Questions S3 (621 questions)

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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)
      \includegraphics[max width=\textwidth, alt={}]{0434a6c1-686a-449d-ba16-dbb8e60288e8-28_2646_1824_105_123}
      VIIIV SIHI NI JIIIM ION OCVIUV SIHI NI JIIAM ION OOVEXV SIHI NI JIIIM IONOO
      VIIIV SIHI NI IIIYM ION OCVIIV SIHI NI JIIIM ION OCVI4V SIHI NI JIIIM I ION OO
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)
Edexcel S3 2006 January Q7
12 marks Standard +0.3
7. The numbers of deaths from pneumoconiosis and lung cancer in a developing country are given in the table.
Age group (years)20-2930-3940-4950-5960-6970 and over
Deaths from pneumoconiosis (1000s)12.55.918.519.431.231.0
Deaths from lung cancer (1000s)3.79.010.219.013.018.0
The correlation between the number of deaths in the different age groups for each disease is to be investigated.
  1. Give one reason why Spearman's rank correlation coefficient should be used.
  2. Calculate Spearman's rank correlation coefficient for these data.
  3. Use a suitable test, at the \(5 \%\) significance level, to interpret your result. State your hypotheses clearly.
    (5)
Edexcel S3 2003 June Q1
8 marks Easy -1.8
  1. Explain how to obtain a sample from a population using
    1. stratified sampling,
    2. quota sampling.
    Give one advantage and one disadvantage of each sampling method.
Edexcel S3 2003 June Q2
8 marks Moderate -0.5
2. A random sample of 30 apples was taken from a batch. The mean weight of the sample was 124 g with standard deviation 20 g .
  1. Find a \(99 \%\) confidence interval for the mean weight \(\mu\) grams of the population of apples. Write down any assumptions you made in your calculations. Given that the actual value of \(\mu\) is 140 ,
  2. state, with a reason, what you can conclude about the sample of 30 apples.
Edexcel S3 2003 June Q3
9 marks Moderate -0.5
3. Given the random variables \(X \sim \mathrm {~N} ( 20,5 )\) and \(Y \sim \mathrm {~N} ( 10,4 )\) where \(X\) and \(Y\) are independent, find
  1. \(\mathrm { E } ( X - Y )\),
  2. \(\operatorname { Var } ( X - Y )\),
  3. \(\mathrm { P } ( 13 < X - Y < 16 )\).
Edexcel S3 2003 June Q4
11 marks Moderate -0.3
4. A new drug to treat the common cold was used with a randomly selected group of 100 volunteers. Each was given the drug and their health was monitored to see if they caught a cold. A randomly selected control group of 100 volunteers was treated with a dummy pill. The results are shown in the table below.
\cline { 2 - 3 } \multicolumn{1}{c|}{}ColdNo cold
Drug3466
Dummy pill4555
Using a \(5 \%\) significance level, test whether or not the chance of catching a cold is affected by taking the new drug. State your hypotheses clearly.
Edexcel S3 2003 June Q5
11 marks Standard +0.3
5. A scientist monitored the levels of river pollution near a factory. Before the factory was closed down she took 100 random samples of water from different parts of the river and found an average weight of pollutants of \(10 \mathrm { mg } \mathrm { l } ^ { - 1 }\) with a standard deviation of \(2.64 \mathrm { mg } \mathrm { l } ^ { - 1 }\). After the factory was closed down the scientist collected a further 120 random samples and found that they contained \(8 \mathrm { mg } \mathrm { l } ^ { - 1 }\) of pollutants on average with a standard deviation of \(1.94 \mathrm { mg } \mathrm { l } ^ { - 1 }\). Test, at the \(5 \%\) level of significance, whether or not the mean river pollution fell after the factory closed down.