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Edexcel S4 2014 June Q1
5 marks Standard +0.3
  1. A production line is designed to fill bottles with oil. The amount of oil placed in a bottle is normally distributed and the mean is set to 100 ml .
The amount of oil, \(x \mathrm { ml }\), in each of 8 randomly selected bottles is recorded, and the following statistics are obtained. $$\bar { x } = 92.875 \quad s = 8.3055$$ Malcolm believes that the mean amount of oil placed in a bottle is less than 100 ml .
Stating your hypotheses clearly, test, at the \(5 \%\) significance level, whether or not Malcolm's belief is supported.
Edexcel S4 2014 June Q2
7 marks Challenging +1.2
2. (a) Define
  1. a Type I error,
  2. a Type II error. Rolls of material, manufactured by a machine, contain defects at a mean rate of 6 per roll. The machine is modified. A single roll is selected at random and a test is carried out to see whether or not the mean number of defects per roll has decreased. The significance level is chosen to be as close as possible to \(5 \%\).
    (b) Calculate the probability of a Type I error for this test.
    (c) Given that the true mean number of defects per roll of material made by the machine is now 4, calculate the probability of a Type II error.
Edexcel S4 2014 June Q3
14 marks Standard +0.8
3. A large number of chicks were fed a special diet for 10 days. A random sample of 9 of these chicks is taken and the weight gained, \(x\) grams, by each chick is recorded. The results are summarised below. $$\sum x = 181 \quad \sum x ^ { 2 } = 3913$$ You may assume that the weights gained by the chicks are normally distributed.
Calculate a 95\% confidence interval for
    1. the mean of the weights gained by the chicks,
    2. the variance of the weights gained by the chicks. A chick which gains less than \(16 g\) has to be given extra feed.
  2. Using appropriate confidence limits from part (a), find the lowest estimate of the proportion of chicks that need extra feed.
Edexcel S4 2014 June Q4
9 marks Challenging +1.3
  1. A random sample of 8 people were given a new drug designed to help people sleep.
In a two-week period the drug was given for one week and a placebo (a tablet that contained no drug) was given for one week. In the first week 4 people, selected at random, were given the drug and the other 4 people were given the placebo. Those who were given the drug in the first week were given the placebo in the second week. Those who were given the placebo in the first week were given the drug in the second week. The mean numbers of hours of sleep per night for each of the people are shown in the table.
Person\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Hours of sleep with drug10.87.28.76.89.410.911.17.6
Hours of sleep with placebo10.06.59.05.68.78.09.86.8
  1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
  2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the drug increases the mean number of hours of sleep per night by more than 10 minutes. State the critical value for this test.
Edexcel S4 2014 June Q5
11 marks Standard +0.3
5. A statistician believes a coin is biased and the probability, \(p\), of getting a head when the coin is tossed is less than 0.5 The statistician decides to test this by tossing the coin 10 times and recording the number, \(X\), of heads. He sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) and rejects the null hypothesis if \(x < 3\)
  1. Find the size of the test.
  2. Show that the power function of this test is $$( 1 - p ) ^ { 8 } \left( 36 p ^ { 2 } + 8 p + 1 \right)$$ Table 1 gives values, to 2 decimal places, of the power function for the statistician's test. \begin{table}[h] \end{table} Table 1
  3. On the axes below draw the graph of the power function for the statistician's test.
  4. Find the range of values of \(p\) for which the probability of accepting the coin as unbiased, when in fact it is biased, is less than or equal to 0.4 \includegraphics[max width=\textwidth, alt={}, center]{1d84c9fc-be67-45be-b439-3111c48ff1cb-09_1143_1209_945_402}
Edexcel S4 2014 June Q6
15 marks Standard +0.3
  1. (a) Explain what is meant by the sampling distribution of an estimator \(T\) of the population parameter \(\theta\).
    (b) Explain what you understand by the statement that \(T\) is a biased estimator of \(\theta\).
