Questions S3 (597 questions)

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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.
Edexcel S3 2014 June Q4
  1. The random variable \(A\) is defined as
$$A = B + 4 C - 3 D$$ where \(B\), \(C\) and \(D\) are independent random variables with $$B \sim \mathrm {~N} \left( 6,2 ^ { 2 } \right) \quad C \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right) \quad D \sim \mathrm {~N} \left( 4,1.5 ^ { 2 } \right)$$ Find \(\mathrm { P } ( A < 45 )\)
Edexcel S3 2014 June Q5
5. A research station is doing some work on the germination of a new variety of genetically modified wheat. They planted 120 rows containing 7 seeds in each row.
The number of seeds germinating in each row was recorded. The results are as follows
Number of seeds germinating in each row01234567
Observed number of rows2611192532169
  1. Write down two reasons why a binomial distribution may be a suitable model.
  2. Show that the probability of a randomly selected seed from this sample germinating is 0.6 The research station used a binomial distribution with probability 0.6 of a seed germinating. The expected frequencies were calculated to 2 decimal places. The results are as follows
    Number of seeds germinating in each row01234567
    Expected number of rows0.202.06\(s\)23.22\(t\)31.3515.683.36
  3. Find the value of \(s\) and the value of \(t\).
  4. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the data can be modelled by a binomial distribution.
Edexcel S3 2014 June Q6
6. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population with mean \(\mu\).
  1. Show that \(\bar { X } = \frac { 1 } { n } \left( X _ { 1 } + X _ { 2 } + \ldots + X _ { n } \right)\) is an unbiased estimator of the population mean \(\mu\). A company produces small jars of coffee. Five jars of coffee were taken at random and weighed. The weights, in grams, were as follows $$\begin{array} { l l l l l } 197 & 203 & 205 & 201 & 195 \end{array}$$
  2. Calculate unbiased estimates of the population mean and variance of the weights of the jars produced by the company. It is known from previous results that the weights are normally distributed with standard deviation 4.8 g . The manager is going to take a second random sample. He wishes to ensure that there is at least a \(95 \%\) probability that the estimate of the population mean is within 1.25 g of its true value.
  3. Find the minimum sample size required.
Edexcel S3 2014 June Q7
7. A machine fills packets with \(X\) grams of powder where \(X\) is normally distributed with mean \(\mu\). Each packet is supposed to contain 1 kg of powder. To comply with regulations, the weight of powder in a randomly selected packet should be such that \(\mathrm { P } ( X < \mu - 30 ) = 0.0005\)
  1. Show that this requires the standard deviation to be 9.117 g to 3 decimal places. A random sample of 10 packets is selected from the machine. The weight, in grams, of powder in each packet is as follows 999.8991 .61000 .31006 .11008 .2997 .0993 .21000 .0997 .11002 .1
  2. Assuming that the standard deviation of the population is 9.117 g , test, at the \(1 \%\) significance level, whether or not the machine is delivering packets with mean weight of less than 1 kg . State your hypotheses clearly.
Edexcel S3 2014 June Q8
8. The heights, in metres, and weights, in kilograms, of a random sample of 9 men are shown in the table below
Man\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
Height \(( x )\)1.681.741.751.761.781.821.841.881.98
Weight \(( y )\)757610077909511096120
  1. Given that \(\mathrm { S } _ { x x } = 0.0632 , \mathrm {~S} _ { y y } = 1957.5556\) and \(\mathrm { S } _ { x y } = 9.3433\) calculate, to 3 decimal places, the product moment correlation coefficient between height and weight for these men.
  2. Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the height and weight of men. Use a \(5 \%\) significance level. State your hypotheses clearly. Peter does not know the heights or weights of the 9 men. He is given photographs of them and asked to put them in order of increasing weight. He puts them in the order $$A C E B G D I F H$$
  3. Find, to 3 decimal places, Spearman's rank correlation coefficient between Peter's order and the actual order.
  4. Use your value of Spearman’s rank correlation coefficient to test for evidence of Peter's ability to correctly order men, by their weight, from their photographs. Use a 5\% significance level and state your hypotheses clearly.
