Questions — Edexcel (10514 questions)

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Edexcel S1 Q4
13 marks Standard +0.3
4. Alan runs on a treadmill each day for as long as he can at 7 miles per hour. The length of time for which he runs is normally distributed with a mean of 21.6 minutes and a standard deviation of 1.8 minutes.
  1. Calculate the probability that on any one day Alan will run for less than 20 minutes.
  2. Estimate the number of times in a ninety-day period that Alan will run for more than 24 minutes.
  3. On a particular day Alan is still running after 22 minutes. Find the probability that he will stop running in the next 2 minutes.
Edexcel S1 Q5
14 marks Easy -1.2
5. In a survey unemployed people were asked how many months it had been, to the nearest month, since they were last employed on a full-time basis. The data collected is summarised in this stem and leaf diagram.
Number of months(2 | 1 means 21 months)Totals
011224446779(11)
102355689( )
21568( )
3079( )
45( )
527(2)
63(1)
70(1)
  1. Write down the values needed to complete the totals column on the stem and leaf diagram.
  2. State the mode of these data.
  3. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
  4. determine if there are any outliers in these data,
  5. draw a box plot representing these data on graph paper,
  6. describe the skewness of these data and suggest a reason for it.
Edexcel S1 Q6
17 marks Easy -1.8
6. In a game two spinners are used. The score on the first spinner is given by the random variable \(A\), which has the following probability distribution:
\(a\)123
\(\mathrm { P } ( A = a )\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
  1. State the name of this distribution.
  2. Write down \(\mathrm { E } ( A )\). The score on the second spinner is given by the random variable \(B\), which has the following probability distribution:
    \(b\)123
    \(\mathrm { P } ( B = b )\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 4 }\)
  3. Find \(\mathrm { E } ( B )\). On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable \(C\).
  4. Show that \(\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }\).
  5. Find the probability distribution of \(C\).
  6. Show that \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).
Edexcel S1 Q1
7 marks Moderate -0.8
  1. A shop recorded the number of pairs of gloves, \(n\), that it sold and the average daytime temperature, \(T ^ { \circ } \mathrm { C }\), for each month over a 12-month period.
The data was then summarised as follows: $$\Sigma T = 124 , \quad \Sigma n = 384 , \quad \Sigma T ^ { 2 } = 1802 , \quad \Sigma n ^ { 2 } = 18518 , \quad \Sigma T n = 2583 .$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Comment on what your value shows and suggest a reason for this.
Edexcel S1 Q2
8 marks Standard +0.3
2. Events \(A\) and \(B\) are independent. Given also that $$\mathrm { P } ( A ) = \frac { 3 } { 4 } \quad \text { and } \quad \mathrm { P } \left( A \cap B ^ { \prime } \right) = \frac { 1 } { 4 }$$ Find
  1. \(\mathrm { P } ( A \cap B )\),
  2. \(\mathrm { P } ( B )\),
  3. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\).
Edexcel S1 Q3
10 marks Easy -1.2
3. The random variable \(X\) is such that $$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$ Find expressions in terms of \(a\) and \(b\) for
  1. \(\mathrm { E } ( 2 X + 3 )\),
  2. \(\quad \operatorname { Var } ( 2 X + 3 )\),
  3. \(\mathrm { E } \left( X ^ { 2 } \right)\).
  4. Show that $$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$
Edexcel S1 Q4
11 marks Standard +0.3
  1. An engineer tested a new material under extreme conditions in a wind tunnel. He recorded the number of microfractures, \(n\), that formed and the wind speed, \(v\) metres per second, for 8 different values of \(v\) with all other conditions remaining constant. He then coded the data using \(x = v - 700\) and \(y = n - 20\) and calculated the following summary statistics.
$$\Sigma x = 100 , \quad \Sigma y = 23 , \quad \Sigma x ^ { 2 } = 215000 , \quad \Sigma x y = 11600 .$$
  1. Find an equation of the regression line of \(y\) on \(x\).
  2. Hence, find an equation of the regression line of \(n\) on \(v\).
  3. Use your regression line to estimate the number of microfractures that would be formed if the material was tested in a wind speed of 900 metres per second with all other conditions remaining constant.
    (2 marks)
Edexcel S1 Q5
12 marks Moderate -0.3
5. An antiques shop recorded the value of items stolen to the nearest pound during each week for a year giving the data in the table below.
