Questions — Edexcel S3 (332 questions)

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Edexcel S3 2021 October Q4
11 marks Moderate -0.3
  1. A local village radio station, LSB, decides to survey adults in its broadcasting area about the programmes it produces. \(L S B\) broadcasts to 4 villages \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .
    The number of households in each of the villages is given below.
VillageNumber of households
A41
B164
C123
D82
LSB decides to take a stratified sample of 200 households.
  1. Explain how to select the households for this stratified sample.
    (3) One of the questions in the survey related to the age group of each member of the household and whether they listen to \(L S B\). The data received are shown below.
    \multirow{2}{*}{}Age group
    18-4950-69Older than 69
    Listen to LSB13016265
    Do not listen to LSB789862
    The data are to be used to determine whether or not there is an association between the age group and whether they listen to \(L S B\).
  2. Calculate the expected frequencies for the age group 50-69 that
    1. listen to \(L S B\)
    2. do not listen to \(L S B\) (2) Given that for the other 4 classes \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 4.657\) to 3 decimal places,
  3. test at the \(5 \%\) level of significance, whether or not there is evidence of an association between age and listening to \(L S B\). Show your working clearly, stating the degrees of freedom and the critical value.
Edexcel S3 2021 October Q5
8 marks Standard +0.3
  1. Assam produces bags of flour. The stated weight printed on the bags of flour is 3 kg . The weights of the bags of flour are normally distributed with standard deviation 0.015 kg .
Assam weighs a random sample of 9 bags of flour and finds their mean weight is 2.977 kg .
  1. Calculate the \(99 \%\) confidence interval for the mean weight of a bag of flour. Give your limits to 3 decimal places. Assam decides to increase the amount of flour put into the bags.
  2. Explain why the confidence interval has led Assam to take this action. After the increase a random sample of \(n\) bags of flour is taken. The sample mean weight of these \(n\) bags is 2.995 kg . A \(95 \%\) confidence interval for \(\mu\) gave a lower limit of less than 2.991 kg .
  3. Find the maximum value of \(n\).
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Edexcel S3 2021 October Q6
12 marks Standard +0.3
6. Amala believes that the resting heart rate is lower in men who exercise regularly compared to men who do not exercise regularly. She measures the resting heart rate, \(h\), of a random sample of 50 men who exercise regularly and a random sample of 40 men who do not exercise regularly. Her results are summarised in the table below.
\cline { 2 - 6 } \multicolumn{1}{c|}{}
Sample
size
\(\sum \boldsymbol { h }\)\(\sum \boldsymbol { h } ^ { 2 }\)
Unbiased
estimate of
the mean
Unbiased
estimate of
the variance
Exercise regularly503270214676\(\alpha\)\(\beta\)
Do not exercise
regularly
40283220166070.829.6
  1. Calculate the value of \(\alpha\) and the value of \(\beta\)
  2. Test, at the \(5 \%\) level of significance, whether there is evidence to support Amala's belief. State your hypotheses clearly.
  3. Explain the significance of the central limit theorem to the test in part (b).
  4. State two assumptions you have made in carrying out the test in part (b).
Edexcel S3 2021 October Q7
17 marks Standard +0.3
  1. A company produces bricks.
The weight of a brick, \(B \mathrm {~kg}\), is such that \(B \sim \mathrm {~N} \left( 1.96 , \sqrt { 0.003 } ^ { 2 } \right)\) Two bricks are chosen at random.
  1. Find the probability that the difference in weight of the 2 bricks is greater than 0.1 kg A random sample of \(n\) bricks is to be taken.
  2. Find the minimum sample size such that the probability of the sample mean being greater than 2 is less than 1\% The bricks are randomly selected and stacked on pallets.
    The weight of an empty pallet, \(E \mathrm {~kg}\), is such that \(E \sim \mathrm {~N} \left( 21.8 , \sqrt { 0.6 } ^ { 2 } \right)\) The random variable \(M\) represents the total weight of a pallet stacked with 500 bricks. The random variable \(T\) represents the total weight of a container of cement.
