Questions Further Statistics (108 questions)

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OCR Further Statistics 2022 June Q3
8 marks Standard +0.8
3 In this question you must show detailed reasoning. A discrete random variable \(V\) has the following probability distribution, where \(p\) and \(q\) are constants.
\(v\)0123
\(\mathrm { P } ( \mathrm { V } = \mathrm { v } )\)\(p\)\(q\)0.120.2
It is given that \(\mathrm { E } ( V ) = \operatorname { Var } ( V )\). Determine the value of \(p\) and the value of \(q\).
OCR Further Statistics 2018 March Q1
6 marks Standard +0.3
1 The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).
  1. Find
    1. \(\mathrm { P } ( X + Y ) > 40\),
    2. \(\operatorname { Var } ( 2 X - Y )\).
    3. State a necessary assumption for your calculations in part (i) to be valid.
OCR Further Statistics 2018 March Q2
5 marks Challenging +1.2
2 A continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \frac { 1 } { x ^ { 3 } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find \(E ( \sqrt { } X )\).
OCR Further Statistics 2018 March Q3
9 marks Standard +0.8
3 Adila has a pack of 50 cards.
  1. Each of the 50 cards is numbered with a different integer from 1 to 50 . Adila selects 5 cards at random without replacement.
    1. Find the probability that exactly 3 of the 5 cards have numbers which are 10 or less.
    2. Adila arranges the 5 cards in a line in a random order. Find the probability that the 5 cards are arranged in numerically increasing order. 10 of the 50 cards are blue and the rest are green.
    3. Adila randomly selects three sets of 10 cards each, without replacement. The sets are labelled \(A , B\) and \(C\). Given that \(A\) contains 3 blue cards and 7 green cards, find the probability that \(B\) contains exactly 2 blue cards and \(C\) contains exactly 3 blue cards.
OCR Further Statistics 2018 March Q4
9 marks Moderate -0.8
4 Sheena travels to school by bus. She records the number of minutes, \(T\), that her bus is late on each of 32 days. She believes that on average \(T\) is greater than 5, and she carries out a significance test at the \(5 \%\) level.
  1. State a condition needed for a Wilcoxon test to be valid in this case. Assume now that this condition is satisfied.
  2. State an advantage of using a Wilcoxon test rather than a sign test.
  3. Calculate the critical region for the test, in terms of a variable which should be defined.
OCR Further Statistics 2018 March Q5
8 marks Standard +0.3
5 A spinner has 5 edges. Each edge is numbered with a different integer from 1 to 5 . When the spinner is spun, it is equally likely to come to rest on any one of the edges. The spinner is spun 100 times. The number of times on which the spinner comes to rest on the edge numbered 5 is denoted by \(X\).
  1. \(\mathrm { E } ( X )\),
  2. \(\operatorname { Var } ( X )\).
    1. Write down
    2. Use a normal distribution with the same mean and variance as in your answers to part (i) to estimate the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\).
    3. Use the binomial distribution to find exactly the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\). Show the values of all relevant calculations.
OCR Further Statistics 2018 March Q6
10 marks Standard +0.3
6 The captain of a sports team analyses the team's results according to the weather conditions, classified as "sunny" and "not sunny". The frequencies are shown in the following table.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Results
\cline { 3 - 5 } \multicolumn{2}{c|}{}WinDrawLose
\multirow{2}{*}{Weather}Sunny1235
\cline { 2 - 5 }Not sunny81210
  1. Test at the \(5 \%\) significance level whether the team's performances are associated with weather conditions.
  2. (a) Identify the cell that gives the largest contribution to the test statistic.
    (b) Interpret your answer to part (ii)(a).
OCR Further Statistics 2018 March Q7
9 marks Standard +0.3
7 The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } x \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(\mathrm { f } ( x )\) satisfies the conditions for a probability density function.
  2. Find the value of \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 15 } { 16 }\).
OCR Further Statistics 2018 March Q8
11 marks Challenging +1.2
8 At a wine-tasting competition, two judges give marks out of 100 to 7 wines as follows.
Wine\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Judge I86.387.587.688.889.489.990.5
Judge II85.388.182.787.789.089.491.5
  1. A spectator claims that there is a high level of agreement between the rank orders of the marks given by the two judges. Test the spectator's claim at the \(1 \%\) significance level.