A population has mean \(\mu\) and variance \(\sigma ^ { 2 }\) A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) is taken from this population.
(c) Calculate the bias of each of the following estimators of \(\mu\). $$\begin{aligned} & \hat { \mu } _ { 1 } = \frac { X _ { 3 } + X _ { 5 } + X _ { 7 } } { 3 } \\ & \hat { \mu } _ { 2 } = \frac { 5 X _ { 1 } + 2 X _ { 2 } + X _ { 9 } } { 6 } \\ & \hat { \mu } _ { 3 } = \frac { 3 X _ { 10 } - X _ { 1 } } { 3 } \end{aligned}$$ (d) Find the variance of each of these three estimators.
(e) State, giving a reason, which of these three estimators for \(\mu\) is
  1. the best estimator,
  2. the worst estimator.
Edexcel S4 2014 June Q7
14 marks Standard +0.3
7. Two groups of students take the same examination. A random sample of students is taken from each of the groups. The marks of the 9 students from Group 1 are as follows $$\begin{array} { l l l l l l l l l } 30 & 29 & 35 & 27 & 23 & 33 & 33 & 35 & 28 \end{array}$$ The marks, \(x\), of the 7 students from Group 2 gave the following statistics $$\bar { x } = 31.29 \quad s ^ { 2 } = 12.9$$ A test is to be carried out to see whether or not there is a difference between the mean marks of the two groups of students. You may assume that the samples are taken from normally distributed populations and that they are independent.
  1. State one other assumption that must be made in order to apply this test and show that this assumption is reasonable by testing it at a \(10 \%\) level of significance. State your hypotheses clearly.
  2. Stating your hypotheses clearly, test, using a significance level of \(5 \%\), whether or not there is a difference between the mean marks of the two groups of students.
Edexcel S4 2015 June Q1
9 marks Standard +0.3
  1. The Sales Manager of a large chain of convenience stores is studying the sale of lottery tickets in her stores. She randomly selects 8 of her stores. From these stores she collects data for the total sales of lottery tickets in the previous January and July. The data are shown below
StoreABCDEFGH
January ticket sales \(( \boldsymbol { \pounds } )\)10801639710110891510661322819
July ticket sales \(( \boldsymbol { \pounds } )\)11131702831104886110901303852
  1. Use a paired \(t\)-test to determine whether or not there is evidence, at the \(5 \%\) level of significance, that the mean sales of lottery tickets in this chain's stores are higher in July than in January. You should state your hypotheses and show your working clearly.
  2. State what assumption the Sales Manager needs to make about the sales of lottery tickets in her stores for the test in part (a) to be valid.
Edexcel S4 2015 June Q2
14 marks Standard +0.8
  1. Fred is a new employee in a delicatessen. He is asked to cut cheese into 100 g blocks. A random sample of 8 of these blocks of cheese is selected. The weight, in grams, of each block of cheese is given below
$$94 , \quad 106 , \quad 115 , \quad 98 , \quad 111 , \quad 104 , \quad 113 , \quad 102$$
  1. Calculate a \(90 \%\) confidence interval for the standard deviation of the weights of the blocks of cheese cut by Fred. Given that the weights of the blocks of cheese are independent,
  2. state what further assumption is necessary for this confidence interval to be valid. The delicatessen manager expects the standard deviation of the weights of the blocks of cheese cut by an employee to be less than 5 g. Any employee who does not achieve this target is given training.
  3. Use your answer from part (a) to comment on Fred's results. A second employee, Olga, has just been given training. Olga is asked to cut cheese into 100 g blocks. A random sample of 20 of these blocks of cheese is selected. The weight of each block of cheese, \(x\) grams, is recorded and the results are summarised below. $$\bar { x } = 102.6 \quad s ^ { 2 } = 19.4$$ Given that the assumption in part (b) is also valid in this case,
  4. test, at a \(10 \%\) level of significance, whether or not the mean weight of the blocks of cheese cut by Olga after her training is 100 g . State your hypotheses clearly.
