Questions S3 (597 questions)

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Edexcel S3 2011 June Q7
  1. 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.
    2. Find the \(98 \%\) confidence interval for the mean weight of a packet of coffee in the supermarket's stock.
    3. State, with a reason, the action you would recommend the manager to take over the weight of a packet of Roastie's Coffee.
    Roastie's Coffee company increase the mean weight of their packets to \(\mu \mathrm { g }\) 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 \(\mu\).
  2. Find the minimum value of \(n\) such that \(\mathrm { P } ( | \bar { X } - \mu ! | < 1 ) \geqslant 0.98\)
Edexcel S3 2012 June Q1
  1. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.
Candidate\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Manager \(X\)6256875465151210
Manager \(Y\)5447715049253044
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Test, at the \(5 \%\) level of significance, whether there is agreement between the rankings awarded by each manager. State your hypotheses clearly. Manager \(Y\) later discovered he had miscopied his score for candidate \(D\) and it should be 54 .
  3. Without carrying out any further calculations, explain how you would calculate Spearman's rank correlation in this case.
Edexcel S3 2012 June Q2
2. A lake contains 3 species of fish. There are estimated to be 1400 trout, 600 bass and 450 pike in the lake. A survey of the health of the fish in the lake is carried out and a sample of 30 fish is chosen.
  1. Give a reason why stratified random sampling cannot be used.
  2. State an appropriate sampling method for the survey.
  3. Give one advantage and one disadvantage of this sampling method.
  4. Explain how this sampling method could be used to select the sample of 30 fish. You must show your working.
Edexcel S3 2012 June Q3
3. (a) Explain what you understand by the Central Limit Theorem. A garage services hire cars on behalf of a hire company. The garage knows that the lifetime of the brake pads has a standard deviation of 5000 miles. The garage records the lifetimes, \(x\) miles, of the brake pads it has replaced. The garage takes a random sample of 100 brake pads and finds that \(\sum x = 1740000\)
(b) Find a 95\% confidence interval for the mean lifetime of a brake pad.
(c) Explain the relevance of the Central Limit Theorem in part (b). Brake pads are made to be changed every 20000 miles on average.
The hire car company complain that the garage is changing the brake pads too soon.
(d) Comment on the hire company's complaint. Give a reason for your answer.
Edexcel S3 2012 June Q4
  1. Two breeds of chicken are surveyed to measure their egg yield. The results are shown in the table below.
\backslashbox{Breed}{Egg yield}LowMediumHigh
Leghorn225226
Cornish14324
Showing each stage of your working clearly, test, at the \(5 \%\) significance level, whether or not there is an association between egg yield and breed of chicken. State your hypotheses clearly.
Edexcel S3 2012 June Q5
5. Mr Alan and Ms Burns are two Mathematics teachers teaching mixed ability groups of students in a large college. At the end of the college year all students took the same examination. A random sample of 29 of Mr Alan's students and a random sample of 26 of Ms Burns' students are chosen. The results are summarised in the table below.
Sample Size, \(n\)Mean, \(\bar { x }\)Standard Deviation, \(s\)
Mr Alan298010
Ms Burns267415
  1. Stating your hypotheses clearly, test, at the \(10 \%\) level of significance whether there is evidence that there is a difference in the mean scores of their students. Ms Burns thinks the comparison was unfair as the examination was set by Mr Alan. She looks up a different set of examination results for these students and, although Mr Alan's sample has a higher mean, she calculates the test statistic for this new set of results to be 1.6 However, Mr Alan now claims that the mean marks of his students are higher than the mean marks of Ms Burns' students.
  2. Test Mr Alan's claim, stating the hypotheses and critical values you would use. Use a \(10 \%\) level of significance.
Edexcel S3 2012 June Q6
6. A total of 100 random samples of 6 items are selected from a production line in a factory and the number of defective items in each sample is recorded. The results are summarised in the table below.