  2. A competitor ranks the wines in a random order. The value of Spearman's rank correlation coefficient between the competitor and Judge I is \(r _ { s }\).
    1. Find the probability that \(r _ { s } = 1\).
    2. Show that \(r _ { s }\) cannot take the value \(\frac { 55 } { 56 }\).
OCR Further Statistics 2018 March Q9
8 marks Challenging +1.2
9 The values of a set of bivariate data \(\left( x _ { i } , y _ { i } \right)\) can be summarised by $$n = 50 , \sum x = 1270 , \sum y = 5173 , \sum x ^ { 2 } = 42767 , \sum y ^ { 2 } = 701301 , \sum x y = 173161 .$$ Ten independent observations of \(Y\) are obtained, all corresponding to \(x = 20\). It may be assumed that the variance of \(Y\) is 1.9 , independently of the value of \(x\). Find a \(95 \%\) confidence interval for the mean \(\bar { Y }\) of the 10 observations of \(Y\). \section*{END OF QUESTION PAPER}
OCR Further Statistics 2018 September Q1
4 marks Moderate -0.8
1 An experiment involves releasing a coin on a sloping plane so that it slides down the slope and then slides along a horizontal plane at the bottom of the slope before coming to rest. The angle \(\theta ^ { \circ }\) of the sloping plane is varied, and for each value of \(\theta\), the distance \(d \mathrm {~cm}\) the coin slides on the horizontal plane is recorded. A scatter diagram to illustrate the results of the experiment is shown below, together with the least squares regression line of \(d\) on \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{28c6a0d9-09a6-4743-af0e-fe2e43e256c9-2_639_972_561_548}
  1. State which two of the following correctly describe the variable \(\theta\).
    Controlled variableCorrelation coefficient
    Dependent variableIndependent variable
    Response variableRegression coefficient
    The least squares regression line of \(d\) on \(\theta\) has equation \(d = 1.96 + 0.11 \theta\).
  2. Use the diagram in the Printed Answer Booklet to explain the term "least squares".
  3. State what difference, if any, it would make to the equation of the regression line if \(d\) were measured in inches rather than centimetres. ( 1 inch \(\approx 2.54 \mathrm {~cm}\) ).
OCR Further Statistics 2018 September Q2
7 marks Standard +0.3
2 Shooting stars occur randomly, independently of one another and at a constant average rate of 12.0 per hour. On each of a series of randomly chosen clear nights I look for shooting stars for 20 minutes at a time. A successful night is a night on which I see at least 8 shooting stars in a 20 -minute period.
From tomorrow, I will count the number, \(X\), of nights on which I look for shooting stars, up to and including the first successful night. Find \(\mathrm { E } ( X )\).
OCR Further Statistics 2018 September Q3
7 marks Standard +0.8
3 A discrete random variable \(X\) has the distribution \(\mathrm { U } ( 11 )\).
The mean of 50 observations of \(X\) is denoted by \(\bar { X }\).
Use an approximate method, which should be justified, to find \(\mathrm { P } ( \bar { X } \leqslant 6.10 )\).
OCR Further Statistics 2018 September Q4
9 marks Standard +0.3
4 A survey is carried out into the length of time for which customers wait for a response on a telephone helpline. A statistician who is analysing the results of the survey starts by modelling the waiting time, \(x\) minutes, by an exponential distribution with probability density function (PDF) $$\mathrm { f } ( x ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ \section*{(i) In this question you must show detailed reasoning.} The mean waiting time is found to be 5.0 minutes. Show that \(\lambda = 0.2\).
(ii) Use the model to calculate the probability that a customer has to wait longer than 20 minutes for a response. In practice it is found that no customer waits for more than 15 minutes for a response. The statistician constructs an improved model to incorporate this fact.
(iii) On the diagram in the Printed Answer Booklet, sketch the following, labelling the curves clearly:
  1. the PDF of the model using the exponential distribution,
  2. a possible PDF for the improved model.
OCR Further Statistics 2018 September Q5
8 marks Standard +0.3
5 Hal designs a 4-edged spinner with edges labelled 1, 2, 3 and 4. He intends that the probability that the spinner will land on any edge should be proportional to the number on that edge. He spins the spinner 20 times and on each spin he records the number of the edge on which it lands. The results are shown in the table.