Edexcel S3 2015 June Q1
  1. A mobile library has 160 books for children on its records. The librarian believes that books with fewer pages are borrowed more often. He takes a random sample of 10 books for children.
    1. Explain how the librarian should select this random sample.
      (2)
    The librarian ranked the 10 books according to how often they had been borrowed, with 1 for the book borrowed the most and 10 for the book borrowed the least. He also recorded the number of pages in each book. The results are in the table below.
    Book\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
    Borrowing rank12345678910
    Number of pages502121158030190356283152317
  2. Calculate Spearman's rank correlation coefficient for these data.
  3. Test the librarian's belief using a \(5 \%\) level of significance. State your hypotheses clearly.
Edexcel S3 2015 June Q2
2. A researcher believes that the mean weight loss of those people using a slimming plan as part of a group is more than 1.5 kg a year greater than the mean weight loss of those using the plan on their own. The mean weight loss of a random sample of 80 people using the plan as part of a group is 8.7 kg with a standard deviation of 2.1 kg . The mean weight loss of a random sample of 65 people using the plan on their own is 6.6 kg with a standard deviation of 1.4 kg .
  1. Stating your hypotheses clearly, test the researcher's claim. Use a \(1 \%\) level of significance.
  2. For the test in part (a), state whether or not it is necessary to assume that the weight loss of a person using this plan has a normal distribution. Give a reason for your answer.
Edexcel S3 2015 June Q3
3. A nursery has 16 staff and 40 children on its records. In preparation for an outing the manager needs an estimate of the mean weight of the people on its records and decides to take a stratified sample of size 14 .
  1. Describe how this stratified sample should be taken. The weights, \(x \mathrm {~kg}\), of each of the 14 people selected are summarised as $$\sum x = 437 \text { and } \sum x ^ { 2 } = 26983$$
  2. Find unbiased estimates of the mean and the variance of the weights of all the people on the nursery's records.
  3. Estimate the standard error of the mean. The estimates of the standard error of the mean for the staff and for the children are 5.11 and 1.10 respectively.
  4. Comment on these values with reference to your answer to part (c) and give a reason for any differences.
Edexcel S3 2015 June Q4
  1. The weights of bags of rice, \(X \mathrm {~kg}\), have a normal distribution with unknown mean \(\mu \mathrm { kg }\) and known standard deviation \(\sigma \mathrm { kg }\). A random sample of 100 bags of rice gave a \(90 \%\) confidence interval for \(\mu\) of \(( 0.4633,0.5127 )\).
    1. Without carrying out any further calculations, use this confidence interval to test whether or not \(\mu = 0.5\)
    State your hypotheses clearly and write down the significance level you have used. A second random sample, of 150 of these bags of rice, had a mean weight of 0.479 kg .
  2. Calculate a \(95 \%\) confidence interval for \(\mu\) based on this second sample.
Edexcel S3 2015 June Q5
    1. The volume, \(B \mathrm { ml }\), in a bottle of Burxton's water has a normal distribution \(B \sim \mathrm {~N} \left( 325,6 ^ { 2 } \right)\) and the volume, \(H \mathrm { ml }\), in a bottle of Hargate's water has a normal distribution \(H \sim \mathrm {~N} \left( 330,4 ^ { 2 } \right)\).
      Rebecca buys 5 bottles of Burxton's water and one bottle of Hargate's water.
      Find the probability that the total volume in the 5 bottles of Burxton's water is more than 5 times the volume in the bottle of Hargate's water.
      (5)
    2. Two independent random samples \(X _ { 1 } , X _ { 2 } , X _ { 3 } , X _ { 4 } , X _ { 5 }\) and \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , Y _ { 4 } , Y _ { 5 }\) are each taken from a normal population with mean \(\mu\) and standard deviation \(\sigma\).