Number of
defective
items
0123456
Number of
samples
616202317108
  1. Show that the mean number of defective items per sample is 2.91 A factory manager suggests that the data can be modelled by a binomial distribution with \(n = 6\). He uses the mean from the sample above and calculates expected frequencies as shown in the table below.
    Number of
    defective
    items
    		0123456
    Expected
    frequency
    		1.8710.5424.82\(a\)22.018.29\(b\)
  2. Calculate the value of \(a\) and the value of \(b\) giving your answers to 2 decimal places.
  3. Test, at the \(5 \%\) level, whether or not the binomial distribution is a suitable model for the number of defective items in samples of 6 items. State your hypotheses clearly.
Edexcel S3 2012 June Q7
7. The heights, in cm, of the male employees in a large company follow a normal distribution with mean 177 and standard deviation 5 The heights, in cm, of the female employees follow a normal distribution with mean 163 and standard deviation 4 A male employee and a female employee are chosen at random.
  1. Find the probability that the male employee is taller than the female employee. Six male employees and four female employees are chosen at random.
  2. Find the probability that their total height is less than 17 m .
Edexcel S3 2013 June Q1
  1. A gym club has 400 members of which 300 are males.
Explain clearly how a stratified sample of size 60 could be taken.
Edexcel S3 2013 June Q2
2. A random sample of size \(n\) is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than \(5 \%\).
Edexcel S3 2013 June Q3
3. The table below shows the population and the number of council employees for different towns and villages.
Town or villagePopulationNumber of council employees
A21110
B3562
C104712
D246321
E489216
F647925
G657167
H657345
I984548
\(J\)1478434
  1. Find, to 3 decimal places, Spearman's rank correlation coefficient between the population and the number of council employees.
  2. Use your value of Spearman's rank correlation coefficient to test for evidence of a positive correlation between the population and the number of council employees. Use a \(2.5 \%\) significance level. State your hypotheses clearly. It is suggested that a product moment correlation coefficient would be a more suitable calculation in this case. The product moment correlation coefficient for these data is 0.627 to 3 decimal places.
  3. Use the value of the product moment correlation coefficient to test for evidence of a positive correlation between the population and the number of council employees. Use a \(2.5 \%\) significance level.
  4. Interpret and comment on your results from part(b) and part(c).
Edexcel S3 2013 June Q4
  1. John thinks that a person's eye colour is related to their hair colour. He takes a random sample of 600 people and records their eye and hair colours. The results are shown in Table 1.
\begin{table}[h]
\multirow{2}{*}{}Hair colour
BlackBrownRedBlondeTotal
\multirow{5}{*}{Eye colour}Brown451251558243
Blue34901058192
Hazel20381626100
Green62972365
Total10528248165600
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} John carries out a \(\chi ^ { 2 }\) test in order to test whether eye colour and hair colour are related. He calculates the expected frequencies shown in Table 2. \begin{table}[h]
\multirow{2}{*}{}Hair colour
BlackBrownRedBlonde
\multirow{4}{*}{Eye colour}Brown42.5114.219.466.8
Blue33.690.215.452.8
Hazel17.547827.5
Green11.430.65.217.9
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. Show how the value 47 in Table 2 has been calculated.
  2. Write down the number of degrees of freedom John should use in this \(\chi ^ { 2 }\) test. Given that the value of the \(\chi ^ { 2 }\) statistic is 20.6 , to 3 significant figures,
  3. find the smallest value of \(\alpha\) for which the null hypothesis will be rejected at the \(\alpha \%\) level of significance.
  4. Use the data from Table 1 to test at the \(5 \%\) level of significance whether or not the proportions of people in the population with black, brown, red and blonde hair are in the ratio 2:6:1:3 State your hypotheses clearly.
Edexcel S3 2013 June Q5
  1. A manufacturer produces circular discs with diameter \(D \mathrm {~mm}\), such that \(D \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). A random sample of discs is taken and, using tables of the normal distribution, a \(90 \%\) confidence interval for \(\mu\) is found to be
    (118.8, 121.2)
    1. Find a 98\% confidence interval for \(\mu\).
    2. Hence write down a 98\% confidence interval for the circumference of the discs.
    Using three different random samples, three \(98 \%\) confidence intervals for \(\mu\) are to be found.