    Given that \(T\) is independent of \(M\) and that \(T \sim \mathrm {~N} \left( 774 , \sqrt { 1.8 } ^ { 2 } \right)\)
  3. calculate \(\mathrm { P } ( 4 T > 100 + 3 M )\)
Edexcel S3 2018 Specimen Q1
5 marks Easy -1.8
  1. The names of the 720 members of a swimming club are listed alphabetically in the club's membership book. The chairman of the swimming club wishes to select a systematic sample of 40 names. The names are numbered from 001 to 720 and a number between 001 and \(w\) is selected at random. The corresponding name and every \(x\) th name thereafter are included in the sample.
    1. Find the value of \(w\).
    2. Find the value of \(x\).
    3. Write down the probability that the sample includes both the first name and the second name in the club's membership book.
    4. State one advantage and one disadvantage of systematic sampling in this case.
Edexcel S3 2018 Specimen Q2
9 marks Standard +0.3
2. Nine dancers, Adilzhan \(( A )\), Bianca \(( B )\), Chantelle \(( C )\), Lee \(( L )\), Nikki \(( N )\), Ranjit \(( R )\), Sergei \(( S )\), Thuy \(( T )\) and Yana \(( Y )\), perform in a dancing competition. Two judges rank each dancer according to how well they perform. The table below shows the rankings of each judge starting from the dancer with the strongest performance.
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly, test at the 1\% level of significance, whether or not the two judges are generally in agreement.
    Rank123456789
    Judge 1\(S\)\(N\)\(B\)\(C\)\(T\)\(A\)\(Y\)\(R\)\(L\)
    Judge 2\(S\)\(T\)\(N\)\(B\)\(C\)\(Y\)\(L\)\(A\)\(R\)
    1. Calculate Spearman's rank correlation coefficient for these data.
Edexcel S3 2018 Specimen Q3
11 marks Standard +0.3
3. The number of accidents on a particular stretch of motorway was recorded each day for 200 consecutive days. The results are summarised in the following table.
Number of accidents012345
Frequency4757463596
  1. Show that the mean number of accidents per day for these data is 1.6 A motorway supervisor believes that the number of accidents per day on this stretch of motorway can be modelled by a Poisson distribution. She uses the mean found in part (a) to calculate the expected frequencies for this model. Her results are given in the following table.
    Number of accidents012345 or more
    Frequency40.3864.61\(r\)27.5711.03\(s\)
  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 \(10 \%\) level of significance to test the motorway supervisor's belief. Show your working clearly.
Edexcel S3 2018 Specimen Q4
11 marks Standard +0.3
4. A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg
  1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
  2. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg
    \includegraphics[max width=\textwidth, alt={}, center]{0434a6c1-686a-449d-ba16-dbb8e60288e8-15_2258_51_313_36}
Edexcel S3 2018 Specimen Q5
12 marks Standard +0.3
  1. A Head of Department at a large university believes that gender is independent of the grade obtained by students on a Business Foundation course. A random sample was taken of 200 male students and 160 female students who had studied the course.
The results are summarised below.
\cline { 3 - 4 } \multicolumn{2}{c|}{}MaleFemale
\multirow{3}{*}{Grade}Distinction\(18.5 \%\)\(27.5 \%\)
\cline { 2 - 4 }Merit\(63.5 \%\)\(60.0 \%\)
\cline { 2 - 4 }Unsatisfactory\(18.0 \%\)\(12.5 \%\)
Stating your hypotheses clearly, test the Head of Department's belief using a \(5 \%\) level of significance. Show your working clearly.
Edexcel S3 2018 Specimen Q6
13 marks Standard +0.3
6. As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, \(x\) minutes, was recorded and the results are summarised by $$\sum x = 1570 \quad \text { and } \quad \sum x ^ { 2 } = 49467.58$$
  1. Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning. An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, \(y\) minutes, was recorded and the results are summarised as $$\bar { y } = 30.9 \quad \text { and } \quad s _ { y } ^ { 2 } = 3.03$$
  2. Test, at the \(5 \%\) level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly.
  3. Explain the relevance of the Central Limit Theorem to the test in part (b).
  4. State an assumption you have made in carrying out the test in part (b).
Edexcel S3 2018 Specimen Q7
5 marks Standard +0.3
  1. A fair six-sided die is labelled with the numbers \(1,2,3,4,5\) and 6
    (b) Find an approximation for the probability that the mean of the 40 scores is less than 3 \includegraphics[max width=\textwidth, alt={}, center]{0434a6c1-686a-449d-ba16-dbb8e60288e8-24_204_714_237_251}
Edexcel S3 2018 Specimen Q8
9 marks Standard +0.3
8. A factory produces steel sheets whose weights \(X \mathrm {~kg}\), are such that \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) A random sample of these sheets is taken and a \(95 \%\) confidence interval for \(\mu\) is found to be (29.74, 31.86)
  1. Find, to 2 decimal places, the standard error of the mean.
  2. Hence, or otherwise, find a \(90 \%\) confidence interval for \(\mu\) based on the same sample of sheets. Using four different random samples, four \(90 \%\) confidence intervals for \(\mu\) are to be found.
