Questions S3 (621 questions)

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Edexcel S3 2011 June Q6
10 marks Standard +0.3
The lifetimes of batteries from manufacturer \(A\) are normally distributed with mean 20 hours and standard deviation 5 hours when used in a camera.
  1. Find the mean and standard deviation of the total lifetime of a pack of 6 batteries from manufacturer \(A\). [2]
Judy uses a camera that takes one battery at a time. She takes a pack of 6 batteries from manufacturer \(A\) to use in her camera on holiday.
  1. Find the probability that the batteries will last for more than 110 hours on her holiday. [2]
The lifetimes of batteries from manufacturer \(B\) are normally distributed with mean 35 hours and standard deviation 8 hours when used in a camera.
  1. Find the probability that the total lifetime of a pack of 6 batteries from manufacturer \(A\) is more than 4 times the lifetime of a single battery from manufacturer \(B\) when used in a camera. [6]
Edexcel S3 2011 June Q7
16 marks Standard +0.3
Roastie's Coffee is sold in packets with a stated weight of 250 g. A supermarket manager claims that the mean weight of the packets is less than the stated weight. She weighs a random sample of 90 packets from their stock and finds that their weights have a mean of 248 g and a standard deviation of 5.4 g.
  1. Using a 5\% level of significance, test whether or not the manager's claim is justified. State your hypotheses clearly. [5]
  2. Find the 98\% confidence interval for the mean weight of a packet of coffee in the supermarket's stock. [4]
  3. State, with a reason, the action you would recommend the manager to take over the weight of a packet of Roastie's Coffee. [2]
Roastie's Coffee company increase the mean weight of their packets to \(\mu\) g and reduce the standard deviation to 3 g. The manager takes a sample of size \(n\) from these new packets. She uses the sample mean \(\bar{X}\) as an estimator of \(\mu\).
  1. Find the minimum value of \(n\) such that P\((|\bar{X} - \mu| < 1) \geq 0.98\) [5]
Edexcel S3 2016 June Q1
Easy -1.8
  1. State two reasons why stratified sampling might be a more suitable sampling method than simple random sampling. (2)
  2. State two reasons why stratified sampling might be a more suitable sampling method than quota sampling. (2)
Edexcel S3 2016 June Q2
Standard +0.3
A new drug to vaccinate against influenza was given to 110 randomly chosen volunteers. The volunteers were given the drug in one of 3 different concentrations, \(A\), \(B\) and \(C\), and then were monitored to see if they caught influenza. The results are shown in the table below.
\(A\)\(B\)\(C\)
Influenza12299
No influenza152322
Test, at the 10\% level of significance, whether or not there is an association between catching influenza and the concentration of the new drug. State your hypotheses and show your working clearly. You should state your expected frequencies to 2 decimal places. (10)
Edexcel S3 2016 June Q3
Moderate -0.3
  1. Describe when you would use Spearman's rank correlation coefficient rather than the product moment correlation coefficient to measure the strength of the relationship between two variables. (1) A shop sells sunglasses and ice cream. For one week in the summer the shopkeeper ranked the daily sales of ice cream and sunglasses. The ranks are shown in the table below.
    SunMonTuesWedsThursFriSat
    Ice cream6475321
    Sunglasses6572341
  2. Calculate Spearman's rank correlation coefficient for these data. (3)
  3. Test, at the 5\% level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. State your hypotheses clearly. (4) The shopkeeper calculates the product moment correlation coefficient from his raw data and finds \(r = 0.65\)
  4. Using this new coefficient, test, at the 5\% level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. (2)
  5. Using your answers to part (c) and part (d), comment on the nature of the relationship between sales of sunglasses and sales of ice cream. (1)
Edexcel S3 2016 June Q4
Standard +0.3
The weights of eggs are normally distributed with mean 60g and standard deviation 5g Sairah chooses 2 eggs at random.
