Questions — Edexcel S3 (332 questions)

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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]
Edexcel S3 2005 June Q1
4 marks Easy -1.8
  1. State two reasons why stratified sampling might be chosen as a method of sampling when carrying out a statistical survey. [2]
  2. State one advantage and one disadvantage of quota sampling. [2]
(Total 4 marks)
Edexcel S3 2005 June Q2
Moderate -0.3
A sample of size 5 is taken from a population that is normally distributed with mean 10 and standard deviation 3. Find the probability that the sample mean lies between 7 and 10. (Total 6 marks)
Edexcel S3 2005 June Q3
Standard +0.3
A researcher carried out a survey of three treatments for a fruit tree disease. The contingency table below shows the results of a survey of a random sample of 60 diseased trees.
No actionRemove diseased branchesSpray with chemicals
Tree died within 1 year1056
Tree survived for 1–4 years597
Tree survived beyond 4 years567
Test, at the 5\% level of significance, whether or not there is any association between the treatment of the trees and their survival. State your hypotheses and conclusion clearly. (Total 11 marks)
Edexcel S3 2005 June Q4
13 marks Standard +0.3
Over a period of time, researchers took 10 blood samples from one patient with a blood disease. For each sample, they measured the levels of serum magnesium, \(s\) mg/dl, in the blood and the corresponding level of the disease protein, \(d\) mg/dl. The results are shown in the table.
\(s\)1.21.93.23.92.54.55.74.01.15.9
\(d\)3.87.011.012.09.012.013.512.22.013.9
[Use \(\sum s^2 = 141.51\), \(\sum d^2 = 1081.74\) and \(\sum sd = 386.32\)]
  1. Draw a scatter diagram to represent these data. [3]
  2. State what is measured by the product moment correlation coefficient. [1]
  3. Calculate \(S_{ss}\), \(S_{dd}\) and \(S_{sd}\). [3]
  4. Calculate the value of the product moment correlation coefficient \(r\) between \(s\) and \(d\). [2]
  5. Stating your hypotheses clearly, test, at the 1\% significance level, whether or not the correlation coefficient is greater than zero. [3]
  6. With reference to your scatter diagram, comment on your result in part (e). [1]
(Total 13 marks)
Edexcel S3 2005 June Q5
Standard +0.3
The number of times per day a computer fails and has to be restarted is recorded for 200 days. The results are summarised in the table.
Number of restartsFrequency
099
165
222
312
42
Test whether or not a Poisson model is suitable to represent the number of restarts per day. Use a 5\% level of significance and state your hypothesis clearly. (Total 12 marks)
Edexcel S3 2005 June Q6
10 marks Standard +0.3
A computer company repairs large numbers of PCs and wants to estimate the mean time to repair a particular fault. Five repairs are chosen at random from the company's records and the times taken, in seconds, are 205 \quad 310 \quad 405 \quad 195 \quad 320.
  1. Calculate unbiased estimates of the mean and the variance of the population of repair times from which this sample has been taken. [4]
It is known from previous results that the standard deviation of the repair time for this fault is 100 seconds. The company manager wants to ensure that there is a probability of at least 0.95 that the estimate of the population mean lies within 20 seconds of its true value.
  1. Find the minimum sample size required. [6]
(Total 10 marks)
Edexcel S3 2005 June Q7
19 marks Standard +0.3
A manufacturer produces two flavours of soft drink, cola and lemonade. The weights, \(C\) and \(L\), in grams, of randomly selected cola and lemonade cans are such that \(C \sim \text{N}(350, 8)\) and \(L \sim \text{N}(345, 17)\).
  1. Find the probability that the weights of two randomly selected cans of cola will differ by more than 6 g. [6]
One can of each flavour is selected at random.
  1. Find the probability that the can of cola weighs more than the can of lemonade. [6]
Cans are delivered to shops in boxes of 24 cans. The weights of empty boxes are normally distributed with mean 100 g and standard deviation 2 g.
  1. Find the probability that a full box of cola cans weighs between 8.51 kg and 8.52 kg. [6]
  2. State an assumption you made in your calculation in part (c). [1]
(Total 19 marks)
Edexcel S3 2006 June Q1
4 marks Easy -2.5
Describe one advantage and one disadvantage of
  1. quota sampling, [2]
  2. simple random sampling. [2]
Edexcel S3 2006 June Q2
6 marks Moderate -0.8
A report on the health and nutrition of a population stated that the mean height of three-year old children is 90 cm and the standard deviation is 5 cm. A sample of 100 three-year old children was chosen from the population.
