Questions S3 (621 questions)

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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]
Edexcel S3 2009 June Q3
11 marks Standard +0.3
A doctor is interested in the relationship between a person's Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual's BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.
Individual\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
BMI17.421.418.924.419.420.122.618.425.828.1
Finishing position35196410278
  1. Calculate Spearman's rank correlation coefficient for these data. [5]
  2. Stating your hypotheses clearly and using a one tailed test with a 5\% 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 2009 June Q4
5 marks Moderate -0.5
A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57. [5]
Edexcel S3 2009 June Q5
12 marks Standard +0.3
The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below.
Number of goalsFrequency
040
133
214
38
45
Table 1
  1. Calculate the mean number of goals scored per game. [2]
The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2.
Number of goalsExpected Frequency
034.994
1\(r\)
2\(s\)
36.752
\(\geqslant 4\)2.221
Table 2
  1. Find the value of \(r\) and the value of \(s\) giving your answers to 3 decimal places. [3]
  2. Stating your hypotheses clearly, use a 5\% level of significance to test the manager's claim. [7]
Edexcel S3 2009 June Q6
10 marks Standard +0.3
The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm. The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm.
  1. Test, using a 5\% level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly. [8]
  2. State two assumptions you made in carrying out the test in part (a). [2]
Edexcel S3 2009 June Q7
11 marks Standard +0.3
A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below. 120.3 \quad 120.1 \quad 120.4 \quad 120.2 \quad 119.9
  1. Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company. [5]
The lengths of climbing rope are known to have a standard deviation of 0.2 m. The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of \(n\), lies within 0.05 m of its true value.
  1. Find the minimum sample size required. [6]
Edexcel S3 2009 June Q8
11 marks Standard +0.3
The random variable \(A\) is defined as $$A = 4X - 3Y$$ where \(X \sim \text{N}(30, 3^2)\), \(Y \sim \text{N}(20, 2^2)\) and \(X\) and \(Y\) are independent. Find
  1. E(\(A\)), [2]
  2. Var(\(A\)). [3]
The random variables \(Y_1\), \(Y_2\), \(Y_3\) and \(Y_4\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as $$B = \sum_{i=1}^{4} Y_i$$
  1. Find P(\(B > A\)). [6]
Edexcel S3 2011 June Q1
3 marks Moderate -0.5
Explain what you understand by the Central Limit Theorem. [3]
Edexcel S3 2011 June Q2
10 marks Standard +0.3
A county councillor is investigating the level of hardship, \(h\), of a town and the number of calls per 100 people to the emergency services, \(c\). He collects data for 7 randomly selected towns in the county. The results are shown in the table below.
Town\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
\(h\)14201618371924
\(c\)52454342618255
  1. Calculate the Spearman's rank correlation coefficient between \(h\) and \(c\). [6]
  2. Test, at the 5\% level of significance, the councillor's claim. State your hypotheses clearly. [4]
After collecting the data, the councillor thinks there is no correlation between hardship and the number of calls to the emergency services.
Edexcel S3 2011 June Q3
10 marks Standard +0.3
A factory manufactures batches of an electronic component. Each component is manufactured in one of three shifts. A component may have one of two types of defect, \(D_1\) or \(D_2\), at the end of the manufacturing process. A production manager believes that the type of defect is dependent upon the shift that manufactured the component. He examines 200 randomly selected defective components and classifies them by defect type and shift. The results are shown in the table below.
\(D_1\)\(D_2\)
First shift4518
Second shift5520
Third shift5012
Stating your hypotheses, test, at the 10\% level of significance, whether or not there is evidence to support the manager's belief. Show your working clearly. [10]
Edexcel S3 2011 June Q4
13 marks Standard +0.3
A shop manager wants to find out if customers spend more money when music is playing in the shop. The amount of money spent by a customer in the shop is £\(x\). A random sample of 80 customers, who were shopping without music playing, and an independent random sample of 60 customers, who were shopping with music playing, were surveyed. The results of both samples are summarised in the table below.
\(\sum x\)\(\sum x^2\)Unbiased estimate of meanUnbiased estimate of variance
Customers shopping without music5320392000\(\bar{x}\)\(s^2\)
Customers shopping with music414031200069.0446.44
  1. Find the values of \(\bar{x}\) and \(s^2\). [5]
  2. Test, at the 5\% level of significance, whether or not the mean money spent is greater when music is playing in the shop. State your hypotheses clearly. [8]
Edexcel S3 2011 June Q5
13 marks Standard +0.3
The number of hurricanes per year in a particular region was recorded over 80 years. The results are summarised in Table 1 below.
No of hurricanes, \(h\)01234567
Frequency0251720121212
Table 1
  1. Write down two assumptions that will support modelling the number of hurricanes per year by a Poisson distribution. [2]
  2. Show that the mean number of hurricanes per year from Table 1 is 4.4875 [2]
  3. Use the answer in part (b) to calculate the expected frequencies \(r\) and \(s\) given in Table 2 below to 2 decimal places. [3]
\(h\)01234567 or more
Expected frequency0.904.04\(r\)13.55\(s\)13.6510.2113.39
Table 2
  1. Test, at the 5\% level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly. [6]