Questions — Edexcel (10514 questions)

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Edexcel S3 Q4
11 marks Standard +0.3
A hospital administrator is assessing staffing needs for its Accident and Emergency Department at different times of day. The administrator already has data on the number of admissions at different times of day but needs to know if the proportion of the cases that are serious remains constant. Staff are asked to assess whether each person arriving at Accident and Emergency has a "minor" or "serious" problem and the results for three different time periods are shown below.
MinorSerious
8 a.m. – 6 p.m.4511
6 p.m. – 2 a.m.4922
2 a.m. – 8 a.m.147
Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of the proportion of serious injuries being different at different times of day. [11]
Edexcel S3 Q5
12 marks Standard +0.3
In a competition, a wine-enthusiast has to rank ten bottles of wine, \(A\) to \(J\), in order starting with the one he thinks is the most expensive. The table below shows his rankings and the actual order according to price.
Rank12345678910
Enthusiast\(D\)\(C\)\(J\)\(A\)\(G\)\(F\)\(B\)\(E\)\(I\)\(H\)
Price\(A\)\(C\)\(D\)\(H\)\(J\)\(B\)\(F\)\(I\)\(G\)\(E\)
  1. Calculate Spearman's rank correlation coefficient for these data. [6]
  2. Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of positive correlation. [4]
  3. Explain briefly how you would have been able to carry out the test if bottles \(B\) and \(F\) had the same price. [2]
Edexcel S3 Q6
13 marks Standard +0.3
A researcher collects data on the height of boys aged between nine and nine and-a-half years and their diet. The data on the height, \(V\) cm, of the 80 boys who had always eaten a vegetarian diet is summarised by $$\Sigma V = 10\,367, \quad \Sigma V^2 = 1\,350\,314.$$
  1. Calculate unbiased estimates of the mean and variance of \(V\). [5]
The researcher calculates unbiased estimates of the mean and variance of the height of boys whose diet has included meat from a sample of size 280, giving values of 130.5 cm and 96.24 cm\(^2\) respectively.
  1. Stating your hypotheses clearly, test at the 1% level whether or not there is a significant difference in the heights of boys of this age according to whether or not they have a vegetarian diet. [8]
Edexcel S3 Q7
13 marks Standard +0.8
An examiner believes that once she has marked the first 20 papers the time it takes her to mark one paper for a particular exam follows a Normal distribution. Having already marked more than 20 papers for each of the \(P1\), \(M1\) and \(S1\) modules set one summer, the mean and standard deviation, in seconds, of the time it takes her to mark a paper for each module are as shown in the table below.
MeanStandard Deviation
\(P1\)25217
\(M1\)31442
\(S1\)28429
  1. Find the probability that the difference in the time it takes her to mark two randomly chosen \(P1\) papers is less than 5 seconds. [6]
  2. Find the probability that it takes her less than 10 hours to mark 45 \(M1\) and 80 \(S1\) papers. [7]
Edexcel S3 Q1
5 marks Easy -1.8
A researcher wishes to take a sample of size 9, without replacement, from a list of 72 people involved in the trial of a new computer keyboard. She numbers the people from 01 to 72 and uses the table of random numbers given in the formula book. She starts with the left-hand side of the sixth row of the table and works across the row. The first two numbers she writes down are 56 and 32.
  1. Find the other six numbers in the sample. [3 marks]
  2. Give one advantage and one disadvantage of using random numbers when taking a sample. [2 marks]
Edexcel S3 Q2
6 marks Moderate -0.8
The length of time that registered customers spend on each visit to a supermarket's website is normally distributed with a mean of 28.5 minutes and a standard deviation of 7.2 minutes. Eight visitors to the site are selected at random and the length of time, \(T\) minutes, that each stays is recorded.
  1. Write down the distribution of \(\overline{T}\), the mean time spent at the site by these eight visitors. [2 marks]
  2. Find \(P(25 < \overline{T} < 30)\). [4 marks]
Edexcel S3 Q3
7 marks Standard +0.3
The discrete random variable \(X\) has the probability distribution given below.
