile.
  2. Find the probability that the total length of the 3 red tiles is less than 4 times the length of the yellow tile. Stefan defines the random variable \(X = a R + b Y\), where \(a\) and \(b\) are constants. He wants to use values of \(a\) and \(b\) such that \(X\) has a mean of 780 and minimum variance.
  3. Find the value of \(a\) and the value of \(b\) that Stefan should use. \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-19_2255_50_314_34}
Edexcel S3 2022 January Q1
8 marks Easy -1.2
  1. The Headteacher of a school is thinking about making changes to the school day. She wants to take a sample of 60 students so that she can find out what the students think about the proposed changes.
The names of the 1200 students of the school are listed alphabetically.
  1. Explain how the Headteacher could take a systematic sample of 60 students.
    1. Explain why systematic sampling is likely to be quicker than simple random sampling in this situation.
    2. With reference to this situation,
      When the Headteacher completes the systematic sample of size 60 she finds that 6 students were to be selected from Year 9. The Head of Mathematics suggests that a stratified sample of size 60 would be a more appropriate method. There were 200 students in Year 9.
  3. Explain why this suggests that a stratified sample of size 60 may be better than the systematic sample taken by the Headteacher.
Edexcel S3 2022 January Q2
8 marks Standard +0.3
2. Krishi owns a farm on which he keeps chickens. He selects, at random, 10 of the eggs produced and weighs each of them.
You may assume that these weights are a random sample from a normal distribution with standard deviation 1.9 g The total weight of these 10 eggs is 537.2 g
  1. Find a \(95 \%\) confidence interval for the mean weight of the eggs produced by Krishi's chickens. Krishi was hoping to obtain a \(99 \%\) confidence interval of width at most 1.5 g
  2. Calculate the minimum sample size necessary to achieve this. \includegraphics[max width=\textwidth, alt={}, center]{fc43aabf-ad04-4852-8539-981cef608f31-04_2662_95_107_1962}
Edexcel S3 2022 January Q3
8 marks Standard +0.3
3. The table shows the time, in seconds, of the fastest qualifying lap for 10 different Formula One racing drivers and their finishing position in the actual race.
Driver\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Fastest
qualifying lap
62.9463.9263.6362.9563.9763.8764.3164.6465.1864.21
Finishing
position
12345678910
  1. Calculate the value of Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly, test at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the fastest qualifying lap time and finishing position for these Formula One racing drivers.
Edexcel S3 2022 January Q4
10 marks Standard +0.3
4. A manager at a large estate agency believes that the type of property affects the time taken to sell it. A random sample of 125 properties sold is shown in the table.
\multirow{2}{*}{}Type of property
BungalowFlatHouseTotal
Sold within three months7294682
Sold in more than three months9191543
Total164861125
Test, at the \(5 \%\) level of significance, whether there is evidence for an association between the type of property and the time taken to sell it. You should state your hypotheses, expected frequencies, test statistic and the critical value used for this test.
Edexcel S3 2022 January Q5
15 marks Standard +0.3
  1. A dog breeder claims that the mean weight of male Great Dane dogs is 20 kg more than the mean weight of female Great Dane dogs.
Tammy believes that the mean weight of male Great Dane dogs is more than 20 kg more than the mean weight of female Great Dane dogs. She takes random samples of 50 male and 50 female Great Dane dogs and records their weights. The results are summarised below, where \(x\) denotes the weight, in kg , of a male Great Dane dog and \(y\) denotes the weight, in kg, of a female Great Dane dog. $$\sum x = 3610 \quad \sum x ^ { 2 } = 260955.6 \quad \sum y = 2585 \quad \sum y ^ { 2 } = 133757.2$$
  1. Find unbiased estimates for the mean and variance of the weights of
    1. the male Great Dane dogs,
    2. the female Great Dane dogs.
  2. Stating your hypotheses clearly, carry out a suitable test to assess Tammy's belief. Use a \(5 \%\) level of significance and state your critical value.
  3. For the test in part (b), state whether or not it is necessary to assume that the weights of the Great Dane dogs are normally distributed. Give a reason for your answer.
  4. State an assumption you have made in carrying out the test in part (b).
Edexcel S3 2022 January Q6
12 marks Standard +0.8
  1. The number of emails per hour received by a helpdesk were recorded. The results for a random sample of 80 one-hour periods are shown in the table.
Number of emails per hour0123456
Frequencies11023151993
  1. Show that the mean number of emails per hour in the sample is 3 The manager believes that the number of emails per hour received could be modelled by a Poisson distribution. The following table shows some of the expected frequencies.
    Number of emails per hourExpected Frequencies
    0\(r\)
    111.949
    217.923
    317.923
    413.443
    5\(s\)
    \(\geqslant 6\)\(t\)
  2. Find the values of \(r , s\) and \(t\), giving your answers to 3 decimal places.
  3. Using a 10\% significance level, test whether or not a Poisson model is reasonable. You should clearly state your hypotheses, test statistic and the critical value used.