Questions — OCR S1 (160 questions)

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OCR S1 2005 January Q1
1 The scatter diagrams below illustrate three sets of bivariate data, \(A , B\) and \(C\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c0a4ca-da0a-4c74-b8b1-bac4fd3f2487-2_440_428_360_317} \captionsetup{labelformat=empty} \caption{Set \(A\)}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c0a4ca-da0a-4c74-b8b1-bac4fd3f2487-2_440_426_360_858} \captionsetup{labelformat=empty} \caption{Set \(B\)}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c0a4ca-da0a-4c74-b8b1-bac4fd3f2487-2_435_424_365_1402} \captionsetup{labelformat=empty} \caption{Set \(C\)}
\end{figure} State, with an explanation in each case, which of the three sets of data has
  1. the largest,
  2. the smallest,
    value of the product moment correlation coefficient.
OCR S1 2005 January Q2
2 The back-to-back stem-and-leaf diagram below shows the number of hours of television watched per week by each of 15 boys and 15 girls. $$\begin{aligned} & \text { Boys Girls }
& \left. \begin{array} { r r r r r r r r | r r r r r r r r r r r r r } & 677664 & 4 & 3 & 0 & 0 & 5 & 5 & 6 & 677888 \end{array} \right\} \end{aligned}$$ Key: 4 | 2 | 2 means a boy who watched 24 hours and a girl who watched 22 hours of television per week.
  1. Find the median and the quartiles of the results for the boys.
  2. Give a reason why the median might be preferred to the mean in using an average to compare the two data sets.
  3. State one advantage, and one disadvantage, of using stem-and-leaf diagrams rather than box-andwhisker plots to represent the data.
OCR S1 2005 January Q3
3 Two commentators gave ratings out of 100 for seven sports personalities. The ratings are shown in the table below.
Personality\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Commentator I73767865868291
Commentator II77787980868995
  1. Calculate Spearman's rank correlation coefficient for these ratings.
  2. State what your answer tells you about the ratings given by the two commentators.
OCR S1 2005 January Q4
4 The table below shows the probability distribution of the random variable \(X\).
\(x\)- 2- 1012
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 5 }\)\(k\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 10 }\)
  1. Find the value of the constant \(k\).
  2. Calculate the values of \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 2005 January Q5
5 On average 1 in 20 members of the population of this country has a particular DNA feature. Members of the population are selected at random until one is found who has this feature.
  1. Find the probability that the first person to have this feature is
    (a) the sixth person selected,
    (b) not among the first 10 people selected.
  2. Find the expected number of people selected.
OCR S1 2005 January Q6
6 Louise and Marie play a series of tennis matches. It is given that, in any match, the probability that Louise wins the first two sets is \(\frac { 3 } { 8 }\).
  1. Find the probability that, in 5 randomly chosen matches, Louise wins the first two sets in exactly 2 of the matches. It is also given that Louise and Marie are equally likely to win the first set.
  2. Show that P (Louise wins the second set, given that she won the first set) \(= \frac { 3 } { 4 }\).
  3. The probability that Marie wins the first two sets is \(\frac { 1 } { 3 }\). Find P(Marie wins the second set, given that she won the first set).
OCR S1 2005 January Q7
7 It is known that, on average, one match box in 10 contains fewer than 42 matches. Eight boxes are selected, and the number of boxes that contain fewer than 42 matches is denoted by \(Y\).
  1. State two conditions needed to model \(Y\) by a binomial distribution. Assume now that a binomial model is valid.
  2. Find
    (a) \(\mathrm { P } ( Y = 0 )\),
    (b) \(\mathrm { P } ( Y \geqslant 2 )\).
  3. On Wednesday 8 boxes are selected, and on Thursday another 8 boxes are selected. Find the probability that on one of these days the number of boxes containing fewer than 42 matches is 0 , and that on the other day the number is 2 or more.
OCR S1 2005 January Q8
8 An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.
  1. If the 8 questions are arranged in a random order, find the probability that the question on geometric distributions is next to the question on binomial distributions. Four of the questions, including the one on geometric distributions, are worth 7 marks each, and the remaining four questions, including the one on binomial distributions, are worth 9 marks each. The 7-mark questions are the first four questions on the paper, but are arranged in random order. The 9-mark questions are the last four questions, but are arranged in random order. Find the probability that
  2. the questions on geometric distributions and on binomial distributions are next to one another,
  3. the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions.
OCR S1 2005 January Q9
9 Five observations of bivariate data produce the following results, denoted as ( \(x _ { i } , y _ { i }\) ) for \(i = 1,2,3,4,5\). $$\begin{aligned} & ( 13,2.7 )
& { \left[ \Sigma x = 90 , \Sigma y = 15.0 , \Sigma x ^ { 2 } = 1720 , \Sigma y ^ { 2 } = 46.86 , \Sigma x y = 264.0 . \right] } \end{aligned}$$
  1. Show that the regression line of \(y\) on \(x\) has gradient - 0.06 , and find its equation in the form \(y = a + b x\).