\(x\)247\(k\)
\(P(X = x)\)0.050.150.30.5
  1. Find the mean of \(X\) in terms of \(k\). [2 marks]
  2. Find the bias in using \((2\overline{X} - 5)\) as an estimator of \(k\). [3 marks]
Fifty observations of \(X\) were made giving a sample mean of 8.34 correct to 3 significant figures.
  1. Calculate an unbiased estimate of \(k\). [2 marks]
Edexcel S3 Q4
7 marks Standard +0.8
The mass of waste in filled large dustbin bags is normally distributed with a mean of 6.8 kg and a standard deviation of 1.5 kg. The mass of waste in filled small dustbin bags is normally distributed with a mean of 3.2 kg and a standard deviation of 0.6 kg. One week there are 8 large and 3 small dustbin bags left for collection outside a block of flats. Find the probability that this waste has a total mass of more than 70 kg. [7 marks]
Edexcel S3 Q5
8 marks Standard +0.3
For a project, a student is investigating whether more athletic individuals have better hand-eye coordination. He records the time it takes a number of students to complete a task testing coordination skills and notes whether or not they play for a school sports team. His results are as follows:
Number of StudentsMeanStandard Deviation
In a School Team5032.8 s4.6 s
Not in a Team19035.1 s8.0 s
Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence that those who play in a school team complete the task more quickly on average. [8 marks]
Edexcel S3 Q6
11 marks Standard +0.3
Two schools in the same town advertise at the same time for new heads of English and History departments. The number of applicants for each post are shown in the table below.
EnglishHistory
Highfield School3214
Rowntree School4826
Stating your hypotheses clearly, test at the 10\% level of significance whether or not there is evidence of the proportion of applicants for each job being different in the two schools. [11 marks]
Edexcel S3 Q7
11 marks Standard +0.3
A sports scientist wishes to examine the link between resting pulse and fitness. He records the resting pulse, \(p\), of 20 volunteers and the length of time, \(t\) minutes, that each one can run comfortably at 4 metres per second on a treadmill. The results are summarised by $$\Sigma p = 1176, \quad \Sigma t = 511, \quad \Sigma p^2 = 70932, \quad \Sigma t^2 = 19213, \quad \Sigma pt = 27188.$$
  1. Calculate the product moment correlation coefficient for these data. [5 marks]
  2. Stating your hypotheses clearly, test at the 1\% level of significance whether there is evidence of people with a lower resting pulse having a higher level of fitness as measured by the test. [4 marks]
  3. State an assumption necessary to carry out the test in part (b) and comment on its validity in this case. [2 marks]
Edexcel S3 Q8
20 marks Standard +0.3
A physicist believes that the number of particles emitted by a radioactive source with a long half-life can be modelled by a Poisson distribution. She records the number of particles emitted in 80 successive 5-minute periods and her results are shown in the table below.
No. of Particles012345 or more
No. of Intervals233214830
  1. Comment on the suitability of a Poisson distribution for this situation. [3 marks]
  2. Show that an unbiased estimate of the mean number of particles emitted in a 5-minute period is 1.2 and find an unbiased estimate of the variance. [5 marks]
  3. Explain how your answers to part (b) support the fitting of a Poisson distribution. [1 mark]
  4. Stating your hypotheses clearly and using a 5\% level of significance, test whether or not these data can be modelled by a Poisson distribution. [11 marks]
Edexcel S3 Q1
5 marks Easy -1.2
A charity has 240 volunteers and wishes to consult a sample of them of size 20.
  1. Explain briefly how a systematic sample can be taken using random numbers. [3]
  2. Give one advantage and one disadvantage of using systematic sampling compared with simple random sampling. [2]
Edexcel S3 Q2
7 marks Standard +0.3
A teacher gives each student in his class a list of 30 numbers. All the numbers have been generated at random by a computer from a normal distribution with a fixed mean and variance. The teacher tells the class that the variance of the distribution is 25 and asks each of them to calculate a 95\% confidence interval based on their list of numbers. The sum of the numbers given to one student is 1419.
