Questions — Edexcel (10514 questions)

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Edexcel S2 Q6
19 marks Standard +0.3
6. The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} \frac { 1 } { 16 } x , & 2 \leq x \leq 6 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Find \(\mathrm { E } ( X )\).
  3. Show that \(\operatorname { Var } ( X ) = \frac { 11 } { 9 }\).
  4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\).
  5. Show that the interquartile range of \(X\) is \(2 ( \sqrt { } 7 - \sqrt { 3 } )\). END
Edexcel S3 Q1
6 marks Moderate -0.5
  1. A museum is open to the public for six hours a day from Monday to Friday every week. The number of visitors, \(V\), to the museum on ten randomly chosen days were as follows:
$$\begin{array} { l l l l l l l l l l } 182 & 172 & 113 & 99 & 168 & 183 & 135 & 129 & 150 & 108 \end{array}$$
  1. Calculate an unbiased estimate of the mean of \(V\). Assuming that \(V\) is normally distributed with a variance of 130 ,
  2. find a 95\% confidence interval for the mean of \(V\).
Edexcel S3 Q2
7 marks Easy -1.3
2.
  1. Explain what is meant by a simple random sample.
  2. Explain briefly how you could use a table of random numbers to select a simple random sample of size 12 from a list of the 70 junior members of a tennis club.
  3. Give an example of a situation in which you might choose to take a stratified sample and explain why.
Edexcel S3 Q3
11 marks Standard +0.3
3. The time that a school pupil spends on French homework each week is normally distributed with a mean of 55 minutes and a standard deviation of 10 minutes. The time that this pupil spends on English homework each week is normally distributed with a mean of 1 hour 30 minutes and a standard deviation of 18 minutes. Find the probability that in a randomly chosen week
  1. the pupil spends more than 2 hours in total doing French and English homework,
  2. the pupil spends more than twice as long doing English homework as he spends doing French homework.
    (6 marks)
Edexcel S3 Q4
11 marks Standard +0.3
4. A group of 40 males and 40 females were asked which of three "Reality TV" shows they liked most - Watched, Stranded or One-2-Win. The results were as follows:
\cline { 2 - 4 } \multicolumn{1}{c|}{}WatchedStrandedOne-2-Win
Males21613
Females151015
Stating your hypotheses clearly, test at the \(10 \%\) level whether or not there is a significant difference in the preferences of males and females.
Edexcel S3 Q5
12 marks Standard +0.3
5. A marathon runner believes that she is more likely to win a medal at her national championships the higher the temperature is on the day of the race. She records the temperature at the start of each of eight races against fields of a similar standard and her finishing position in each race. Her results are shown in the table below.
Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\)1691157211215
Finishing position215519104611
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Using a 5\% level of significance and stating your hypotheses clearly, interpret your result. Another runner suggests that she should use her time in each race instead of her finishing position and calculate the product moment correlation coefficient for the data.
  3. Comment on this suggestion.
Edexcel S3 Q6
12 marks Standard +0.3
6. The weight of a particular electrical component is normally distributed with a mean of 46.7 grams and a variance of 1.8 grams \(^ { 2 }\). The component is sold in boxes of 12 .
  1. State the distribution of the mean weight of the components in one box.
  2. Find the probability that the mean weight of the components in a randomly chosen box is more than 47 grams.
    (3 marks)
    After a break in production the component manufacturer wishes to find out if the mean weight of the components has changed. A random sample of 30 components is found to have a mean weight of 46.5 grams.
  3. Assuming that the variance of the weight of the components is unchanged, test at the \(5 \%\) level of significance if there has been any change in the mean weight of the components.
    (7 marks)
Edexcel S3 Q7
16 marks Standard +0.3
7. A student collects data on whether competitors in local tennis tournaments are right, or left-handed. The table below shows the number of left-handed players who reached the last 16 for fifty tournaments.
No. of Left-handed Players01234\(\geq 5\)
No. of Tournaments412181150
The student believes that a binomial distribution with \(n = 16\) and \(p = 0.1\) could be a suitable model for these data.