  3. Calculate the probability that at least 3 of these intervals will contain \(\mu\). \section*{8. A factory produces steel sheets whose weights \(X \mathrm { gg }\), are such \(X \sim N ( \mu , \sigma ) ^ { 2 }\)} A. A. A random sample of these sheets is taken and a \(95 \%\) confidence interval for \(\mu\) is found to
    be \(( 29.74,31.86 )\)
    1. Find, to 2 decimal places, the standard error of the mean.
    2. Hence, or otherwise, find a \(90 \%\) confidence interval for \(\mu\) based on the same sample
      of sheets. (3)
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Edexcel S3 Specimen Q1
7 marks Moderate -0.3
  1. A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.
Test, at the \(5 \%\) level of significance, the managing director's claim. State your hypotheses clearly.
Edexcel S3 Specimen Q2
9 marks Standard +0.3
2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.
  1. Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
  2. find the probability that Philip beats James in the race by more than 2 minutes.
Edexcel S3 Specimen Q3
10 marks Moderate -0.3
3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .
  1. Find the probability that \(X\) is within 0.6 mm of \(w\). The same board is measured 16 times and the results are recorded.
  2. Find the probability that the mean of these results is within 0.3 mm of \(w\). Given that the mean of these 16 measurements is 35.6 mm ,
  3. find a 98\% confidence interval for \(w\).
Edexcel S3 Specimen Q4
10 marks Standard +0.3
  1. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b \mathrm {~cm}\), and the depth of a river, \(s \mathrm {~cm}\), at seven positions. The results are shown in the table below.
PositionAB\(C\)D\(E\)\(F\)G
Distance from inner bank \(b \mathrm {~cm}\)100200300400500600700
Depth \(s \mathrm {~cm}\)60758576110120104
  1. Calculate Spearman's rank correlation coefficient between \(b\) and \(s\).
  2. Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a \(1 \%\) level of significance.
Edexcel S3 Specimen Q5
10 marks Standard +0.3
5. A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below.
\backslashbox{Annual income}{Finances}WorseSameBetter
Under £15 00014119
£15000 and above172029
Test, at the \(5 \%\) level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly. \includegraphics[max width=\textwidth, alt={}, center]{304e58fa-eb82-4e2d-83f4-848f3eb461c8-15_2576_1774_141_159}
Edexcel S3 Specimen Q6
12 marks Standard +0.8
6. A total of 228 items are collected from an archaeological site. The distance from the centre of the site is recorded for each item. The results are summarised in the table below.
Distance from the
centre of the site (m)
\(0 - 1\)\(1 - 2\)\(2 - 4\)\(4 - 6\)\(6 - 9\)\(9 - 12\)
Number of items221544375258
Test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a continuous uniform distribution. State your hypotheses clearly.
Edexcel S3 Specimen Q7
17 marks Moderate -0.3
  1. A large company surveyed its staff to investigate the awareness of company policy. The company employs 6000 full-time staff and 4000 part-time staff.
    1. Describe how a stratified sample of 200 staff could be taken.
    2. Explain an advantage of using a stratified sample rather than a simple random sample.
    A random sample of 80 full-time staff and an independent random sample of 80 part-time staff were given a test of policy awareness. The results are summarised in the table below.
    Mean score \(( \bar { x } )\)
    Variance of
    scores \(\left( s ^ { 2 } \right)\)
    Full-time staff5221
    Part-time staff5019
  2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean policy awareness scores for full-time and part-time staff are different.
  3. Explain the significance of the Central Limit Theorem to the test in part (c).
  4. State an assumption you have made in carrying out the test in part (c). After all the staff had completed a training course the 80 full-time staff and the 80 part-time staff were given another test of policy awareness. The value of the test statistic \(z\) was 2.53
  5. Comment on the awareness of company policy for the full-time and part-time staff in light of this result. Use a \(1 \%\) level of significance.