  1. Find the confidence interval that should be obtained by this student. [5]
Assuming that all the students calculate their confidence intervals correctly,
  1. state the proportion of the students you would expect to have a confidence interval that includes the true mean of the distribution, [1]
  2. explain why the probability of any one student's confidence interval including the true mean is not 0.95 [1]
Edexcel S3 Q3
10 marks Standard +0.3
A newly promoted manager is present when an experienced manager interviews six candidates, \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\) for a job. Both managers rank the candidates in order of preference, starting with the best candidate, giving the following lists: Experienced Manager: \(B\) \(F\) \(A\) \(C\) \(E\) \(D\) New Manager: \(F\) \(C\) \(B\) \(D\) \(E\) \(A\)
  1. Calculate Spearman's rank correlation coefficient for these data. [5]
  2. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence of positive correlation. [4]
  3. Comment on whether the new manager needs training in the assessment of candidates at interview. [1]
Edexcel S3 Q4
10 marks Standard +0.3
A student collected data on the number of text messages, \(t\), sent by 30 students in her year group in the previous week. Her results are summarised as follows: \(\Sigma t = 1039\), \(\Sigma t^2 = 65393\).
  1. Calculate unbiased estimates of the mean and variance of the number of text messages sent by these students per week. [4]
Another student collected similar data for 20 different students and calculated unbiased estimates of the mean and variance of 32.0 and 963.4 respectively.
  1. Calculate unbiased estimates of the mean and variance for the combined sample of 50 students. [6]
Edexcel S3 Q5
11 marks Standard +0.8
An organic farm produces eggs which it sells through a local shop. The weight of the eggs produced on the farm are normally distributed with a mean of 55 grams and a standard deviation of 3.9 grams.
  1. Find the probability that two of the farm's eggs chosen at random differ in weight by more than 4 grams. [5]
The farm sells boxes of six eggs selected at random. The weight of the boxes used are normally distributed with a mean of 28 grams and a standard deviation of 1.2 grams.
  1. Find the probability that a randomly chosen box with six eggs in weighs less than 350 grams. [6]
Edexcel S3 Q6
15 marks Standard +0.3
A survey found that of the 320 people questioned who had passed their driving test aged under twenty-five, 104 had been involved in an accident in the two years following their test. Of the 80 people in the survey who were aged twenty-five or over when they passed their test, 16 had been involved in an accident in the following two years.
  1. Draw up a contingency table showing this information. [2]
It is desired to test whether the proportion of drivers having accidents within two years of passing their test is different for those who were aged under twenty-five at the time of passing their test than for those aged twenty-five or over.
    1. Stating your hypotheses clearly, carry out the test at the 5\% level of significance.
    2. Explain clearly why there is only one degree of freedom. [11]
It is found that 12 people who were aged under twenty-five when they took their test and had been involved in an accident in the following two years had been omitted from the information given.
  1. Explain why you do not need to repeat the calculation to know the correct result of the test. [2]
Edexcel S3 Q7
17 marks Standard +0.3
A shoe manufacturer sees a report from another country stating that the length of adult male feet is normally distributed with a mean of 22.4 cm and a standard deviation of 2.8 cm. The manufacturer wishes to see if this model is appropriate for his customers and collects data on the length, correct to the nearest cm, of the right foot of a random sample of 200 males giving the following results:
Length (cm)\(\leq 18\)\(19 - 21\)\(22 - 24\)\(25 - 27\)\(\geq 28\)
No. of Men2448694118
The expected frequencies for the \(\leq 18\) and \(19 - 21\) groups are calculated as 16.46 and 58.44 respectively, correct to 2 decimal places.