    (6)
Edexcel S4 2015 June Q3
14 marks Standard +0.8
  1. As part of their research two sports science students, Ali and Bea, select a random sample of 10 adult male swimmers and a random sample of 13 adult male athletes from local sports clubs. They measure the arm span, \(x \mathrm {~cm}\), of each person selected.
    The data are summarised in the table below
\(n\)\(s ^ { 2 }\)\(\bar { x }\)
Swimmers1048195
Athletes13161186
The students know that the arm spans of adult male swimmers and of adult male athletes may each be assumed to be normally distributed.
They decide to share out the data analysis, with Ali investigating the means of the two distributions and Bea investigating the variances of the two distributions. Ali assumes that the variances of the two distributions are equal. She calculates the pooled estimate of variance, \(s _ { p } { } ^ { 2 }\)
  1. Show that \(s _ { p } { } ^ { 2 } = 112.6\) to 1 decimal place. Ali claims that there is no difference in the mean arm spans of adult male swimmers and of adult male athletes.
  2. Stating your hypotheses clearly, test this claim at the \(10 \%\) level of significance.
    (5) Bea believes that the variances of the arm spans of adult male swimmers and adult male athletes are not equal.
  3. Show that, at the \(10 \%\) level of significance, the data support Bea's belief. State your hypotheses and show your working clearly. Ali and Bea combine their work and present their results to their tutor, Clive.
  4. Explain why Clive is not happy with their research and state, with a reason, which of the tests in parts (b) and (c) is not valid.
Edexcel S4 2015 June Q4
11 marks Challenging +1.8
4. A poultry farm produces eggs which are sold in boxes of 6 . The farmer believes that the proportion, \(p\), of eggs that are cracked when they are packed in the boxes is approximately 5\%. She decides to test the hypotheses $$\mathrm { H } _ { 0 } : p = 0.05 \text { against } \mathrm { H } _ { 1 } : p > 0.05$$ To test these hypotheses she randomly selects a box of eggs and rejects \(\mathrm { H } _ { 0 }\) if the box contains 2 or more eggs that are cracked. If the box contains 1 egg that is cracked, she randomly selects a second box of eggs and rejects \(\mathrm { H } _ { 0 }\) if it contains at least 1 egg that is cracked. If the first or the second box contains no cracked eggs, \(\mathrm { H } _ { 0 }\) is immediately accepted and no further boxes are sampled.
  1. Show that the power function of this test is $$1 - ( 1 - p ) ^ { 6 } - 6 p ( 1 - p ) ^ { 11 }$$
  2. Calculate the size of this test. Given that \(p = 0.1\)
  3. find the expected number of eggs inspected each time this test is carried out, giving your answer correct to 3 significant figures,
  4. calculate the probability of a Type II error. Given that \(p = 0.1\) is an unacceptably high value for the farmer,
  5. use your answer from part (d) to comment on the farmer's test.
Edexcel S4 2015 June Q5
9 marks Standard +0.3
  1. A researcher is investigating the accuracy of IQ tests. One company offers IQ tests that it claims will give any individual's IQ with a standard deviation of 5
The researcher takes these tests 9 times with the following results $$123 , \quad 118 , \quad 127 , \quad 120 , \quad 134 , \quad 120 , \quad 118 , \quad 135 , \quad 121$$
  1. Find the sample mean, \(\bar { x }\), and the sample variance, \(s ^ { 2 }\), of these scores.
    (2) Given that any individual's IQ scores on these tests are independent and have a normal distribution,
  2. use the hypotheses $$\mathrm { H } _ { 0 } : \sigma ^ { 2 } = 25 \text { against } \mathrm { H } _ { 1 } : \sigma ^ { 2 } > 25$$ to test the company's claim at the \(5 \%\) significance level.