  1. Find the probability that the difference in weight of these 2 eggs is more than 2g (5) Sairah is packing eggs into cartons. The weight of an empty egg carton is normally distributed with mean 40g and standard deviation 1.5g
  2. Find the distribution of the total weight of a carton filled with 12 randomly chosen eggs. (3)
  3. Find the probability that a randomly chosen carton, filled with 12 randomly chosen eggs, weighs more than 800g (2)
Edexcel S3 2016 June Q5
Standard +0.3
A doctor claims there is a higher mean lung capacity in people who exercise regularly compared to people who do not exercise regularly. He measures the lung capacity, \(x\), of 35 people who exercise regularly and 42 people who do not exercise regularly. His results are summarised in the table below.
\(n\)\(\bar{x}\)\(s^2\)
Exercise regularly3526.312.2
Do not exercise regularly4224.810.1
  1. Test, at the 5\% level of significance, the doctor's claim. State your hypotheses clearly. (6)
  2. State any assumptions you have made in testing the doctor's claim. (2) The doctor decides to add another person who exercises regularly to his data. He measures the person's lung capacity and finds \(x = 31.7\)
  3. Find the unbiased estimate of the variance for the sample of 36 people who exercise regularly. Give your answer to 3 significant figures. (4)
Edexcel S3 2016 June Q6
Standard +0.3
An airport manager carries out a survey of families and their luggage. Each family is allowed to check in a maximum of 4 suitcases. She observes 50 families at the check-in desk and counts the total number of suitcases each family checks in. The data are summarised in the table below.
Number of suitcases01234
Frequency6251261
The manager claims that the data can be modelled by a binomial distribution with \(p = 0.3\)
  1. Test the manager's claim at the 5\% level of significance. State your hypotheses clearly. Show your working clearly and give your expected frequencies to 2 decimal places. (8) The manager also carries out a survey of the time taken by passengers to check in. She records the number of passengers that check in during each of 100 five-minute intervals. The manager makes a new claim that these data can be modelled by a Poisson distribution. She calculates the expected frequencies given in the table below.
    Number of passengers012345 or more
    Observed frequency540311860
    Expected frequency16.5329.75\(r\)\(s\)7.233.64
  2. Find the value of \(r\) and the value of \(s\) giving your answers to 2 decimal places. (3)
  3. Stating your hypotheses clearly, use a 1\% level of significance to test the manager's new claim. (6)
Edexcel S3 2016 June Q7
Standard +0.3
A restaurant states that its hamburgers contain 20\% fat. Paul claims that the mean fat content of their hamburgers is less than 20\%. Paul takes a random sample of 50 hamburgers from the restaurant and finds that they contain a mean fat content of 19.5\% with a standard deviation of 1.5\% You may assume that the fat content of hamburgers is normally distributed.
  1. Find the 90\% confidence interval for the mean fat content of hamburgers from the restaurant. (4)
  2. State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers. (2) The restaurant changes the mean fat content of their hamburgers to \(\mu\)\% and adjusts the standard deviation to 2\%. Paul takes a sample of size \(n\) from this new batch of hamburgers. He uses the sample mean \(\bar{X}\) as an estimator of \(\mu\).
  3. Find the minimum value of \(n\) such that \(\mathrm{P}(|\bar{X} - \mu| < 0.5) \geq 0.9\) (5)
Edexcel S3 Q1
5 marks Moderate -0.8
A random sample \(X_1, X_2, \ldots, X_{10}\) is taken from a normal population with mean 100 and standard deviation 14.
  1. Write down the distribution of \(\overline{X}\), the mean of this sample. [2]
  2. Find \(\text{Pr}(|\overline{X} - 100| > 5)\). [3]
Edexcel S3 Q2
6 marks Standard +0.3
A random sample of the invoices, for books purchased by the customers of a large bookshop, was classified by book cover (hardback, paperback) and type of book (novel, textbook, general interest). As part of the analysis of these invoices, an approximate \(\chi^2\) statistic was calculated and found to be 11.09. Assuming that there was no need to amalgamate any of the classifications, carry out an appropriate test to determine whether or not there was any association between book cover and type of book. State your hypotheses clearly and use a 5% level of significance. [6]
Edexcel S3 Q3
11 marks Standard +0.3
As part of a research project into the role played by cholesterol in the development of heart disease a random sample of 100 patients was put on a special fish-based diet. A different random sample of 80 patients was kept on a standard high-protein low-fat diet. After several weeks their blood cholesterol was measured and the results summarised in the table below.