  1. Calculate expected frequencies for the other three classes. [7]
  2. Stating your hypotheses clearly, test at the 10\% level of significance whether or not this data can be modelled by the distribution N(22.4, 2.8²). [7]
The manufacturer wishes to refine the model by not assuming a mean and standard deviation.
  1. Explain briefly how the manufacturer should proceed. [3]
Edexcel S4 Q1
6 marks Standard +0.3
A beach is divided into two areas \(A\) and \(B\). A random sample of pebbles is taken from each of the two areas and the length of each pebble is measured. A sample of size 26 is taken from area \(A\) and the unbiased estimate for the population variance is \(s_A^2 = 0.495 \text{ mm}^2\). A sample of size 25 is taken from area \(B\) and the unbiased estimate for the population variance is \(s_B^2 = 1.04 \text{ mm}^2\).
  1. Stating your hypotheses clearly test, at the 10\% significance level, whether or not there is a difference in variability of pebble length between area \(A\) and area \(B\). [5]
  2. State the assumption you have made about the populations of pebble lengths in order to carry out this test. [1]
Edexcel S4 Q2
9 marks Standard +0.3
A random sample of 10 mustard plants had the following heights, in mm, after 4 days growth. 5.0, 4.5, 4.8, 5.2, 4.3, 5.1, 5.2, 4.9, 5.1, 5.0 Those grown previously had a mean height of 5.1 mm after 4 days. Using a 2.5\% significance level, test whether or not the mean height of these plants is less than that of those grown previously. (You may assume that the height of mustard plants after 4 days follows a normal distribution.) [9]
Edexcel S4 Q3
9 marks Standard +0.8
A train company claims that the probability \(p\) of one of its trains arriving late is 10\%. A regular traveller sets up the hypothesis \(H_0: p = 0.1\) and decides that the probability is greater than 10\% and decides to test this by randomly selecting 12 trains and recording the number \(X\) of trains that were late. The traveller sets up the hypotheses \(H_0: p = 0.1\) and \(H_1: p > 0.1\) and decides to reject \(H_0\) if \(x \ge 2\).
  1. Find the size of the test. [1]
  2. Show that the power function of the test is $$1 - (1 - p)^{10}(1 + 10p + 55p^2).$$ [4]
  3. Calculate the power of the test when
    1. \(p = 0.2\),
    2. \(p = 0.6\). [3]
  4. Comment on your results from part (c). [1]
Edexcel S4 Q4
9 marks Standard +0.8
A random sample of 15 tomatoes is taken and the weight \(x\) grams of each tomato is found. The results are summarised by \(\sum x = 208\) and \(\sum x^2 = 2962\).
  1. Assuming that the weights of the tomatoes are normally distributed, calculate the 90\% confidence interval for the variance \(\sigma^2\) of the weights of the tomatoes. [7]
  2. State with a reason whether or not the confidence interval supports the assertion \(\sigma^2 = 3\). [2]
Edexcel S4 Q5
11 marks Standard +0.3
  1. Define
    1. a Type I error,
    2. a Type II error. [2]
A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.
  1. [(b)] Assuming that the number of chicks reared per year follows a Poisson distribution test, at the 5\% significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly. [4]
  2. Calculate the probability of the Type I error for this test. [3]
  3. Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8, calculate the probability of a Type II error. [2]
Edexcel S4 Q6
14 marks Standard +0.3
A random sample of three independent variables \(X_1, X_2\) and \(X_3\) is taken from a distribution with mean \(\mu\) and variance \(\sigma^2\).
  1. Show that \(\frac{1}{3}X_1 + \frac{1}{3}X_2 + \frac{1}{3}X_3\) is an unbiased estimator for \(\mu\). [3]
An unbiased estimator for \(\mu\) is given by \(\hat{\mu} = aX_1 + bX_2\) where \(a\) and \(b\) are constants.
  1. [(b)] Show that Var(\(\hat{\mu}\)) = \((2a^2 - 2a + 1)\sigma^2\). [6]
  2. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat{\mu}\) has minimum variance. [5]