      1. Find the distribution of the random variable \(D = Y _ { 1 } - \bar { X }\)
    3. Hence show that \(\mathrm { P } \left( Y _ { 1 } > \bar { X } + \sigma \right) = 0.181\) correct to 3 decimal places.
    Ankit believes that \(\mathrm { P } \left( U _ { 1 } > \bar { U } + \sigma \right) = 0.181\) correct to 3 decimal places, for any random sample \(U _ { 1 } , U _ { 2 } , U _ { 3 } , U _ { 4 } , U _ { 5 }\) taken from a normal population with mean \(\mu\) and standard deviation \(\sigma\).
  2. Explain briefly why the result from part (b) should not be used to confirm Ankit's belief.
  3. Find, correct to 3 decimal places, the actual value of \(\mathrm { P } \left( U _ { 1 } > \bar { U } + \sigma \right)\).
Edexcel S3 2015 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{740f7555-3a9a-4526-9048-39908aa8f8dd-10_684_694_239_625} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The sketch in Figure 1 represents a target which consists of 4 regions formed from 4 concentric circles of radii \(4 \mathrm {~cm} , 7 \mathrm {~cm} , 9 \mathrm {~cm}\) and 10 cm . The regions are coloured as labelled in Figure 1.
A random sample of 100 children each choose a point on the target and their results are summarised in the table below. (b) Find the value of \(r\) and the value of \(s\). Henry obtained a test statistic of 6.188 and no groups were pooled.
(c) State what conclusion Henry should make about his claim. Phoebe believes that the children chose the region of the target according to colour. She believes that boys and girls would favour different colours and splits the original data by gender to obtain the following table. \section*{Observed frequencies}
Colour of regionGreenRedBlueYellowTotal
Boys101210335
Girls1227151165
(d) State suitable hypotheses to test Phoebe's belief. Phoebe calculated the following expected frequencies to carry out a suitable test. \section*{Expected frequencies}
Colour of regionGreenRedBlueYellow
Boys7.713.658.754.9
Girls14.325.3516.259.1
(e) Show how the value of 25.35 was obtained. Phoebe carried out the test using 2 degrees of freedom and a \(10 \%\) level of significance. She obtained a test statistic of 1.411
(f) Explain clearly why Phoebe used 2 degrees of freedom.
(g) Stating your critical value clearly, determine whether or not these data support Phoebe's belief.
Edexcel S3 2016 June Q1
  1. (a) State two reasons why stratified sampling might be a more suitable sampling method than simple random sampling.
    (b) State two reasons why stratified sampling might be a more suitable sampling method than quota sampling.
  2. 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.
\cline { 2 - 4 } \multicolumn{1}{c|}{}ABC
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
3. (a) 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
(b) Calculate Spearman's rank correlation coefficient for these data.
(c) 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. The shopkeeper calculates the product moment correlation coefficient from his raw data and finds \(r = 0.65\)
(d) 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.
(e) 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.
Edexcel S3 2016 June Q4
4. The weights of eggs are normally distributed with mean 60 g and standard deviation 5 g Sairah chooses 2 eggs at random.
  1. Find the probability that the difference in weight of these 2 eggs is more than 2 g
    (5) Sairah is packing eggs into cartons. The weight of an empty egg carton is normally distributed with mean 40 g and standard deviation 1.5 g
  2. Find the distribution of the total weight of a carton filled with 12 randomly chosen eggs.
  3. Find the probability that a randomly chosen carton, filled with 12 randomly chosen eggs, weighs more than 800 g
Edexcel S3 2016 June Q5
5. 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.
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(n\)\(\bar { x }\)\(s ^ { 2 }\)
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.
  2. State any assumptions you have made in testing the doctor's claim. 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.
Edexcel S3 2016 June Q6
6. 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. Stating your hypotheses clearly, use a \(1 \%\) level of significance to test the manager's new claim.
Edexcel S3 2016 June Q7
7. 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.
  2. State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers. 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 \(\mathrm { P } ( | \bar { X } - \mu | < 0.5 ) \geqslant 0.9\)