  1. Write down the approximate distribution of the sample mean height. Give a reason for your answer. [3]
  2. Hence find the probability that the sample mean height is at least 91 cm. [3]
Edexcel S3 2006 June Q3
9 marks Standard +0.3
A biologist investigated whether or not the diet of chickens influenced the amount of cholesterol in their eggs. The cholesterol content of 70 eggs selected at random from chickens fed diet A had a mean value of 198 mg and a standard deviation of 47 mg. A random sample of 90 eggs from chickens fed diet B had a mean cholesterol content of 201 mg and a standard deviation of 23 mg.
  1. Stating your hypotheses clearly and using a 5\% level of significance, test whether or not there is a difference between the mean cholesterol content of eggs laid by chickens fed on these two diets. [7]
  2. State, in the context of this question, an assumption you have made in carrying out the test in part (a). [2]
Edexcel S3 2006 June Q4
9 marks Standard +0.3
The table below shows the price of an ice cream and the distance of the shop where it was purchased from a particular tourist attraction.
ShopDistance from tourist attraction (m)Price (£)
A501.75
B1751.20
C2702.00
D3751.05
E4250.95
F5801.25
G7100.80
H7900.75
I8901.00
J9800.85
  1. Find, to 3 decimal places, the Spearman rank correlation coefficient between the distance of the shop from the tourist attraction and the price of an ice cream. [5]
  2. Stating your hypotheses clearly and using a 5\% one-tailed test, interpret your rank correlation coefficient. [4]
Edexcel S3 2006 June Q5
9 marks Standard +0.3
The workers in a large office block use a lift that can carry a maximum load of 1090 kg. The weights of the male workers are normally distributed with mean 78.5 kg and standard deviation 12.6 kg. The weights of the female workers are normally distributed with mean 62.0 kg and standard deviation 9.8 kg. Random samples of 7 males and 8 females can enter the lift.
  1. Find the mean and variance of the total weight of the 15 people that enter the lift. [4]
  2. Comment on any relationship you have assumed in part (a) between the two samples. [1]
  3. Find the probability that the maximum load of the lift will be exceeded by the total weight of the 15 people. [4]
Edexcel S3 2006 June Q6
11 marks Standard +0.3
A research worker studying colour preference and the age of a random sample of 50 children obtained the results shown below.
Age in yearsRedBlueTotals
412618
810717
126915
Totals282250
Using a 5\% significance level, carry out a test to decide whether or not there is an association between age and colour preference. State your hypotheses clearly. [11]
Edexcel S3 2006 June Q7
14 marks Moderate -0.3
A machine produces metal containers. The weights of the containers are normally distributed. A random sample of 10 containers from the production line was weighed, to the nearest 0.1 kg, and gave the following results 49.7, 50.3, 51.0, 49.5, 49.9 50.1, 50.2, 50.0, 49.6, 49.7.
  1. Find unbiased estimates of the mean and variance of the weights of the population of metal containers. [5]
The machine is set to produce metal containers whose weights have a population standard deviation of 0.5 kg.
  1. Estimate the limits between which 95\% of the weights of metal containers lie. [4]
  2. Determine the 99\% confidence interval for the mean weight of metal containers. [5]
Edexcel S3 2006 June Q8
13 marks Standard +0.3
Five coins were tossed 100 times and the number of heads recorded. The results are shown in the table below.
Number of heads012345
Frequency6182934103
  1. Suggest a suitable distribution to model the number of heads when five unbiased coins are tossed. [2]
  2. Test, at the 10\% level of significance, whether or not the five coins are unbiased. State your hypotheses clearly. [11]
Edexcel S3 2009 June Q1
6 marks Easy -1.8
A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.
  1. Explain in detail how the researcher should obtain such a sample. [2]
  2. Give one advantage and one disadvantage of
    1. quota sampling,
    2. systematic sampling.
    [4]
Edexcel S3 2009 June Q2
9 marks Moderate -0.3
The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm. The heights of the orchids are normally distributed. Given that the population standard deviation is 0.5 cm,
  1. estimate limits between which 95\% of the heights of the orchids lie, [3]
  2. find a 98\% confidence interval for the mean height of the orchids. [4]
A grower claims that the mean height of this type of orchid is 19.5 cm.
  1. Comment on the grower's claim. Give a reason for your answer. [2]