Questions — Edexcel (10514 questions)

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Edexcel S2 Specimen Q3
7 marks Moderate -0.3
A manufacturer of chocolates produces 3 times as many soft centred chocolates as hard centred ones. Assuming that chocolates are randomly distributed within boxes of chocolates, find the probability that in a box containing 20 chocolates there are
  1. equal numbers of soft centred and hard centred chocolates, [3]
  2. fewer than 5 hard centred chocolates. [2]
A large box of chocolates contains 100 chocolates.
  1. Write down the expected number of hard centred chocolates in a large box. [2]
Edexcel S2 Specimen Q4
11 marks Standard +0.3
A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:
No. of errors012345
No. of pages376560492712
  1. Show that the mean number of errors per page in this sample of pages is 2. [2]
  2. Find the variance of the number of errors per page in this sample. [2]
  3. Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution. [1]
Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2.
  1. Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a 5\% level of significance. [6]
Edexcel S2 Specimen Q5
12 marks Standard +0.3
In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30. During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.
  1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. [5]
Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10. The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.
  1. Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
  2. State the significance level of this test giving your answer to 2 significant figures. [1]
Edexcel S2 Specimen Q6
14 marks Standard +0.3
A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25. The sheep are randomly spread throughout the field.
  1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. [1]
Calculate the probability that a randomly selected sample square contains
  1. no sheep, [1]
  2. more than 2 sheep. [4]
A sheepdog has been sent into the field to round up the sheep.
  1. Explain why the model may no longer be applicable. [1]
In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.
  1. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep. [7]
Edexcel S2 Specimen Q7
20 marks Standard +0.3
The continuous random variable \(X\) has probability density function f(\(x\)) given by $$\text{f}(x) = \begin{cases} \frac{1}{20}x^3, & 1 \leq x \leq 3 \\ 0, & \text{otherwise} \end{cases}$$
  1. Sketch f(\(x\)) for all values of \(x\). [3]
  2. Calculate E(\(X\)). [3]
  3. Show that the standard deviation of \(X\) is 0.459 to 3 decimal places. [3]
  4. Show that for \(1 \leq x \leq 3\), P(\(X \leq x\)) is given by \(\frac{1}{80}(x^4 - 1)\) and specify fully the cumulative distribution function of \(X\). [5]
  5. Find the interquartile range for the random variable \(X\). [4]
Some statisticians use the following formula to estimate the interquartile range: $$\text{interquartile range} = \frac{4}{3} \times \text{standard deviation}.$$
  1. Use this formula to estimate the interquartile range in this case, and comment. [2]
Edexcel S3 2015 June Q1
5 marks Easy -1.8
The names of the 720 members of a swimming club are listed alphabetically in the club's membership book. The chairman of the swimming club wishes to select a systematic sample of 40 names. The names are numbered from 001 to 720 and a number between 001 and \(w\) is selected at random. The corresponding name and every \(x\)th name thereafter are included in the sample.
  1. Find the value of \(w\). [1]
  2. Find the value of \(x\). [1]
  3. Write down the probability that the sample includes both the first name and the second name in the club's membership book. [1]
  4. State one advantage and one disadvantage of systematic sampling in this case. [2]
Edexcel S3 2015 June Q2
9 marks Standard +0.3
Nine dancers, Adilzhan (\(A\)), Bianca (\(B\)), Chantelle (\(C\)), Lee (\(L\)), Nikki (\(N\)), Ranjit (\(R\)), Sergei (\(S\)), Thuy (\(T\)) and Yana (\(Y\)), perform in a dancing competition. Two judges rank each dancer according to how well they perform. The table below shows the rankings of each judge starting from the dancer with the strongest performance.
Rank123456789
Judge 1\(S\)\(N\)\(B\)\(C\)\(T\)\(A\)\(Y\)\(R\)\(L\)
Judge 2\(S\)\(T\)\(N\)\(B\)\(C\)\(Y\)\(L\)\(A\)\(R\)
  1. Calculate Spearman's rank correlation coefficient for these data. [5]
  2. Stating your hypotheses clearly, test at the 1\% level of significance, whether or not the two judges are generally in agreement. [4]
Edexcel S3 2015 June Q3
11 marks Standard +0.3
The number of accidents on a particular stretch of motorway was recorded each day for 200 consecutive days. The results are summarised in the following table.
Number of accidents012345
Frequency4757463596
  1. Show that the mean number of accidents per day for these data is 1.6 [1]
A motorway supervisor believes that the number of accidents per day on this stretch of motorway can be modelled by a Poisson distribution. She uses the mean found in part (a) to calculate the expected frequencies for this model. Her results are given in the following table.