Edge number1234
Frequency3746
Test at the \(10 \%\) significance level whether the results are consistent with the intended probabilities.
OCR Further Statistics 2018 September Q6
10 marks Standard +0.8
6 A bag contains 7 red counters and 5 blue counters.
  1. Fred chooses 4 counters at random, without replacement. Show that the probability that Fred chooses exactly 2 red counters is \(\frac { 14 } { 33 }\).
  2. Lina chooses 4 counters at random from the bag, records whether or not exactly 2 red counters are chosen, and returns the counters to the bag. She carries out this experiment 99 times.
    1. Find the mean of the number of experiments that result in choosing exactly 2 red counters.
    2. Find the variance of the number of experiments that result in choosing exactly 2 red counters.
    3. Alex arranges all 12 counters in a random order in a straight line. A is the event: no two blue counters are next to one another. B is the event: all the blue counters are next to one another. Find \(\mathrm { P } ( A \cup B )\).
OCR Further Statistics 2018 September Q7
11 marks Standard +0.3
7 The table shows the values of 5 observations of bivariate data \(( x , y )\).
\(x\)4.65.96.57.88.3
\(y\)15.610.810.410.19.7
$$n = 5 , \Sigma x = 33.1 , \Sigma y = 56.6 , \Sigma x ^ { 2 } = 227.95 , \Sigma y ^ { 2 } = 664.26 , \Sigma x y = 362.37$$
  1. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
  2. State what this value of \(r\) tells you about a scatter diagram illustrating the data.
  3. Test at the \(5 \%\) significance level whether there is association between \(x\) and \(y\).
  4. State the value of Spearman's rank correlation coefficient \(r _ { s }\) for the data.
  5. State whether \(r , r _ { s }\), or both or neither is changed when the values of \(x\) are replaced by
    1. \(3 x - 2\),
    2. \(\sqrt { x }\).
OCR Further Statistics 2018 September Q8
8 marks Standard +0.3
8 In an experiment to investigate the effect of background music in carrying out work, ten students were each given a task. Five of the students did the task in silence and the other five did the task with background music. The scores on the tasks were as follows.
Silence4346555861
Background music1931385270
  1. Use a Wilcoxon rank-sum test to test at the 10\% level whether the presence of background music affects scores.
  2. A statistician suggests that the experiment is redesigned so that each student takes one task in silence and another task with background music. The differences in the test scores would then be analysed using a paired-sample method. State an advantage in redesigning the experiment in this way.
OCR Further Statistics 2018 September Q9
11 marks Standard +0.3
9 The continuous random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The sum of a random sample of 16 observations of \(C\) is 224.0 .
  1. Find an unbiased estimate of \(\mu\).
  2. It is given that an unbiased estimate of \(\sigma ^ { 2 }\) is 0.24. Find the value of \(\Sigma c ^ { 2 }\). \(D\) is the sum of 10 independent observations of \(C\).
  3. Explain whether \(D\) has a normal distribution. The continuous random variable \(F\) is normally distributed with mean 15.0, and it is known that \(\mathrm { P } ( F < 13.2 ) = 0.115\).
  4. Use the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) to find \(\mathrm { P } ( D + F > 157.0 )\). \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR Further Statistics 2018 December Q1
5 marks Standard +0.8
1 The performance of a piece of music is being recorded. The piece consists of three sections, \(A , B\) and \(C\). The times, in seconds, taken to perform the three sections are normally distributed random variables with the following means and standard deviations.
SectionMeanStandard deviation
\(A\)26413
\(B\)1739
\(C\)26413
  1. Assume first that the times for the three sections are independent. Find the probability that the total length of the performance is greater than 720.0 seconds.
  2. In fact sections \(A\) and \(C\) are musically identical, and the recording is made by using a single performance of section \(A\) twice, together with a performance of section \(B\). In this case find the probability that the total length of the performance is greater than 720.0 seconds.
OCR Further Statistics 2018 December Q2
7 marks Moderate -0.3
2 In a fairground game a competitor scores \(0,1,2\) or 3 with probabilities given in the following table, where \(a\) and \(b\) are constants.