  6. Interpret your answers to part (c) and part (f).
Edexcel S3 2006 January Q1
7 marks Easy -1.2
  1. A school has 15 classes and a sixth form. In each class there are 30 students. In the sixth form there are 150 students. There are equal numbers of boys and girls in each class. There are equal numbers of boys and girls in the sixth form. The head teacher wishes to obtain the opinions of the students about school uniforms.
Explain how the head teacher would take a stratified sample of size 40.
(7)
Edexcel S3 2006 January Q2
9 marks Moderate -0.5
2. A workshop makes two types of electrical resistor. The resistance, \(X\) ohms, of resistors of Type A is such that \(X \sim \mathrm {~N} ( 20,4 )\).
The resistance, \(Y\) ohms, of resistors of Type B is such that \(Y \sim \mathrm {~N} ( 10,0.84 )\).
When a resistor of each type is connected into a circuit, the resistance \(R\) ohms of the circuit is given by \(R = X + Y\) where \(X\) and \(Y\) are independent. Find
  1. \(\mathrm { E } ( R )\),
  2. \(\operatorname { Var } ( R )\),
  3. \(\mathrm { P } ( 28.9 < R < 32.64 )\) (6)
Edexcel S3 2006 January Q3
12 marks Moderate -0.3
3. The drying times of paint can be assumed to be normally distributed. A paint manufacturer paints 10 test areas with a new paint. The following drying times, to the nearest minute, were recorded. $$82 , \quad 98 , \quad 140 , \quad 110 , \quad 90 , \quad 125 , \quad 150 , \quad 130 , \quad 70 , \quad 110 .$$
  1. Calculate unbiased estimates for the mean and the variance of the population of drying times of this paint. Given that the population standard deviation is 25 ,
  2. find a 95\% confidence interval for the mean drying time of this paint. Fifteen similar sets of tests are done and the \(95 \%\) confidence interval is determined for each set.
  3. Estimate the expected number of these 15 intervals that will enclose the true value of the population mean \(\mu\).
Edexcel S3 2006 January Q4
9 marks Moderate -0.3
4. People over the age of 65 are offered an annual flu injection. A health official took a random sample from a list of patients who were over 65 . She recorded their gender and whether or not the offer of an annual flu injection was accepted or rejected. The results are summarised below.
GenderAcceptedRejected
Male170110
Female280140
Using a \(5 \%\) significance level, test whether or not there is an association between gender and acceptance or rejection of an annual flu injection. State your hypotheses clearly.
Edexcel S3 2006 January Q5
13 marks Standard +0.3
5. Upon entering a school, a random sample of eight girls and an independent random sample of eighty boys were given the same examination in mathematics. The girls and boys were then taught in separate classes. After one year, they were all given another common examination in mathematics. The means and standard deviations of the boys' and the girls' marks are shown in the table.
Examination marks
\multirow{2}{*}{}Upon entryAfter 1 year
MeanStandard deviationMeanStandard deviation
Boys5012596
Girls5312626
You may assume that the test results are normally distributed.
  1. Test, at the \(5 \%\) level of significance, whether or not the difference between the means of the boys' and girls' results was significant when they entered school.
  2. Test, at the \(5 \%\) level of significance, whether or not the mean mark of the boys is significantly less than the mean mark of the girls in the 'After 1 year' examination.
  3. Interpret the results found in part (a) and part (b).
Edexcel S3 2006 January Q6
13 marks Standard +0.3
6. An area of grass was sampled by placing a \(1 \mathrm {~m} \times 1 \mathrm {~m}\) square randomly in 100 places. The numbers of daisies in each of the squares were counted. It was decided that the resulting data could be modelled by a Poisson distribution with mean 2. The expected frequencies were calculated using the model. The following table shows the observed and expected frequencies.
Number of daisiesObserved frequencyExpected frequency
0813.53
13227.07
227\(r\)
318\(s\)
4109.02
533.61
611.20
700.34
\(\geq 8\)1\(t\)
  1. Find values for \(r , s\) and \(t\).
  2. Using a \(5 \%\) significance level, test whether or not this Poisson model is suitable. State your hypotheses clearly. An alternative test might have been to estimate the population mean by using the data given.
  3. Explain how this would have affected the test.
    (2)