    (4) Gurdip works for the company and has taken these IQ tests 12 times. Gurdip claims that the sample variance of these 12 scores is \(s ^ { 2 } = 8.17\)
  3. Use this value of \(s ^ { 2 }\) to calculate a \(95 \%\) confidence interval for the variance of Gurdip's IQ test scores.
    [0pt] [You may use \(\mathrm { P } \left( \chi _ { 11 } ^ { 2 } > 3.816 \right) = 0.975\) and \(\mathrm { P } \left( \chi _ { 11 } ^ { 2 } > 21.920 \right) = 0.025\) ]
  4. Assuming that \(\sigma ^ { 2 } = 25\), comment on Gurdip's claim.
Edexcel S4 2015 June Q6
18 marks Challenging +1.2
6. A random sample \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 2 n }\) is taken from a population with mean \(\frac { \mu } { 3 }\) and variance \(3 \sigma ^ { 2 }\). A second random sample \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , \ldots , Y _ { n }\) is taken from a population with mean \(\frac { \mu } { 2 }\) and variance \(\frac { \sigma ^ { 2 } } { 2 }\), where the \(X\) and \(Y\) variables are all independent. \(A\), \(B\) and \(C\) are possible estimators of \(\mu\), where $$\begin{aligned} & A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 2 } \\ & B = \frac { 3 X _ { 1 } } { 2 } + \frac { 2 Y _ { 1 } } { 3 } \\ & C = \frac { 3 X _ { 1 } + 4 Y _ { 1 } } { 3 } \end{aligned}$$
  1. Show that two of \(A , B\) and \(C\) are unbiased estimators of \(\mu\) and find the bias of the third estimator of \(\mu\).
  2. Showing your working clearly, find which of \(A\), \(B\) and \(C\) is the best estimator of \(\mu\). The estimator $$D = \frac { 1 } { k } \left( \sum _ { i = 1 } ^ { 2 n } X _ { i } + \sum _ { i = 1 } ^ { n } Y _ { i } \right)$$ is an unbiased estimator of \(\mu\).
  3. Find \(k\) in terms of \(n\).
  4. Show that \(D\) is also a consistent estimator of \(\mu\).
  5. Find the least value of \(n\) for which \(D\) is a better estimator of \(\mu\) than any of \(A\), \(B\) or \(C\).
Edexcel S4 2016 June Q1
9 marks Standard +0.3
  1. A new diet has been designed. Its designers claim that following the diet for a month will result in a mean weight loss of more than 2 kg . In a trial, a random sample of 10 people followed the new diet for a month. Their weights, in kg, before starting the diet and their weights after following the diet for a month were recorded. The results are given in the table below.
Person\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Weight before diet (kg)96110116981219198106110116
Weight after diet (kg)91101111961219190101104110
  1. Using a suitable \(t\)-test, at the \(5 \%\) level of significance, state whether or not the trial supports the designers’ claim. State your hypotheses and show your working clearly.
  2. State an assumption necessary for the test in part (a).
Edexcel S4 2016 June Q2
13 marks Standard +0.3
2. The weights of piglets at birth, \(M \mathrm {~kg}\), are normally distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) A random sample of 9 piglets is taken and their weights at birth, \(m \mathrm {~kg}\), are recorded. The results are summarised as $$\sum m = 11.6 \quad \sum m ^ { 2 } = 15.2$$ Stating your hypotheses clearly, test at the 5\% level of significance
  1. whether or not the mean weight of piglets at birth is greater than 1.2 kg ,
  2. whether or not the standard deviation of the weights of piglets at birth is different from 0.3 kg .
Edexcel S4 2016 June Q3
6 marks Standard +0.3
3. A jar contains a large number of sweets which have either soft centres or hard centres. The jar is thought to contain equal proportions of sweets with soft centres and sweets with hard centres. A random sample of 20 sweets is taken from the jar and the number of sweets with hard centres is recorded.