Score0123
Probability\(a\)\(b\)\(b\)\(b\)
The competitor's expected score is 0.9 .
  1. Show that \(b = 0.15\).
  2. Find the variance of the score.
  3. The competitor has to pay \(\pounds 2.50\) to take part, and wins a prize of \(\pounds 2 X\), where \(X\) is the score achieved. Find the expectation of the competitor's loss.
OCR Further Statistics 2018 December Q3
7 marks Standard +0.8
3
  1. Alex places 20 black counters and 8 white counters into a bag. She removes 8 counters at random without replacement. Find the probability that the bag now contains exactly 5 white counters.
  2. Bill arranges 8 blue counters and 4 green counters in a random order in a straight line. Find the probability that exactly three of the green counters are next to one another.
OCR Further Statistics 2018 December Q4
8 marks Moderate -0.3
4 Leyla investigates the number of shoppers who visit a shop between 10.30 am and 11 am on Saturday mornings. She makes the following assumptions.
  • Shoppers visit the shop independently of one another.
  • The average rate at which shoppers visit the shop between these times is constant.
    1. State an appropriate distribution with which Leyla could model the number of shoppers who visit the shop between these times.
Leyla uses this distribution, with mean 14, as her model.
  • Calculate the probability that, between 10.35 am and 10.50 am on a randomly chosen Saturday, at least 10 shoppers visit the shop. Leyla chooses 25 Saturdays at random.
  • Find the expected number of Saturdays, out of 25, on which there are no visitors to the shop between 10.35 am and 10.50 am .
  • In fact on 5 of these Saturdays there were no visitors to the shop between 10.35 am and 10.50 am . Use this fact to comment briefly on the validity of the model that Leyla has used.
    •
    OCR Further Statistics 2018 December Q5
    10 marks Moderate -0.3
    5 The birth rate, \(x\) per thousand members of the population, and the life expectancy at birth, \(y\) years, in 14 randomly selected African countries are given in the table.
    Country\(x\)\(y\)Country\(x\)\(y\)
    Benin4.859.2Mozambique5.454.63
    Cameroon4.754.87Nigeria5.752.29
    Congo4.961.42Senegal5.165.81
    Gambia5.759.83Somalia6.554.88
    Liberia4.760.25Sudan4.463.08
    Malawi5.160.97Uganda5.857.25
    Mauretania4.662.77Zambia5.458.75
    \(n = 14 , \sum x = 72.8 , \sum y = 826 , \sum x ^ { 2 } = 392.96 , \sum y ^ { 2 } = 48924.54 , \sum x y = 4279.16\)
    1. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
    2. State what would be the effect on the value of \(r\) if the birth rate were given per hundred and not per thousand.
    3. Explain what the sign of \(r\) tells you about the relationship between life expectancy and birth rate for these countries.
    4. Test at the \(5 \%\) significance level whether there is correlation between birth rate and life expectancy at birth in African countries.
    5. A researcher wants to estimate the life expectancy at birth in Zimbabwe, where the birth rate is 3.9 per thousand. Explain whether a reliable estimate could be obtained using the regression line of \(y\) on \(x\) for the given data.
    OCR Further Statistics 2018 December Q6
    15 marks Standard +0.3
    6 The reaction times, in milliseconds, of all adult males in a standard experiment have a symmetrical distribution with mean and median both equal to 700 and standard deviation 125. The reaction times of a random sample of 6 international athletes are measured and the results are as follows: \(\begin{array} { l l l l l l } 702 & 631 & 540 & 714 & 575 & 480 \end{array}\) It is required to test whether international athletes have a mean reaction time which is less than 700.
    1. Assume first that the reaction times of international athletes have the distribution \(\mathrm { N } \left( \mu , 125 ^ { 2 } \right)\). Test at the \(5 \%\) significance level whether \(\mu < 700\).
    2. Now assume only that the distribution of the data is symmetrical, but not necessarily normal.
      1. State with a reason why a Wilcoxon test is preferable to a sign test.
      2. Use an appropriate Wilcoxon test at the \(5 \%\) significance level to test whether the median reaction time of international athletes is less than 700 .
    3. Explain why the significance tests in part (a) and part (b)(ii) could produce different results.