  2. The regression line is used to estimate the value of \(y\) corresponding to \(x = 20\), but the value \(x = 20\) is accurate only to the nearest whole number. Calculate the difference between the largest and the smallest values that the estimated value of \(y\) could take. The numbers \(e _ { 1 } , e _ { 2 } , e _ { 3 } , e _ { 4 } , e _ { 5 }\) are defined by $$e _ { i } = a + b x _ { i } - y _ { i } \quad \text { for } i = 1,2,3,4,5$$
  3. The values of \(e _ { 1 } , e _ { 2 }\) and \(e _ { 3 }\) are \(0.6 , - 0.7\) and 0.2 respectively. Calculate the values of \(e _ { 4 }\) and \(e _ { 5 }\).
  4. Calculate the value of \(e _ { 1 } ^ { 2 } + e _ { 2 } ^ { 2 } + e _ { 3 } ^ { 2 } + e _ { 4 } ^ { 2 } + e _ { 5 } ^ { 2 }\) and explain the relevance of this quantity to the regression line found in part (i).
  5. Find the mean and the variance of \(e _ { 1 } , e _ { 2 } , e _ { 3 } , e _ { 4 } , e _ { 5 }\).
OCR S1 2007 January Q1
1 Part of the probability distribution of a variable, \(X\), is given in the table.
\(x\)0123
\(\mathrm { P } ( X = x )\)\(\frac { 3 } { 10 }\)\(\frac { 1 } { 5 }\)\(\frac { 2 } { 5 }\)
  1. Find \(\mathrm { P } ( X = 0 )\).
  2. Find \(\mathrm { E } ( X )\).
OCR S1 2007 January Q2
2 The table contains data concerning five households selected at random from a certain town.
Number of people in the household23357
Number of cars belonging to people in the household11324
  1. Calculate the product moment correlation coefficient, \(r\), for the data in the table.
  2. Give a reason why it would not be sensible to use your answer to draw a conclusion about all the households in the town.
OCR S1 2007 January Q3
3 The digits 1, 2, 3, 4 and 5 are arranged in random order, to form a five-digit number.
  1. How many different five-digit numbers can be formed?
  2. Find the probability that the five-digit number is
    (a) odd,
    (b) less than 23000 .
OCR S1 2007 January Q4
4 Each of the variables \(W , X , Y\) and \(Z\) takes eight integer values only. The probability distributions are illustrated in the following diagrams.
\includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_437_394_397_280}
\includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_433_380_397_685}
\includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_428_383_402_1082}
\includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_425_376_402_1482}
  1. For which one or more of these variables is
    (a) the mean equal to the median,
    (b) the mean greater than the median?
  2. Give a reason why none of these diagrams could represent a geometric distribution.
  3. Which one of these diagrams could not represent a binomial distribution? Explain your answer briefly.
OCR S1 2007 January Q5
5 A chemical solution was gradually heated. At five-minute intervals the time, \(x\) minutes, and the temperature, \(y ^ { \circ } \mathrm { C }\), were noted.
\(x\)05101520253035
\(y\)0.83.06.810.915.619.623.426.7
$$\left[ n = 8 , \Sigma x = 140 , \Sigma y = 106.8 , \Sigma x ^ { 2 } = 3500 , \Sigma y ^ { 2 } = 2062.66 , \Sigma x y = 2685.0 . \right]$$
  1. Calculate the equation of the regression line of \(y\) on \(x\).
  2. Use your equation to estimate the temperature after 12 minutes.
  3. It is given that the value of the product moment correlation coefficient is close to + 1 . Comment on the reliability of using your equation to estimate \(y\) when
    (a) \(x = 17\),
    (b) \(x = 57\).
OCR S1 2007 January Q6
6 A coin is biased so that the probability that it will show heads on any throw is \(\frac { 2 } { 3 }\). The coin is thrown repeatedly. The number of throws up to and including the first head is denoted by \(X\). Find
  1. \(\mathrm { P } ( X = 4 )\),
  2. \(\mathrm { P } ( X < 4 )\),
  3. \(\mathrm { E } ( X )\).
OCR S1 2007 January Q7
7 A bag contains three 1 p coins and seven 2 p coins. Coins are removed at random one at a time, without replacement, until the total value of the coins removed is at least 3p. Then no more coins are removed.
  1. Copy and complete the probability tree diagram. First coin
    \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-4_350_317_1279_568} Find the probability that
  2. exactly two coins are removed,
  3. the total value of the coins removed is 4p.
OCR S1 2007 January Q8
8 In the 2001 census, the household size (the number of people living in each household) was recorded. The percentages of households of different sizes were then calculated. The table shows the percentages for two wards, Withington and Old Moat, in Manchester.