  2. Calculate the probability that all the intervals will contain \(\mu\).
Edexcel S3 2013 June Q6
6. The continuous random variable \(X\) is uniformly distributed over the interval $$[ a - 1 , a + 5 ]$$ where \(a\) is a constant.
Fifty observations of \(X\) are taken, giving a sample mean of 17.2
  1. Use the Central Limit Theorem to find an approximate distribution for \(\bar { X }\).
  2. Hence find a 95\% confidence interval for \(a\).
Edexcel S3 2013 June Q7
7. A farmer monitored the amount of lead in soil in a field next to a factory. He took 100 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be \(67 \mathrm { mg } / \mathrm { kg }\) with standard deviation \(25 \mathrm { mg } / \mathrm { kg }\).
After the factory closed, the farmer took 150 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be \(60 \mathrm { mg } / \mathrm { kg }\) with standard deviation \(10 \mathrm { mg } / \mathrm { kg }\).
  1. Test at the \(5 \%\) level of significance whether or not the mean weight of lead in the soil decreased after the factory closed. State your hypotheses clearly.
  2. Explain the significance of the Central Limit Theorem to the test in part(a).
  3. State an assumption you have made to carry out this test.
Edexcel S3 2013 June Q8
8. A farmer supplies both duck eggs and chicken eggs. The weights of duck eggs, \(D\) grams, and chicken eggs, \(C\) grams, are such that $$D \sim \mathrm {~N} \left( 54,1.2 ^ { 2 } \right) \text { and } C \sim \mathrm {~N} \left( 44,0.8 ^ { 2 } \right)$$
  1. Find the probability that the weights of 2 randomly selected duck eggs will differ by more than 3 g .
  2. Find the probability that the weight of a randomly selected chicken egg is less than \(\frac { 4 } { 5 }\) of the weight of a randomly selected duck egg. Eggs are packed in boxes which contain either 6 randomly selected duck eggs or 6 randomly selected chicken eggs. The weight of an empty box has distribution \(\mathrm { N } \left( 28 , \sqrt { 5 } ^ { 2 } \right)\).
  3. Find the probability that a full box of duck eggs weighs at least 50 g more than a full box of chicken eggs.
Edexcel S3 2013 June Q1
  1. A doctor takes a random sample of 100 patients and measures their intake of saturated fats in their food and the level of cholesterol in their blood. The results are summarised in the table below.
\backslashbox{Intake of saturated fats}{Cholesterol level}HighLow
High128
Low2654
Using a \(5 \%\) level of significance, test whether or not there is an association between cholesterol level and intake of saturated fats. State your hypotheses and show your working clearly.
Edexcel S3 2013 June Q2
2. The table below shows the number of students per member of staff and the student satisfaction scores for 7 universities.
University\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Number of
students per
member of staff
14.213.113.311.710.515.910.8
Student
satisfaction
score
4.14.23.84.03.94.33.7
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence of a correlation between the number of students per member of staff and the student satisfaction score.
Edexcel S3 2013 June Q3
3. A college manager wants to survey students' opinions of enrichment activities. She decides to survey the students on the courses summarised in the table below.
CourseNumber of students enrolled
Leisure and Sport420
Information Technology337
Health and Social Care200
Media Studies43
Each student takes only one course.
The manager has access to the college's information system that holds full details of each of the enrolled students including name, address, telephone number and their course of study. She wants to compare the opinions of students on each course and has a generous budget to pay for the cost of the survey.
  1. Give one advantage and one disadvantage of carrying out this survey using
    1. quota sampling,
    2. stratified sampling. The manager decides to take a stratified sample of 100 students.