Value of goods stolen (£)Number of weeks
0-19931
200-3996
400-5993
600-7994
800-9995
1000-19992
2000-29991
Letting \(x\) represent the mid-point of each group and using the coding \(y = \frac { x - 699.5 } { 200 }\),
  1. find \(\sum\) fy.
  2. estimate to the nearest pound the mean and standard deviation of the value of the goods stolen each week using your value for \(\sum f y\) and \(\sum f y ^ { 2 } = 424\).
    (6 marks)
    The median for these data is \(\pounds 82\).
  3. Explain why the manager of the shop might be reluctant to use either the mean or the median in summarising these data.
    (3 marks)
Edexcel S1 Q6
12 marks Moderate -0.3
6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.
  1. Show that the probability that the first two contestants to be eliminated are both male is \(\frac { 2 } { 15 }\).
  2. Find the probability that more females than males are eliminated in the first three rounds of the game.
  3. Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
    (3 marks)
Edexcel S1 Q7
15 marks Moderate -0.8
7. A cyber-cafe recorded how long each user stayed during one day giving the following results.
Length of stay
(minutes)
\(0 -\)\(30 -\)\(60 -\)\(90 -\)\(120 -\)\(240 -\)\(360 -\)
Number of users153132231720
  1. Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
  2. Find the median and quartiles that this model would predict.
  3. Comment on the suitability of the suggested model in the light of the new results.
Edexcel S2 Q1
4 marks Easy -1.8
  1. Briefly describe the difference between a census and a sample survey.
  2. Illustrate the difference by considering the case of a village council which has to decide whether or not to build a new village hall. Given that the council decides to use a sample survey,
  3. suggest suitable sampling units.
Edexcel S2 Q2
6 marks Moderate -0.3
2. The number of copies of The Statistician that a newsagent sells each week is modelled by a Poisson distribution. On average, he sells 1.5 copies per week.
  1. Find the probability that he sells no copies in a particular week.
  2. If he stocks 5 copies each week, find the probability he will not have enough copies to meet that week's demand.
  3. Find the minimum number of copies that he should stock in order to have at least a \(95 \%\) probability of being able to satisfy the week's demand.
Edexcel S2 Q3
10 marks Standard +0.3
3. A die is rolled 60 times, and results in 16 sixes.
  1. Use a suitable approximation to test, at the \(5 \%\) significance level, whether the probability of scoring a six is \(\frac { 1 } { 6 }\) or not. State your hypotheses clearly.
  2. Describe how you would change the test if you wished to investigate whether the probability of scoring a six is greater than \(\frac { 1 } { 6 }\). Carry out this modified test.
Edexcel S2 Q4
12 marks Standard +0.3
4. A continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l } 0 \\ \frac { 1 } { 84 } \left( x ^ { 2 } - 16 \right) \\ 1 \end{array} \right.$$ $$\begin{aligned} & x < 4 , \\ & 4 \leq x \leq 10 , \\ & x > 10 . \end{aligned}$$
  1. Find the median value of \(X\).
  2. Find the interquartile range for \(X\).
  3. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
  4. Sketch the graph of \(\mathrm { f } ( x )\) and hence write down the mode of \(X\), explaining how you obtain your answer from the graph. \section*{STATISTICS 2 (A) TEST PAPER 1 Page 2}
Edexcel S2 Q5
12 marks Standard +0.3
Lupin seeds are sold in packets of 15 . On average, 9 seeds in a packet are green and 6 are red. Find, to 2 decimal places, the probability that in any particular packet there are
  1. less than 2 red seeds,
  2. more red than green seeds. The seeds from 10 packets are then combined together.
  3. Use a suitable approximation to find the probability that the total number of green seeds is more than 100 .
Edexcel S2 Q6
14 marks Standard +0.3
6. Patients suffering from 'flu are treated with a drug. The number of days, \(t\), that it then takes for them to recover is modelled by the continuous random variable \(T\) with the probability density function $$\begin{array} { l l } \mathrm { f } ( t ) = \frac { 3 t ^ { 2 } ( 4 - t ) } { 64 } & 0 \leq t \leq 4 \\ \mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$
  1. Find the mean and standard deviation of \(T\).
  2. Find the probability that a patient takes more than 3 days to recover.
  3. Two patients are selected at random. Find the probability that they both recover within three days.
  4. Comment on the suitability of the model.
Edexcel S2 Q7
17 marks Standard +0.8
7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per \(\mathrm { cm } ^ { 2 }\). One particular patch, of area \(1 \mathrm {~cm} ^ { 2 }\), is selected at random. Assuming that the number of daisies per \(\mathrm { cm } ^ { 2 }\) has a Poisson distribution,
  1. find the probability that the chosen patch contains
    1. no daisies,
    2. one daisy. Ten such patches are chosen. Using your answers to part (a),
  2. find the probability that the total number of daisies is less than two.