  1. Stating your hypotheses clearly test the student's model at the \(5 \%\) level of significance.
    (13 marks)
    To improve the model the student decides to estimate \(p\) using the data in the table. Using this value of \(p\) to calculate expected frequencies the student had 5 classes after combining and calculated that \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 2.127\)
  2. Test at the \(5 \%\) level of significance whether or not the binomial distribution is a suitable model for the number of left-handed players who reach the last 16 in local tennis tournaments. \section*{END}
Edexcel S3 Q1
4 marks Easy -1.2
  1. A Veterinary Surgeon wishes to survey a stratified sample of size 100 from those people who have pets registered at her surgery. The list below shows the strata to be used and the number in each group.
  • people who own just dogs - 165 ,
  • people who own just cats - 140 ,
  • people who own just small mammals - 105,
  • others, including those who own more than one type of pet - 90 .
    1. Find how many members of each group should be included in the sample.
    2. Give two advantages of using stratified sampling.
Edexcel S3 Q2
9 marks Standard +0.3
  1. A psychologist is investigating the numbers people choose when asked to pick a number at random in a given interval. He finds that when asked to pick a number between 0 and 100 people are less likely to pick certain numbers, such as multiples of ten. He believes, however that if people are asked to pick an odd number between 0 and 100 they are equally likely to pick a number ending in any of the digits \(1,3,5,7\) or 9 .
To test this theory he asks 80 people to pick an odd number between 0 and 100 and records the last digit of the numbers chosen. His results are shown in the table below.
Last Digit13579
Frequency1620141713
Stating your hypotheses clearly and using a 10\% level of significance test the psychologist's theory.
(9 marks)
Edexcel S3 Q3
10 marks Standard +0.3
3. A clothes manufacturer wishes to find out if adult females have become taller on average since twenty years ago when their mean height was 5 ft 6 inches. Studies over time have shown that the standard deviation of the height of adult females has been fairly constant at 2.3 inches. The manager wishes to test if the mean height is now more than 5 ft 6 inches and takes a sample of 150 adult females.
  1. Stating your hypotheses clearly, find the critical region for the mean height of the sample for a test at the \(5 \%\) level of significance. The total height of the females in the sample is 832 ft .
  2. Carry out the test making your conclusion clear.
Edexcel S3 Q4
12 marks Standard +0.3
4. For a project a student collects data on engine size and sales over a period of time for the models of cars made by one particular manufacturer. Her results are shown in the table below.
Engine Capacity
(litres)
1.11.31.62.12.42.62.83.0
Sales527632840619350425487401
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is any evidence of correlation.
  3. Explain why it is more appropriate to use Spearman's rank correlation coefficient for this test than the product moment correlation coefficient.
    (2 marks)
Edexcel S3 Q5
12 marks Standard +0.3
5. A child is playing with a set of red and blue wooden cubes. The side length of the red cubes is normally distributed with a mean of 14.5 cm and a variance of \(16.0 \mathrm {~cm} ^ { 2 }\). The side length of the blue cubes is normally distributed with a mean of 12.2 cm and a variance of \(9.0 \mathrm {~cm} ^ { 2 }\).
  1. Find the probability that a randomly chosen red cube will have a side length of more than 3 cm greater than a randomly chosen blue cube. The child makes two towers, one from 4 red cubes and one from 5 blue cubes. Assuming that the cubes for each colour of tower were chosen at random,
  2. find the probability that the red tower is taller than the blue tower.
  3. Explain why the assumption that the cubes for each tower were chosen at random is unlikely to be realistic.
Edexcel S3 Q6
14 marks Standard +0.3
6. A market researcher recorded the number of adverts for vehicles in each of three categories on ITV, Channel 4 and Channel 5 over a period of time. The results are shown in the table below.
ITVChannel 4Channel 5
Family Saloon693528
Sports Car202818
Off-road Vehicle12228
  1. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the proportion of adverts for each type of vehicle being dependent on the channel.
  2. Suggest a reason for your result in part (a).
Edexcel S3 Q7
14 marks Standard +0.3
7.