GroupSample sizeMean drop in cholesterol (mg/dl)Standard deviation
Special diet1007522
Standard diet806431
  1. Stating your hypotheses clearly and using a 5% level of significance, test whether or not the special diet is more effective in reducing blood cholesterol levels than the standard diet. [9]
  2. Explain briefly any assumptions you made in order to carry out this test. [2]
Edexcel S3 Q4
13 marks Standard +0.8
Breakdowns on a certain stretch of motorway were recorded each day for 80 consecutive days. The results are summarised in the table below.
Number of breakdowns012\(>2\)
Frequency3832100
It is suggested that the number of breakdowns per day can be modelled by a Poisson distribution. Using a 5% level of significance, test whether or not the Poisson distribution is a suitable model for these data. State your hypotheses clearly. [13]
Edexcel S3 Q5
12 marks Moderate -0.3
The random variable \(R\) is defined as \(R = X + 4Y\) where \(X \sim \text{N}(8, 2^2)\), \(Y \sim \text{N}(14, 3^2)\) and \(X\) and \(Y\) are independent. Find
  1. E\((R)\), [2]
  2. Var\((R)\), [3]
  3. P\((R < 41)\) [3]
The random variables \(Y_1\), \(Y_2\) and \(Y_3\) are independent and each has the same distribution as \(Y\). The random variable \(S\) is defined as $$S = \sum_{i=1}^{3} Y_i - \frac{1}{2}X.$$
  1. Find Var\((S)\). [4]
Edexcel S3 Q6
12 marks Moderate -0.8
As part of her statistics project, Deepa decided to estimate the amount of time A-level students at her school spend on private study each week. She took a random sample of students from those studying Arts subjects, Science subjects and a mixture of Arts and Science subjects. Each student kept a record of the time they spent on private study during the third week of term.
  1. Write down the name of the sampling method used by Deepa. [1]
  2. Give a reason for using this method and give one advantage this method has over simple random sampling. [2]
The results Deepa obtained are summarised in the table below.
Type of studentSample sizeMean number of hours
Arts1212.6
Science1214.1
Mixture810.2
  1. Show that an estimate of the mean time spent on private study by A level students at Deepa's school, based on these 32 students is 12.56, to 2 decimal places. [3]
The standard deviation of the time spent on private study by students at the school was 2.48 hours.
  1. Assuming that the number of hours spent on private study is normally distributed, find a 95% confidence interval for the mean time spent on private study by A level students at Deepa's school. [4]
A member of staff at the school suggested that A level students should spend on average 12 hours each week on private study.
  1. Comment on this suggestion in the light of your interval. [2]
Edexcel S3 Q7
16 marks Standard +0.3
For one of the activities at a gymnastics competition, 8 gymnasts were awarded marks out of 10 for each of artistic performance and technical ability. The results were as follows.
Gymnast\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Technical ability8.58.69.57.56.89.19.49.2
Artistic performance6.27.58.26.76.07.28.09.1
The value of the product moment correlation coefficient for these data is 0.774.
  1. Stating your hypotheses clearly and using a 1% level of significance, interpret this value. [5]
  2. Calculate the value of the rank correlation coefficient for these data. [6]
  3. Stating your hypotheses clearly and using a 1% level of significance, interpret this coefficient. [3]
  4. Explain why the rank correlation coefficient might be the better one to use with these data. [2]
Edexcel S3 Specimen Q1
4 marks Easy -1.8
The 240 members of a bowling club are listed alphabetically in the club's membership book. The committee wishes to select a sample of 30 members to fill in a questionnaire about the facilities the club offers.
  1. Explain how the committee could use a table of random numbers to take a systematic sample. [3]
  2. Give one advantage of this method over taking a simple random sample. [1]
Edexcel S3 Specimen Q2
5 marks Moderate -0.8
The weights of pears, \(P\) grams, are normally distributed with a mean of 110 and a standard deviation of 8. Geoff buys a bag of 16 pears.
  1. Write down the distribution of \(\overline{P}\), the mean weight of the 16 pears. [2]
  2. Find P\((110 < \overline{P} < 113)\). [3]
Edexcel S3 Specimen Q3
10 marks Standard +0.3
The three tasks most frequently carried out in a garage are \(A\), \(B\) and \(C\). For each of the tasks the times, in minutes, taken by the garage mechanics are assumed to be normally distributed with means and standard deviations given in the following table.