  3. By considering the distribution of daisies over patches of \(10 \mathrm {~cm} ^ { 2 }\), use the Poisson distribution to find the probability that a particular area of \(10 \mathrm {~cm} ^ { 2 }\) contains no more than one daisy.
  4. Compare your answers to parts (b) and (c).
  5. Use a suitable approximation to find the probability that a patch of area \(1 \mathrm {~m} ^ { 2 }\) contains more than 8100 daisies.
Edexcel S2 Q1
4 marks Easy -1.8
Explain what is meant by
  1. a population,
  2. a sampling unit. Suggest suitable sampling frames for surveys of
  3. families who have holidays in Greece,
  4. mothers with children under two years old.
Edexcel S2 Q2
6 marks Easy -1.3
2. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k & 5 \leq x \leq 15 , \\ \mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$
  1. Find \(k\) and specify the cumulative density function \(\mathrm { F } ( x )\).
  2. Write down the value of \(\mathrm { P } ( X < 8 )\).
Edexcel S2 Q3
8 marks Moderate -0.3
3. A coin is tossed 20 times, giving 16 heads.
  1. Test at the \(1 \%\) significance level whether the coin is fair, stating your hypotheses clearly.
  2. Find the critical region for the same test at the \(0.1 \%\) significance level.
Edexcel S2 Q4
9 marks Standard +0.8
4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is \(p\).
  1. Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
  2. Evaluate this probability when \(p = 0.6\).
Edexcel S2 Q5
15 marks Moderate -0.3
5. In a certain school, \(32 \%\) of Year 9 pupils are left-handed. A random sample of 10 Year 9 pupils is chosen.
  1. Find the probability that none are left-handed.
  2. Find the probability that at least two are left-handed.
  3. Use a suitable approximation to find the probability of getting more than 5 but less than 15 left-handed pupils in a group of 35 randomly selected Year 9 pupils.
    Explain what adjustment is necessary when using this approximation. \section*{STATISTICS 2 (A) TEST PAPER 3 Page 2}
Edexcel S2 Q6
15 marks Standard +0.3
A sample of radioactive material decays randomly, with an approximate mean of 1.5 counts per minute.
  1. Name a distribution that would be suitable for modelling the number of counts per minute. Give any parameters required for the model.
  2. Find the probability of at least 4 counts in a randomly chosen minute.
  3. Find the probability of 3 counts or fewer in a random interval lasting 5 minutes. More careful measurements, over 50 one-minute intervals, give the following data for \(x\), the number of counts per minute: $$\sum x = 84 , \quad \sum x ^ { 2 } = 226$$
  4. Decide whether these data support your answer to part (a).
  5. Use the improved data to find probability of exactly two counts in a given one-minute interval.
Edexcel S2 Q7
18 marks Standard +0.3
7. Each day on the way to work, a commuter encounters a similar traffic jam. The length of time, in 10-minute units, spent waiting in the traffic jam is modelled by the random variable \(T\) with the cumulative distribution function: $$\begin{array} { l l } \mathrm { F } ( t ) = 0 & t < 0 , \\ \mathrm {~F} ( t ) = \frac { t ^ { 2 } \left( 3 t ^ { 2 } - 16 t + 24 \right) } { 16 } & 0 \leq t \leq 2 , \\ \mathrm {~F} ( t ) = 1 & t > 2 . \end{array}$$
  1. Show that 0.77 is approximately the median value of \(T\).
  2. Given that he has already waited for 12 minutes, find the probability that he will have to wait another 3 minutes.
  3. Find, and sketch, the probability density function of \(T\).
  4. Hence find the modal value of \(T\).
  5. Comment on the validity of this model.
Edexcel S2 Q1
4 marks Easy -1.8
A random sample is to be taken from the A-level results obtained by the final-year students in a Sixth Form College. Suggest
  1. suitable sampling units,
  2. a suitable sampling frame.
  3. Would it be advisable simply to use the results of all those doing A-level Maths? Explain your answer.