Questions S1 (2020 questions)

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OCR S1 2007 January Q3
6 marks Moderate -0.8
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
    1. odd,
    2. less than 23000 .
OCR S1 2007 January Q4
5 marks Moderate -0.3
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
    1. the mean equal to the median,
    2. the mean greater than the median?
    3. Give a reason why none of these diagrams could represent a geometric distribution.
    4. Which one of these diagrams could not represent a binomial distribution? Explain your answer briefly.
OCR S1 2007 January Q5
8 marks Moderate -0.8
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
    1. \(x = 17\),
    2. \(x = 57\).
OCR S1 2007 January Q6
8 marks Moderate -0.8
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
11 marks Standard +0.3
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
13 marks Moderate -0.3
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
11 marks Standard +0.3
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
7 marks Easy -1.3
1
  1. The letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E are arranged in a straight line.
    1. How many different arrangements are possible?
    2. In how many of these arrangements are the letters A and B next to each other?
    3. 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
5 marks Moderate -0.8
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
6 marks Standard +0.3
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 marks Easy -1.2
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
8 marks Moderate -0.8
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
    1. \(\mathrm { P } ( U \leqslant 5 )\),
    2. \(\quad \mathrm { P } ( U \geqslant 3 )\).
    3. \(30 \%\) 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
11 marks Moderate -0.3
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
8 marks Standard +0.3
7
  1. Andrew plays 10 tennis matches. In each match he either wins or loses.
    1. State, in this context, two conditions needed for a binomial distribution to arise.
    2. Assuming these conditions are satisfied, define a variable in this context which has a binomial distribution.
    3. 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 \(p\).
OCR S1 2008 January Q8
12 marks Easy -1.3
8 The stem-and-leaf diagram shows the age in completed years of the members of a sports club. \section*{Male} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Female}
8876166677889
7655332121334578899
98443323347
5214018
9050
\end{table} Key: 1 | 4 | 0 represents a male aged 41 and a female aged 40.
  1. Find the median and interquartile range for the males.
  2. The median and interquartile range for the females are 27 and 15 respectively. Make two comparisons between the ages of the males and the ages of the females.
  3. The mean age of the males is 30.7 and the mean age of the females is 27.5 , each correct to 1 decimal place. Give one advantage of using the median rather than the mean to compare the ages of the males with the ages of the females. A record was kept of the number of hours, \(X\), spent by each member at the club in a year. The results were summarised by $$n = 49 , \quad \Sigma ( x - 200 ) = 245 , \quad \Sigma ( x - 200 ) ^ { 2 } = 9849 .$$
  4. Calculate the mean and standard deviation of \(X\).
OCR S1 2008 January Q9
11 marks Moderate -0.8
9 It is thought that the pH value of sand (a measure of the sand's acidity) may affect the extent to which a particular species of plant will grow in that sand. A botanist wished to determine whether there was any correlation between the pH value of the sand on certain sand dunes, and the amount of each of two plant species growing there. She chose random sections of equal area on each of eight sand dunes and measured the pH values. She then measured the area within each section that was covered by each of the two species. The results were as follows.
\cline { 2 - 10 } \multicolumn{1}{c|}{}Dune\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
\cline { 2 - 10 } \multicolumn{1}{c|}{}pH value, \(x\)8.58.59.58.56.57.58.59.0
\multirow{2}{*}{
Area, \(y \mathrm {~cm} ^ { 2 }\)
covered
}		Species \(P\)1501505753304515340330
\cline { 2 - 10 }Species \(Q\)1701580230752500
The results for species \(P\) can be summarised by $$n = 8 , \quad \Sigma x = 66.5 , \quad \Sigma x ^ { 2 } = 558.75 , \quad \Sigma y = 1935 , \quad \Sigma y ^ { 2 } = 711275 , \quad \Sigma x y = 17082.5 .$$
  1. Give a reason why it might be appropriate to calculate the equation of the regression line of \(y\) on \(x\) rather than \(x\) on \(y\) in this situation.
  2. Calculate the equation of the regression line of \(y\) on \(x\) for species \(P\), in the form \(y = a + b x\), giving the values of \(a\) and \(b\) correct to 3 significant figures.
  3. Estimate the value of \(y\) for species \(P\) on sand where the pH value is 7.0 . The values of the product moment correlation coefficient between \(x\) and \(y\) for species \(P\) and \(Q\) are \(r _ { P } = 0.828\) and \(r _ { Q } = 0.0302\).
  4. Describe the relationship between the area covered by species \(Q\) and the pH value.
  5. State, with a reason, whether the regression line of \(y\) on \(x\) for species \(P\) will provide a reliable estimate of the value of \(y\) when the pH value is
    1. 8,
    2. 4 .
    3. Assume that the equation of the regression line of \(y\) on \(x\) for species \(Q\) is also known. State, with a reason, whether this line will provide a reliable estimate of the value of \(y\) when the pH value is 8 .