TaskMeanStandard deviation
\(A\)22538
\(B\)16523
\(C\)18527
Assuming that the times for the three tasks are independent, calculate the probability that
  1. the total time taken by a single randomly chosen mechanic to carry out all three tasks lies between 533 and 655 minutes, [5]
  2. a randomly chosen mechanic takes longer to carry out task \(B\) than task \(C\). [5]
Edexcel S3 Specimen Q4
11 marks Standard +0.3
At the end of a season a league of eight ice hockey clubs produced the following table showing the position of each club in the league and the average attendances (in hundreds) at home matches.
Club\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Position12345678
Average3738192734262232
  1. Calculate the Spearman rank correlation coefficient between position in the league and average home attendance. [5]
  2. Stating clearly your hypotheses and using a 5\% two-tailed test, interpret your rank correlation coefficient. [4]
Many sets of data include tied ranks.
  1. Explain briefly how tied ranks can be dealt with. [2]
Edexcel S3 Specimen Q5
11 marks Moderate -0.3
For a six-sided die it is assumed that each of the sides has an equal chance of landing uppermost when the die is rolled.
  1. Write down the probability function for the random variable \(X\), the number showing on the uppermost side after the die has been rolled. [2]
  2. State the name of the distribution. [1]
A student wishing to check the above assumption rolled the die 300 times and for the sides 1 to 6, obtained the frequencies 41, 49, 52, 58, 37 and 63 respectively.
  1. Analyse these data and comment on whether or not the assumption is valid for this die. Use a 5\% level of significance and state your hypotheses clearly. [8]
Edexcel S3 Specimen Q6
11 marks Standard +0.3
A sociologist was studying the smoking habits of adults. A random sample of 300 adult smokers from a low income group and an independent random sample of 400 adult smokers from a high income group were asked what their weekly expenditure on tobacco was. The results are summarised below.
\(N\)means.d.
Low income group300£6.40£6.69
High income group400£7.42£8.13
  1. Using a 5\% significance level, test whether or not the two groups differ in the mean amounts spent on tobacco. [9]
  2. Explain briefly the importance of the central limit theorem in this example. [2]
Edexcel S3 Specimen Q7
11 marks Moderate -0.3
A survey in a college was commissioned to investigate whether or not there was any association between gender and passing a driving test. A group of 50 male and 50 female students were asked whether they passed or failed their driving test at the first attempt. All the students asked had taken the test. The results were as follows.
PassFail
Male2327
Female3218
Stating your hypotheses clearly test, at the 10\% level, whether or not there is any evidence of an association between gender and passing a driving test at the first attempt. [11]
Edexcel S3 Specimen Q8
12 marks Moderate -0.3
Observations have been made over many years of \(T\), the noon temperature in °C, on 21st March at Sunnymere. The records for a random sample of 12 years are given below. 5.2, 3.1, 10.6, 12.4, 4.6, 8.7, 2.5, 15.3, \(-1.5\), 1.8, 13.2, 9.3.
  1. Find unbiased estimates of the mean and variance of \(T\). [5]
Over the years, the standard deviation of \(T\) has been found to be 5.1.
  1. Assuming a normal distribution find a 90\% confidence interval for the mean of \(T\). [5]
A meteorologist claims that the mean temperature at noon in Sunnymere on 21st March is 4 °C.
  1. Use your interval to comment on the meteorologist's claim. [2]
AQA S3 2016 June Q1
8 marks Standard +0.3
In advance of a referendum on independence, the regional assembly of an eastern province of a particular country carried out an opinion poll to assess the strength of the 'Yes' vote. Of the 480 men polled, 264 indicated that they intended to vote 'Yes', and of the 500 women polled, 220 indicated that they intended to vote 'Yes'.
  1. Construct an approximate 95\% confidence interval for the difference between the proportion of men who intend to vote 'Yes' and the proportion of women who intend to vote 'Yes'. [6 marks]
  2. Comment on a claim that, in the forthcoming referendum, the percentage of men voting 'Yes' will exceed the percentage of women voting 'Yes' by at least 2.5 per cent. Justify your answer. [2 marks]