Number of accidents012345 or more
Frequency40.3864.61\(r\)27.5711.03\(s\)
  1. Find the value of \(r\) and the value of \(s\), giving your answers to 2 decimal places. [3]
  2. Stating your hypotheses clearly, use a 10\% level of significance to test the motorway supervisor's belief. Show your working clearly. [7]
Edexcel S3 2015 June Q4
11 marks Standard +0.3
A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg
  1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg [6]
Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
  1. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg [5]
Edexcel S3 2015 June Q5
12 marks Standard +0.3
A Head of Department at a large university believes that gender is independent of the grade obtained by students on a Business Foundation course. A random sample was taken of 200 male students and 160 female students who had studied the course. The results are summarised below.
MaleFemale
Distinction18.5\%27.5\%
Merit63.5\%60.0\%
Unsatisfactory18.0\%12.5\%
Stating your hypotheses clearly, test the Head of Department's belief using a 5\% level of significance. Show your working clearly. [12]
Edexcel S3 2015 June Q6
13 marks Standard +0.3
As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, \(x\) minutes, was recorded and the results are summarised by $$\sum x = 1570 \quad \text{and} \quad \sum x^2 = 49467.58$$
  1. Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning. [4]
An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, \(y\) minutes, was recorded and the results are summarised as $$\bar{y} = 30.9 \quad \text{and} \quad s_y^2 = 3.03$$
  1. Test, at the 5\% level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly. [7]
  2. Explain the relevance of the Central Limit Theorem to the test in part (b). [1]
  3. State an assumption you have made in carrying out the test in part (b). [1]
Edexcel S3 2015 June Q7
5 marks Moderate -0.3
A fair six-sided die is labelled with the numbers 1, 2, 3, 4, 5 and 6. The die is rolled 40 times and the score, \(S\), for each roll is recorded.
  1. Find the mean and the variance of \(S\). [2]
  2. Find an approximation for the probability that the mean of the 40 scores is less than 3 [3]
Edexcel S3 2015 June Q8
9 marks Standard +0.3
A factory produces steel sheets whose weights \(X\) kg, are such that \(X \sim \text{N}(\mu, \sigma^2)\) A random sample of these sheets is taken and a 95\% confidence interval for \(\mu\) is found to be (29.74, 31.86)
  1. Find, to 2 decimal places, the standard error of the mean. [3]
  2. Hence, or otherwise, find a 90\% confidence interval for \(\mu\) based on the same sample of sheets. [3]
Using four different random samples, four 90\% confidence intervals for \(\mu\) are to be found.
  1. Calculate the probability that at least 3 of these intervals will contain \(\mu\). [3]
Edexcel S3 Q1
5 marks Easy -2.0
A hotel has 160 rooms of which 20 are classified as De-luxe, 40 Premier and 100 as Standard. The manager wants to obtain information about room usage in the hotel by taking a 10\% sample of the rooms.
  1. Suggest a suitable sampling method. [1]
  2. Explain in detail how the manager should obtain the sample. [4]
Edexcel S3 Q2
9 marks Standard +0.3
A random sample of 100 classical CDs produced by a record company had a mean playing time of 70.6 minutes and a standard deviation of 9.1 minutes. An independent random sample of 80 CDs produced by a different company had a mean playing time of 67.2 minutes with a standard deviation of 8.4 minutes.
  1. Using a 1\% level of significance, test whether or not there is a difference in the mean playing times of the CDs produced by these two companies. State your hypotheses clearly. [8]
  2. State an assumption you made in carrying out the test in part (a). [1]
Edexcel S3 Q3
10 marks Standard +0.3
The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.
  1. Write down the distribution of \(M\), the mean weight, in kg, of this sample. [2]
  2. Find P(\(M < 78.5\)). [3]
The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.
  1. Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]
Edexcel S3 Q4
11 marks Standard +0.3
At the end of a season an athletics coach graded a random sample of ten athletes according to their performances throughout the season and their dedication to training. The results, expressed as percentages, are shown in the table below.
AthletePerformanceDedication
A8672
B6069
C7859
D5668
E8080
F6684
G5165
H5955
I7379
J4953
  1. Calculate the Spearman rank correlation coefficient between performance and dedication. [5]
  2. Stating clearly your hypotheses and using a 10\% level of significance, interpret your rank correlation coefficient. [5]
  3. Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data. [1]
Edexcel S3 Q5
11 marks Standard +0.3
The manager of a leisure centre collected data on the usage of the facilities in the centre by its members. A random sample from her records is summarised below.