  2. Calculate the number of students to be sampled from each course.
  3. Describe how to choose students for the stratified sample.
Edexcel S3 2013 June Q4
4. Customers at a post office are timed to see how long they wait until being served at the counter. A random sample of 50 customers is chosen and their waiting times, \(x\) minutes, are summarised in Table 1. \begin{table}[h]
Waiting time in minutes \(( x )\)Frequency
\(0 - 3\)8
\(3 - 5\)12
\(5 - 6\)13
\(6 - 8\)9
\(8 - 12\)8
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Show that an estimate of \(\bar { x } = 5.49\) and an estimate of \(s _ { x } ^ { 2 } = 6.88\) The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
    Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]
    Waiting Time\(x < 3\)\(3 - 5\)\(5 - 6\)\(6 - 8\)\(x > 8\)
    Expected Frequency8.5612.737.56\(a\)\(b\)
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  2. Find the value of \(a\) and the value of \(b\).
  3. Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.
Edexcel S3 2013 June Q5
  1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
One large and 3 small bottles of Blumen are chosen at random.
  1. Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
  2. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
Edexcel S3 2013 June Q6
6. Fruit-n-Veg4U Market Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is \(0.5 \mathrm {~kg} ^ { 2 }\) and the variance of the yield for the new variety of plant is \(0.75 \mathrm {~kg} ^ { 2 }\). A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .
  1. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
  2. Explain the relevance of the Central Limit Theorem to the test in part (a).
Edexcel S3 2013 June Q7
  1. Lambs are born in a shed on Mill Farm. The birth weights, \(x \mathrm {~kg}\), of a random sample of 8 newborn lambs are given below.
$$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$
  1. Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
  2. Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of Mill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
  3. Calculate a \(95 \%\) confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
Edexcel S3 2014 June Q1
  1. A journalist is investigating factors which influence people when they buy a new car. One possible factor is fuel efficiency. The journalist randomly selects 8 car models. Each model's annual sales and fuel efficiency, in km/litre, are shown in the table below.
Car model\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Annual sales18005400181007100930048001220010700
Fuel efficiency5.218.614.813.218.311.916.517.7
  1. Calculate Spearman's rank correlation coefficient for these data. The journalist believes that car models with higher fuel efficiency will achieve higher sales.
  2. Stating your hypotheses clearly, test whether or not the data support the journalist's belief. Use a \(5 \%\) level of significance.
  3. State the assumption necessary for a product moment correlation coefficient to be valid in this case.
  4. The mean and median fuel efficiencies of the car models in the random sample are 14.5 km /litre and 15.65 km /litre respectively. Considering these statistics, as well as the distribution of the fuel efficiency data, state whether or not the data suggest that the assumption in part (c) might be true in this case. Give a reason for your answer. (No further calculations are required.)
Edexcel S3 2014 June Q2
  1. A survey asked a random sample of 200 people their age and the main use of their mobile phone.
The results are shown in Table 1 below. \begin{table}[h]
\multirow{2}{*}{}Main use of their mobile phone
InternetTextsPhone calls
\multirow{3}{*}{Age}Under 2027149
From 20 to 40323429
Over 40151921
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} The data are to be used to test whether or not age and main use of their mobile phone are independent. Table 2 shows the expected frequencies for each group, assuming people's age and main use of their mobile phone are independent. \begin{table}[h]
\multirow{2}{*}{}Main use of their mobile phone
InternetTextsPhone calls
\multirow{3}{*}{Age}Under 2018.516.7514.75
From 20 to 4035.1531.82528.025
Over 4020.3518.42516.225
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. For users under 20 choosing the Internet as the main use of their mobile phone,
    1. verify that the expected frequency is 18.5
    2. show that the contribution to the \(\chi ^ { 2 }\) test statistic is 3.91 to 3 significant figures.
  2. Given that the \(\chi ^ { 2 }\) test statistic for the data is 9.893 to 3 decimal places, test at the \(5 \%\) level of significance whether or not age and main use of their mobile phone are independent. State your hypotheses clearly.