  1. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that there are equal proportions of sweets with soft centres and sweets with hard centres in the jar.
  2. Calculate the probability of a Type I error for this test. Given that there are 3 times as many sweets with soft centres as there are sweets with hard centres,
  3. calculate the probability of a Type II error for this test.
Edexcel S4 2016 June Q4
9 marks Standard +0.8
  1. A manufacturer produces boxes of screws containing short screws and long screws. The manufacturer claims that the probability, \(p\), of a randomly selected screw being long, is 0.5
A shopkeeper does not believe the manufacturer's claim. He designs two tests, \(A\) and \(B\), to test the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) In test \(A\), a random sample of 10 screws is taken from a box of screws and \(\mathrm { H } _ { 0 }\) is rejected if there are fewer than 3 long screws. In test \(B\), a random sample of 5 screws is taken from a box of screws and \(\mathrm { H } _ { 0 }\) is rejected if there are no long screws, otherwise a second random sample of 5 screws is taken from a box of screws. If there are no long screws in this second sample \(\mathrm { H } _ { 0 }\) is rejected, otherwise it is accepted.
  1. Find the size of test \(A\).
  2. Find the size of test \(B\).
  3. Find an expression for the power function of test \(B\) in terms of \(p\). Some values, to 2 decimal places, of the power function for test \(A\) and the power function for test \(B\) are given in the table below.
    \(p\)0.10.20.30.4
    Power test \(A\)0.93\(r\)0.380.17
    Power test \(B\)0.830.550.310.15
  4. Find the value of \(r\). The shopkeeper believes that the value of \(p\) is less than 0.4
  5. Suggest which of the tests the shopkeeper should use. Give a reason for your answer.
Edexcel S4 2016 June Q5
14 marks Standard +0.8
5. Fire brigades in cities \(X\) and \(Y\) are in similar locations. The response times, in minutes, during a particular month, for randomly selected calls are summarised in the table below.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Sample sizeSample mean
Standard deviation
\(S\)
\(X\)914.86.76
\(Y\)67.25.42
You may assume that the response times are from independent normal distributions.
Stating your hypotheses and showing your working clearly
  1. test, at the \(10 \%\) level of significance, whether or not the variances of the populations from which the response times are drawn are the same,
    (5)
  2. test, at the \(5 \%\) level of significance, whether or not the mean response time for the fire brigade in city \(X\) is more than 5 minutes longer than the mean response time for the fire brigade in city \(Y\).
  3. Explain why your result in part (a) enables you to carry out the test in part (b).
Edexcel S4 2016 June Q6
15 marks Challenging +1.2
6. A random sample of size \(n\) is taken from the random variable \(X\), which has a continuous uniform distribution over the interval [ \(0 , a\) ], \(a > 0\) The sample mean is denoted by \(\bar { X }\)
  1. Show that \(Y = 2 \bar { X }\) is an unbiased estimator of \(a\) The maximum value, \(M\), in the sample has probability density function $$f ( m ) = \left\{ \begin{array} { c c } \frac { n m ^ { n - 1 } } { a ^ { n } } & 0 \leqslant m \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$
  2. Find E(M)
  3. Show that \(\operatorname { Var } ( M ) = \frac { n a ^ { 2 } } { ( n + 2 ) ( n + 1 ) ^ { 2 } }\) The estimator \(S\) is defined by \(S = \frac { n + 1 } { n } M\) Given that \(n > 1\)
  4. state which of \(Y\) or \(S\) is the better estimator for \(a\). Give a reason for your answer.
Edexcel S4 2016 June Q7
9 marks Challenging +1.8
7. The times taken to travel to school by sixth form students are normally distributed. A head teacher records the times taken to travel to school, in minutes, of a random sample of 10 sixth form students from her school. Based on this sample, the \(95 \%\) confidence interval for the mean time taken to travel to school for sixth form students from her school is
[0pt] [28.5, 48.7] Calculate a 99\% confidence interval for the variance of the time taken to travel to school for sixth form students from her school.