OCR S1 2005 June Q1
6 marks Moderate -0.8
1
  1. Calculate the value of Spearman's rank correlation coefficient between the two sets of rankings, \(A\) and \(B\), shown in Table 1. \begin{table}[h]
    \(A\)12345
    \(B\)41325
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
  2. The value of Spearman's rank correlation coefficient between the set of rankings \(B\) and a third set of rankings, \(C\), is known to be - 1 . Copy and complete Table 2 showing the set of rankings \(C\). \begin{table}[h]
    \(B\)41325
    \(C\)
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
OCR S1 2005 June Q2
8 marks Moderate -0.8
2 The probability that a certain sample of radioactive material emits an alpha-particle in one unit of time is 0.14 . In one unit of time no more than one alpha-particle can be emitted. The number of units of time up to and including the first in which an alpha-particle is emitted is denoted by \(T\).
  1. Find the value of
    1. \(\mathrm { P } ( T = 5 )\),
    2. \(\mathrm { P } ( T < 8 )\).
    3. State the value of \(\mathrm { E } ( T )\).
OCR S1 2005 June Q3
8 marks Moderate -0.8
3 In a supermarket the proportion of shoppers who buy washing powder is denoted by \(p .16\) shoppers are selected at random.
  1. Given that \(p = 0.35\), use tables to find the probability that the number of shoppers who buy washing powder is
    1. at least 8,
    2. between 4 and 9 inclusive.
    3. Given instead that \(p = 0.38\), find the probability that the number of shoppers who buy washing powder is exactly 6 .
OCR S1 2005 June Q4
9 marks Moderate -0.3
4 The table shows the latitude, \(x\) (in degrees correct to 3 significant figures), and the average rainfall \(y\) (in cm correct to 3 significant figures) of five European cities.
City\(x\)\(y\)
Berlin52.558.2
Bucharest44.458.7
Moscow55.853.3
St Petersburg60.047.8
Warsaw52.356.6
$$\left[ n = 5 , \Sigma x = 265.0 , \Sigma y = 274.6 , \Sigma x ^ { 2 } = 14176.54 , \Sigma y ^ { 2 } = 15162.22 , \Sigma x y = 14464.10 . \right]$$
  1. Calculate the product moment correlation coefficient.
  2. The values of \(y\) in the table were in fact obtained from measurements in inches and converted into centimetres by multiplying by 2.54 . State what effect it would have had on the value of the product moment correlation coefficient if it had been calculated using inches instead of centimetres.
  3. It is required to estimate the annual rainfall at Bergen, where \(x = 60.4\). Calculate the equation of an appropriate line of regression, giving your answer in simplified form, and use it to find the required estimate.
OCR S1 2005 June Q5
13 marks Easy -1.2
5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve. \includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-4_1344_1335_386_425} Use the curve to estimate
  1. the interquartile range of the marks,
  2. \(x\), if \(40 \%\) of the candidates scored more than \(x\) marks,
  3. the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
  4. Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored \(35,36,37 , \ldots , 55\).
  5. What does this information suggest about the estimate of the interquartile range found in part (i)?
OCR S1 2005 June Q6
14 marks Moderate -0.3
6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\). The tree diagram shows the different combinations of three coloured discs chosen. \includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-5_863_986_559_612}
  1. Write down the values of \(a , b , c , d , e\) and \(f\). The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).
    \(r\)0123
    \(\mathrm { P } ( R = r )\)\(\frac { 1 } { 10 }\)\(k\)\(\frac { 9 } { 20 }\)\(\frac { 1 } { 5 }\)
  2. Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
  3. Find the value of \(k\).
  4. Calculate the mean and variance of \(R\).
OCR S1 2006 June Q1
6 marks Moderate -0.8
1 Some observations of bivariate data were made and the equations of the two regression lines were found to be as follows. $$\begin{array} { c c } y \text { on } x : & y = - 0.6 x + 13.0 \\ x \text { on } y : & x = - 1.6 y + 21.0 \end{array}$$
  1. State, with a reason, whether the correlation between \(x\) and \(y\) is negative or positive.
  2. Neither variable is controlled. Calculate an estimate of the value of \(x\) when \(y = 7.0\).
  3. Find the values of \(\bar { x }\) and \(\bar { y }\).
OCR S1 2006 June Q2
7 marks Moderate -0.8
2 A bag contains 5 black discs and 3 red discs. A disc is selected at random from the bag. If it is red it is replaced in the bag. If it is black, it is not replaced. A second disc is now selected at random from the bag. Find the probability that
  1. the second disc is black, given that the first disc was black,
  2. the second disc is black,
  3. the two discs are of different colours.
OCR S1 2006 June Q3
8 marks Moderate -0.8
3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.
  1. How many different arrangements of the letters are possible?
  2. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
  3. Find the probability that at least one of these 2 cards has D printed on it.