FacilityMaleFemale
Pool4068
Jacuzzi2633
Gym5231
Making your method clear, test whether or not there is any evidence of an association between gender and use of the club facilities. State your hypotheses clearly and use a 5\% level of significance. [11]
Edexcel S3 Q6
12 marks Standard +0.3
Data were collected on the number of female puppies born in 200 litters of size 8. It was decided to test whether or not a binomial model with parameters \(n = 8\) and \(p = 0.5\) is a suitable model for these data. The following table shows the observed frequencies and the expected frequencies, to 2 decimal places, obtained in order to carry out this test.
Number of femalesObserved number of littersExpected number of litters
010.78
196.25
22721.88
346\(R\)
449\(S\)
535\(T\)
62621.88
756.25
820.78
  1. Find the values of \(R\), \(S\) and \(T\). [4]
  2. Carry out the test to determine whether or not this binomial model is a suitable one. State your hypotheses clearly and use a 5\% level of significance. [7]
An alternative test might have involved estimating \(p\) rather than assuming \(p = 0.5\).
  1. Explain how this would have affected the test. [1]
Edexcel S3 Q7
17 marks Standard +0.3
The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows. $$498 \quad 502 \quad 500 \quad 496 \quad 509 \quad 504 \quad 511 \quad 497 \quad 506 \quad 499$$
  1. Find unbiased estimates of the mean and the variance of the population from which this sample was taken. [5]
Given that the population standard deviation is 5.0 g,
  1. estimate limits, to 2 decimal places, between which 90\% of the weights of the tubs lie, [2]
  2. find a 95\% confidence interval for the mean weight of the tubs. [5]
A second random sample of 15 tubs was found to have a mean weight of 501.9 g.
  1. Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the mean weight of these tubs is greater than 500 g. [5]
Edexcel S3 2002 June Q2
9 marks Standard +0.3
A random sample of 100 classical CDs produced by a record company had a mean playing time of 70.6 minutes and a standard deviation of 9.1 minutes. An independent random sample of 120 CDs produced by a different company had a mean playing time of 67.2 minutes with a standard deviation of 8.4 minutes.
  1. Using a 1\% level of significance, test whether or not there is a difference in the mean playing times of the CDs produced by these two companies. State your hypotheses clearly. [8]
  2. State an assumption you made in carrying out the test in part (a). [1]
Edexcel S3 2002 June Q3
10 marks Standard +0.3
The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.
  1. Write down the distribution of \(\bar{M}\), the mean weight, in kg, of this sample. [2]
  2. Find P(\(\bar{M} < 78.5\)). [3]
The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.
  1. Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]
Edexcel S3 2002 June Q4
11 marks Standard +0.3
At the end of a season an athletics coach graded a random sample of ten athletes according to their performances throughout the season and their dedication to training. The results, expressed as percentages, are shown in the table below.
AthletePerformanceDedication
A8672
B6069
C7859
D5668
E8080
F6684
G3165
H5955
I7379
J4953
  1. Calculate the Spearman rank correlation coefficient between performance and dedication. [5]
  2. Stating clearly your hypotheses and using a 10\% level of significance, interpret your rank correlation coefficient. [5]
  3. Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data. [1]
Edexcel S3 2002 June Q6
12 marks Standard +0.3
Data were collected on the number of female puppies born in 200 litters of size 8. It was decided to test whether or not a binomial model with parameters \(n = 8\) and \(p = 0.5\) is a suitable model for these data. The following table shows the observed frequencies and the expected frequencies, to 2 decimal places, obtained in order to carry out this test.
Number of femalesObserved number of littersExpected number of litters
010.78
196.25
22721.88
346\(R\)
449\(S\)
535\(T\)
62621.88
756.25
820.78
  1. Find the values of \(R\), \(S\) and \(T\). [4]
  2. Carry out the test to determine whether or not this binomial model is a suitable one. State your hypotheses clearly and use a 5\% level of significance. [7]
An alternative test might have involved estimating \(p\) rather than assuming \(p = 0.5\).
  1. Explain how this would have affected the test. [1]
Edexcel S3 2002 June Q7
17 marks Standard +0.3
The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows. 498 502 500 496 509 504 511 497 506 499
  1. Find unbiased estimates of the mean and the variance of the population from which this sample was taken. [5]
Given that the population standard deviation is 5.0 g,
  1. estimate limits, to 2 decimal places, between which 90\% of the weights of the tubs lie, [2]
  2. find a 95\% confidence interval for the mean weight of the tubs. [5]
A second random sample of 15 tubs was found to have a mean weight of 501.9 g.
  1. Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the mean weight of these tubs is greater than 500 g. [5]