(9)
Edexcel S4 2017 June Q1
14 marks Challenging +1.2
  1. The times taken by children to run 150 m are normally distributed. The times taken, \(x\) seconds, by a random sample of 9 boys and an independent random sample of 6 girls are recorded. The following statistics are obtained.
Number of childrenSample mean \(\bar { x }\)\(\sum x ^ { 2 }\)
Boys922.84693.60
Girls629.55236.12
  1. Test, at the \(10 \%\) level of significance, whether or not the variances of the two distributions are equal. State your hypotheses clearly. The Headteacher claims that the mean time taken for the girls is more than 5 seconds greater than the mean time taken for the boys.
  2. Stating your hypotheses clearly, test the Headteacher's claim. Use a \(1 \%\) level of significance and show your working clearly.
Edexcel S4 2017 June Q2
8 marks Standard +0.8
  1. The number of accidents per year in Daftstown follows a Poisson distribution with mean \(\lambda\). The value of \(\lambda\) has previously been 6 but Jonty claims that since the Council increased the speed limit, the value of \(\lambda\) has increased.
Jonty records the number of accidents in Daftstown in the first year after the speed limit was increased. He plans to test, at the \(5 \%\) significance level, whether or not there is evidence of an increase in the mean number of accidents in Daftstown per year.
  1. Stating your hypotheses clearly, calculate the probability of a Type I error for this test. Given that there were 9 accidents in the first year after the speed limit was increased,
  2. state, giving a reason, whether or not there is evidence to support Jonty's claim.
  3. Given that the value of \(\lambda\) has actually increased to 8, calculate the probability of drawing the conclusion, using this test, that the number of accidents per year in Daftstown has not increased.
Edexcel S4 2017 June Q3
11 marks Standard +0.3
3. The lengths, \(X \mathrm {~mm}\), of the wings of adult blackbirds follow a normal distribution. A random sample of 5 adult blackbirds is taken and the lengths of the wings are measured. The results are summarised below $$\sum x = 655 \text { and } \sum x ^ { 2 } = 85845$$
  1. Test, at the \(10 \%\) level of significance, whether or not the mean length of an adult blackbird's wing is less than 135 mm . State your hypotheses clearly.
  2. Find the \(90 \%\) confidence interval for the variance of the lengths of adult blackbirds' wings. Show your working clearly.
Edexcel S4 2017 June Q4
12 marks Standard +0.3
4. A coach believes that the average score in the final round of a golf tournament is more than one point below the average score in the first round. To test this belief, the scores of 8 randomly selected players are recorded. The results are given in the table below.
Player\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
First round7680727883888172
Final round7078757579848369
    1. State why a paired \(t\)-test is suitable for use with these data.
    2. State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
  2. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the coach's belief. Show your working clearly.
  3. Explain, in the context of the coach's belief, what a Type II error would be in this case.
Edexcel S4 2017 June Q5
11 marks Challenging +1.2
  1. Jamland and Goodjam are two suppliers of jars of jam. The weights of the jars of jam produced by each supplier can be assumed to be normally distributed with unknown, but equal, variances. A random sample of 20 jars of jam is taken from those supplied by Jamland.
Based on this sample, the 95\% confidence interval for the mean weight of a jar of Jamland jam, in grams, is
[0pt] [ 492, 507 ] A random sample of 10 jars of jam is selected from those supplied by Goodjam. The weight of each jar of Goodjam jam, \(y\) grams, is recorded. The results are summarised as follows $$\bar { y } = 480 \quad s _ { y } ^ { 2 } = 280$$ Find a 90\% confidence interval for the value by which the mean weight of a jar of jam supplied by Jamland exceeds the mean weight of a jar of jam supplied by Goodjam.