\cline { 2 - 8 } \multicolumn{1}{c|}{}Household size
\cline { 2 - 8 } \multicolumn{1}{c|}{}1234567 or more
Withington34.126.112.712.88.24.02.1
Old Moat35.127.114.711.47.62.81.3
  1. Calculate the median and interquartile range of the household size for Withington.
  2. Making an appropriate assumption for the last class, which should be stated, calculate the mean and standard deviation of the household size for Withington. Give your answers to an appropriate degree of accuracy. The corresponding results for Old Moat are as follows.
    Median
    Interquartile
    range
    		Mean
    Standard
    deviation
    222.41.5
  3. State one advantage of using the median rather than the mean as a measure of the average household size.
  4. By comparing the values for Withington with those for Old Moat, explain briefly why the interquartile range may be less suitable than the standard deviation as a measure of the variation in household size.
  5. For one of the above wards, the value of Spearman's rank correlation coefficient between household size and percentage is - 1 . Without any calculation, state which ward this is. Explain your answer.
OCR S1 2007 January Q9
9 A variable \(X\) has the distribution \(\mathrm { B } ( 11 , p )\).
  1. Given that \(p = \frac { 3 } { 4 }\), find \(\mathrm { P } ( X = 5 )\).
  2. Given that \(\mathrm { P } ( X = 0 ) = 0.05\), find \(p\).
  3. Given that \(\operatorname { Var } ( X ) = 1.76\), find the two possible values of \(p\).
OCR S1 2008 January Q1
1
  1. The letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E are arranged in a straight line.
    (a) How many different arrangements are possible?
    (b) In how many of these arrangements are the letters A and B next to each other?
  2. From the letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , two different letters are selected at random. Find the probability that these two letters are A and B .
OCR S1 2008 January Q2
2 A random variable \(T\) has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 5 } \right)\). Find
  1. \(\mathrm { P } ( T = 4 )\),
  2. \(\mathrm { P } ( T > 4 )\),
  3. \(\mathrm { E } ( T )\).
OCR S1 2008 January Q3
3 A sample of bivariate data was taken and the results were summarised as follows. $$n = 5 \quad \Sigma x = 24 \quad \Sigma x ^ { 2 } = 130 \quad \Sigma y = 39 \quad \Sigma y ^ { 2 } = 361 \quad \Sigma x y = 212$$
  1. Show that the value of the product moment correlation coefficient \(r\) is 0.855 , correct to 3 significant figures.
  2. The ranks of the data were found. One student calculated Spearman's rank correlation coefficient \(r _ { s }\), and found that \(r _ { s } = 0.7\). Another student calculated the product moment coefficient, \(R\), of these ranks. State which one of the following statements is true, and explain your answer briefly.
    (A) \(R = 0.855\)
    (B) \(R = 0.7\)
    (C) It is impossible to give the value of \(R\) without carrying out a calculation using the original data.
  3. All the values of \(x\) are now multiplied by a scaling factor of 2 . State the new values of \(r\) and \(r _ { s }\).
OCR S1 2008 January Q4
4 A supermarket has a large stock of eggs. 40\% of the stock are from a firm called Eggzact. 12\% of the stock are brown eggs from Eggzact. An egg is chosen at random from the stock. Calculate the probability that
  1. this egg is brown, given that it is from Eggzact,
  2. this egg is from Eggzact and is not brown.
OCR S1 2008 January Q5
5
  1. \(20 \%\) of people in the large town of Carnley support the Residents' Party. 12 people from Carnley are selected at random. Out of these 12 people, the number who support the Residents' Party is denoted by \(U\). Find
    (a) \(\mathrm { P } ( U \leqslant 5 )\),
    (b) \(\quad \mathrm { P } ( U \geqslant 3 )\).
  2. \(30 \%\) of people in Carnley support the Commerce Party. 15 people from Carnley are selected at random. Out of these 15 people, the number who support the Commerce Party is denoted by \(V\). Find \(\mathrm { P } ( V = 4 )\).
OCR S1 2008 January Q6
6 The probability distribution for a random variable \(Y\) is shown in the table.
\(y\)123
\(\mathrm { P } ( Y = y )\)0.20.30.5
  1. Calculate \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). Another random variable, \(Z\), is independent of \(Y\). The probability distribution for \(Z\) is shown in the table.
    \(z\)123
    \(\mathrm { P } ( Z = z )\)0.10.250.65
    One value of \(Y\) and one value of \(Z\) are chosen at random. Find the probability that
  2. \(Y + Z = 3\),
  3. \(Y \times Z\) is even.
OCR S1 2008 January Q7
7
  1. Andrew plays 10 tennis matches. In each match he either wins or loses.
    (a) State, in this context, two conditions needed for a binomial distribution to arise.
    (b) Assuming these conditions are satisfied, define a variable in this context which has a binomial distribution.
  2. The random variable \(X\) has the distribution \(\mathrm { B } ( 21 , p )\), where \(0 < p < 1\). Given that \(\mathrm { P } ( X = 10 ) = \mathrm { P } ( X = 9 )\), find the value of \(p\).