Questions — Edexcel S3 (313 questions)

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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.
Edexcel S3 2014 June Q3
  1. A company produces two types of milk powder, 'Semi-Skimmed' and 'Full Cream'. In tests, each type of milk powder is used to make a large number of cups of coffee. The mass, \(S\) grams, of 'Semi-Skimmed' milk powder used in one cup of coffee is modelled by \(S \sim \mathrm {~N} \left( 4.9,0.8 ^ { 2 } \right)\). The mass, \(C\) grams, of 'Full Cream' milk powder used in one cup of coffee is modelled by \(C \sim \mathrm {~N} \left( 2.5,0.4 ^ { 2 } \right)\)
    1. Two cups of coffee, one with each type of milk powder, are to be selected at random. Find the probability that the mass of 'Semi-Skimmed' milk powder used will be at least double that of the 'Full Cream' milk powder used.
    2. 'Semi-Skimmed' milk powder is sold in 500 g packs. Find the probability that one pack will be sufficient for 100 cups of coffee.
Edexcel S3 2014 June Q4
4. A manufacturing company produces solar panels. The output of each solar panel is normally distributed with standard deviation 6 watts. It is thought that the mean output, \(\mu\), is 160 watts. A researcher believes that the mean output of the solar panels is greater than 160 watts. He writes down the output values of 5 randomly selected solar panels. He uses the data to carry out a hypothesis test at the \(5 \%\) level of significance. He tests \(\mathrm { H } _ { 0 } : \mu = 160\) against \(\mathrm { H } _ { 1 } : \mu > 160\)
On reporting to his manager, the researcher can only find 4 of the output values. These are shown below $$\begin{array} { l l l l } 168.2 & 157.4 & 173.3 & 161.1 \end{array}$$ Given that the result of the hypothesis test is that there is significant evidence to reject \(\mathrm { H } _ { 0 }\) at the \(5 \%\) level of significance, calculate the minimum possible missing output value, \(\alpha\). Give your answer correct to 1 decimal place.
Edexcel S3 2014 June Q5
5. A student believes that there is a difference in the mean lengths of English and French films. He goes to the university video library and randomly selects a sample of 120 English films and a sample of 70 French films. He notes the length, \(x\) minutes, of each of the films in his samples. His data are summarised in the table below.
\(\Sigma x\)\(\Sigma x ^ { 2 }\)\(s ^ { 2 }\)\(n\)
English films1065095690998.5120
French films651061584915170
  1. Verify that the unbiased estimate of the variance, \(s ^ { 2 }\), of the lengths of English films is 98.5 minutes \({ } ^ { 2 }\)
  2. Stating your hypotheses clearly, test, at the 1\% level of significance, whether or not the mean lengths of English and French films are different.
  3. Explain the significance of the Central Limit Theorem to the test in part (b).
  4. The university video library contained 724 English films and 473 French films. Explain how the student could have taken a stratified sample of 190 of these films.
Edexcel S3 2014 June Q6
6. Bags of \(\pounds 1\) coins are paid into a bank. Each bag contains 20 coins. The bank manager believes that \(5 \%\) of the \(\pounds 1\) coins paid into the bank are fakes. He decides to use the distribution \(X \sim \mathrm {~B} ( 20,0.05 )\) to model the random variable \(X\), the number of fake \(\pounds 1\) coins in each bag.
  1. State the assumptions necessary for the binomial distribution to be an appropriate model in this case. The bank manager checks a random sample of 150 bags of \(\pounds 1\) coins and records the number of fake coins found in each bag. His results are summarised in Table 1. \begin{table}[h]
    Number of fake coins in each bag01234 or more
    Observed frequency436226136
    Expected frequency53.856.6\(r\)8.9\(s\)
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
  2. Calculate the values of \(r\) and \(s\), giving your answers to 1 decimal place.
  3. Carry out a hypothesis test, at the \(5 \%\) significance level, to see if the data supports the bank manager's statistical model. State your hypotheses clearly. Question 6 parts (d) and (e) are continued on page 24 The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than \(5 \%\). She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2. \begin{table}[h]
    Number of fake coins in each bag01234 or more
    Observed frequency436226136
    Expected frequency44.555.733.212.54.1
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  4. Explain why there are 2 degrees of freedom in this case.
  5. Given that she obtains a \(\chi ^ { 2 }\) test statistic of 2.67 , test the assistant manager's hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel S3 2014 June Q7
7. A petrol pump is tested regularly to check that the reading on its gauge is accurate. The random variable \(X\), in litres, is the quantity of petrol actually dispensed when the gauge reads 10.00 litres. \(X\) is known to have distribution \(X \sim \mathrm {~N} \left( \mu , 0.08 ^ { 2 } \right)\)
  1. Eight random tests gave the following values of \(x\) $$\begin{array} { l l l l l l l l } 10.01 & 9.97 & 9.93 & 9.99 & 9.90 & 9.95 & 10.13 & 9.94 \end{array}$$
    1. Find a 95\% confidence interval for \(\mu\) to 2 decimal places.
    2. Use your result to comment on the accuracy of the petrol gauge.
  2. A sample mean of 9.96 litres was obtained from a random sample of \(n\) tests. A \(90 \%\) confidence interval for \(\mu\) gave an upper limit of less than 10.00 litres. Find the minimum value of \(n\).
Edexcel S3 2014 June Q1
  1. (a) Explain what you understand by a random sample from a finite population.
    (b) Give an example of a situation when it is not possible to take a random sample.
A college lecturer specialising in shoe design wants to change the way in which she organises practical work. She decides to gather ideas from her 75 students. She plans to give a questionnaire to a random sample of 8 of these students.
(c) (i) Describe the sampling frame that she should use.
(ii) Explain in detail how she should use a table of random numbers to obtain her sample.
Edexcel S3 2014 June Q2
2. The weights of pears in an orchard are assumed to have unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A random sample of 20 pears is taken and their weights recorded.
The sample is represented by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 20 }\). State whether or not the following are statistics. Give reasons for your answers.
    1. \(\frac { X _ { 1 } + 3 X _ { 20 } } { 2 }\)
    2. \(\sum _ { i = 1 } ^ { 20 } \left( X _ { i } - \mu \right)\)
    3. \(\sum _ { i = 1 } ^ { 20 } \left( \frac { X _ { i } - \mu } { \sigma } \right)\)
  2. Find the mean and variance of \(\frac { 3 X _ { 1 } - X _ { 20 } } { 2 }\)
Edexcel S3 2014 June Q3
3. A number of males and females were asked to rate their happiness under the headings "not happy", "fairly happy" and "very happy". The results are shown in the table below
Happiness\multirow{2}{*}{Total}
\cline { 3 - 5 } \multicolumn{2}{|c|}{}Not happyFairly happyVery happy
\multirow{2}{*}{Gender}Female9433486
\cline { 2 - 6 }Male13251654
Total226850140
Stating your hypotheses, test at the \(5 \%\) level of significance, whether or not there is evidence of an association between happiness and gender. Show your working clearly.