  1. Briefly state the central limit theorem. A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.
  2. Write down the distribution of the number of sixes obtained when the ten dice are thrown.
  3. Find the mean and variance of this distribution. The student throws the ten dice 100 times, recording the number of sixes showing each time.
  4. Find the probability that the mean number of sixes obtained is more than 1.8
Edexcel S3 Q1
5 marks Easy -1.8
A personnel manager has details on all company employees and wishes to consult a sample of them on a possible change to the company's hours of business. She decides to take a stratified sample based on different age groups.
  1. Give one advantage of using stratified sampling in this situation. The manager needs to select a sample of size 10 , without replacement, from a list of 65 employees aged 16 to 25 . She numbers these employees from 01 to 65 in alphabetical order and uses the table of random numbers given in the formula book. She starts with the top of the sixth two-digit column and works down. The first two numbers she writes down are 30 and 47.
  2. Find the other eight numbers in the sample.
  3. Suggest another factor that might be useful to consider in deciding on the strata.
    (1 mark)
Edexcel S3 Q2
6 marks Standard +0.3
2. A Geography teacher is interested in the link between mathematical ability and the ability to visualise three-dimensional situations. He gives a group of 15 students a test and records each student's score, \(m\), on the mathematics questions and each student's score, \(v\), on the visiospatial questions. He calculates the following summary statistics: $$S _ { m m } = 3747.73 , \quad S _ { v v } = 2791.33 , \quad S _ { m v } = 2564.33$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Stating your hypotheses clearly and using a \(5 \%\) level of significance test the theory that students who are good at Mathematics tend to have better visio-spatial awareness.
    (4 marks)
Edexcel S3 Q3
9 marks Standard +0.3
3. A random variable \(X\) is distributed normally with a standard deviation of 6.8 Sixty observations of \(X\) are made and found to have a mean of 31.4
  1. Find a 90\% confidence interval for the mean of \(X\).
  2. How many observations of \(X\) would be needed in order to obtain a \(90 \%\) confidence interval for the mean of \(X\) with a width of less than 1.5
    (5 marks)
Edexcel S3 Q4
12 marks Standard +0.3
4. A paranormal investigator invites couples who believe they have a telepathic connection to participate in a trial. With each couple one person looks at a card with one of five shapes on it and the other person says which of the shapes they think it is. This is repeated six times and the number of correct answers recorded. The results from 120 couples are given below.
Number Correct0123456
Number of Couples2656288200
The investigator wishes to see if this data fits a binomial distribution with parameters \(n = 6\) and \(p = \frac { 1 } { 5 }\) and calculates to 2 decimal places the expected frequencies given below.
Number Correct0123456
Expected Frequency9.831.840.180.01
  1. Find the other expected frequencies.
  2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not the distribution is an appropriate model.
  3. Comment on your findings.
Edexcel S3 Q5
13 marks Standard +0.3
5. A Policy Unit wished to find out whether attitudes to the European Union varied with age. It conducted a survey asking 200 individuals to which of three age groups they belonged and whether they regarded themselves as generally pro-Europe or Eurosceptic. The results are shown in the table below.
\cline { 2 - 3 } \multicolumn{1}{c|}{}Pro-EuropeEurosceptic
\(18 - 34\) years4321
\(35 - 54\) years3036
55 years or over2743
  1. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether attitudes to Europe are associated with age.
    (11 marks)
    The survey also asked people if they voted at the last election. When the above test was repeated using only the results from those who had voted a value of 4.872 was calculated for \(\sum \frac { ( O - E ) ^ { 2 } } { E }\). No classes were combined.
  2. Find if this value leads to a different result.
Edexcel S3 Q6
14 marks Challenging +1.2
6. Four swimmers, \(A , B , C\) and \(D\), are to be used in a \(4 \times 100\) metres freestyle relay. The time for each swimmer to complete a leg follows a normal distribution. The mean and standard deviation, in seconds, of the time for each swimmer to complete a leg and the order in which they are to swim are shown in the table below.
meanstandard deviation
\(1 ^ { \text {st } }\) leg \(- A\)63.11.2
\(2 ^ { \text {nd } }\) leg \(- B\)65.71.5
\(3 ^ { \text {rd } } \operatorname { leg } - C\)65.41.8
\(4 ^ { \text {th } }\) leg - \(D\)62.50.9
  1. Find the probability that the total time for first two legs is less than the total time for the last two.
    (6 marks)
    The total time for another team to complete this relay is normally distributed with a mean of 259.0 seconds and a standard deviation of 3.4 seconds. The two teams are to compete over four races.
  2. Find the probability that the first team wins all four races, assuming that the team's performances are not affected by previous results.
    (8 marks)
Edexcel S3 Q7
16 marks Standard +0.3
7. A telephone company believes that, for young people, the average length of a telephone call on a land line is longer than on a mobile, due to the difference in price. The company collected data on the time, \(t\) minutes, of 500 calls made by young people on mobiles and the data is summarised by $$\Sigma t = 7335 , \quad \Sigma t ^ { 2 } = 172040 .$$
  1. Calculate unbiased estimates of the mean and variance of \(t\). For 200 calls made on land lines by the same young people, unbiased estimates of the mean and variance of the call length were 15.9 minutes and 108.5 minutes \({ } ^ { 2 }\) respectively.
  2. Stating your hypotheses clearly, test at the \(5 \%\) level whether or not there is evidence that longer calls are made on land lines than on mobiles.
    (9 marks)
  3. Explain the importance of the central limit theorem in carrying out the test in part (b).
Edexcel S4 2006 January Q1
8 marks Standard +0.3
  1. A diabetic patient records her blood glucose readings in \(\mathrm { mmol } / \mathrm { l }\) at random times of day over several days. Her readings are given below.
$$\begin{array} { l l l l l l l } 5.3 & 5.7 & 8.4 & 8.7 & 6.3 & 8.0 & 7.2 \end{array}$$ Assuming that the blood glucose readings are normally distributed calculate
  1. an unbiased estimate for the variance \(\sigma ^ { 2 }\) of the blood glucose readings,
  2. a \(90 \%\) confidence interval for the variance \(\sigma ^ { 2 }\) of blood glucose readings.
  3. State whether or not the confidence interval supports the assertion that \(\sigma = 0.9\). Give a reason for your answer.
Edexcel S4 2006 January Q2
13 marks Standard +0.3
2. (a) Define
  1. a Type I error,
  2. a Type II error. A manufacturer sells socks in boxes of 50 .
    The mean number of faulty socks per box is 7.5 . In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.
    (b) (i) Assuming that the number of faulty socks per box follows a binomial distribution derive a critical region needed to test whether or not there is evidence that the new machine has reduced the mean number of faulty socks per box. Use a \(5 \%\) significance level.
  3. Stating your hypotheses clearly, carry out the test in part (i).
    (c) Find the probability of the Type I error for this test.
    (d) Given that the true mean number of faulty socks per box on the new machine is 5 , calculate the probability of a Type II error for this test.
    (e) Explain what would have been the effect of changing the significance level for the test in part (b) to \(2 \frac { 1 } { 2 } \%\).
Edexcel S4 2006 January Q3
7 marks Standard +0.3
3. A population has mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of size 3 is to be taken from this population and \(\bar { X }\) denotes its sample mean. A second random sample of size 4 is to be taken from this population and \(\bar { Y }\) denotes its sample mean.
  1. Show that unbiased estimators for \(\mu\) are given by
    1. \(\hat { \mu } _ { 1 } = \frac { 1 } { 3 } \bar { X } + \frac { 2 } { 3 } \bar { Y }\),
    2. \(\hat { \mu } _ { 2 } = \frac { 5 \bar { X } + 4 \bar { Y } } { 9 }\).
  2. Calculate Var \(\left( \hat { \mu } _ { 1 } \right)\)
  3. Given that \(\operatorname { Var } \left( \hat { \mu } _ { 2 } \right) = \frac { 37 } { 243 } \sigma ^ { 2 }\), state, giving a reason, which of these two estimators should be
    used. used.