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OCR S4 2013 June Q1
8 marks Standard +0.3
1
\(S\)
012
0\(\frac { 1 } { 8 }\)\(\frac { 1 } { 8 }\)0
1\(\frac { 1 } { 8 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 8 }\)
20\(\frac { 1 } { 8 }\)\(\frac { 1 } { 8 }\)
An unbiased coin is tossed three times. The random variables \(F\) and \(S\) denote the total number of heads that occur in the first two tosses and the total number of heads that occur in the last two tosses respectively. The table above shows the joint probability distribution of \(F\) and \(S\).
  1. Show how the entry \(\frac { 1 } { 4 }\) in the table is obtained.
  2. Find \(\operatorname { Cov } ( F , S )\).
OCR S4 2013 June Q2
7 marks Standard +0.3
2 Two drugs, I and II, for alleviating hay fever are trialled in a hospital on each of 12 volunteer patients. Each received drug I on one day and drug II on a different day. After receiving a drug, the number of times each patient sneezed over a period of one hour was noted. The results are given in the table.
Patient123456789101112
Drug I1134191610296172013425
Drug II122010183219131019912
The patients may be considered to be a random sample of all hay fever sufferers.
A researcher believes that patients taking drug II sneeze less than patients taking drug I.
Test this belief using the Wilcoxon signed rank test at the \(5 \%\) significance level.
OCR S4 2013 June Q3
9 marks Standard +0.8
3 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } x \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0 \\ 0 & \text { otherwise } . \end{cases}$$
  1. Show that the moment generating function of \(X\) is \(( 1 - 2 t ) ^ { - 2 }\) for \(t < \frac { 1 } { 2 }\), and state why the condition \(t < \frac { 1 } { 2 }\) is necessary.
  2. Use the moment generating function to find \(\operatorname { Var } ( X )\).
OCR S4 2013 June Q4
10 marks Standard +0.3
4 The effect of water salinity on the growth of a type of grass was studied by a biologist. A random sample of 22 seedlings was divided into two groups \(A\) and \(B\), each of size 11 .
Group \(A\) was treated with water of \(0 \%\) salinity and group \(B\) was treated with water of \(0.5 \%\) salinity. After three weeks the height (in cm) of each seedling was measured with the following results, which are ordered for convenience.
Group \(A\)8.69.49.79.810.110.511.011.211.812.7
Group \(B\)7.48.48.58.89.29.39.59.910.011.1
Jeffery was asked to test whether the two treatments resulted, on average, in a difference in growth. He chose the Wilcoxon rank sum test.
  1. Justify Jeffery's choice of test.
  2. Carry out the test at the \(5 \%\) significance level.
OCR S4 2013 June Q5
12 marks Standard +0.8
5 The discrete random variable \(U\) has probability distribution given by $$\mathrm { P } ( U = r ) = \begin{cases} \frac { 1 } { 16 } \binom { 4 } { r } & r = 0,1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find and simplify the probability generating function (pgf) of \(U\).
  2. Use the pgf to find \(\mathrm { E } ( U )\) and \(\operatorname { Var } ( U )\).
  3. Identify the distribution of \(U\), giving the values of any parameters.
  4. Obtain the pgf of \(Y\), where \(Y = U ^ { 2 }\).
  5. State, giving a reason, whether you can obtain the pgf of \(U + Y\) by multiplying the pgf of \(U\) by the pgf of \(Y\).
OCR S4 2013 June Q6
13 marks Challenging +1.2
6 The continuous random variable \(X\) has mean \(\mu\) and variance \(\sigma ^ { 2 }\), and the independent continuous random variable \(Y\) has mean \(2 \mu\) and variance \(3 \sigma ^ { 2 }\). Two observations of \(X\) and three observations of \(Y\) are taken and are denoted by \(X _ { 1 } , X _ { 2 } , Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) respectively.
  1. Find the expectation of the sum of these 5 observations and hence construct an unbiased estimator, \(T _ { 1 }\), of \(\mu\).
  2. The estimator \(T _ { 2 }\), where \(T _ { 2 } = X _ { 1 } + X _ { 2 } + c \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)\), is an unbiased estimator of \(\mu\). Find the value of the constant \(c\).
  3. Determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is more efficient.
  4. Find the values of the constants \(a\) and \(b\) for which $$a \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } \right) + b \left( Y _ { 1 } ^ { 2 } + Y _ { 2 } ^ { 2 } + Y _ { 3 } ^ { 2 } \right)$$ is an unbiased estimator of \(\sigma ^ { 2 }\).
OCR S4 2013 June Q7
13 marks Standard +0.3
7 Each question on a multiple-choice examination paper has \(n\) possible responses, only one of which is correct. Joni takes the paper and has probability \(p\), where \(0 < p < 1\), of knowing the correct response to any question, independently of any other. If she knows the correct response she will choose it, otherwise she will choose randomly from the \(n\) possibilities. The events \(K\) and \(A\) are 'Joni knows the correct response' and 'Joni answers correctly' respectively.
  1. Show that \(\mathrm { P } ( A ) = \frac { q + n p } { n }\), where \(q = 1 - p\).
  2. Find \(P ( K \mid A )\). A paper with 100 questions has \(n = 4\) and \(p = 0.5\). Each correct response scores 1 and each incorrect response scores - 1 .
  3. (a) Joni answers all the questions on the paper and scores 40 . How many questions did she answer correctly?
    (b) By finding the distribution of the number of correct answers, or otherwise, find the probability that Joni scores at least 40 on the paper using her strategy.
OCR S4 2014 June Q1
8 marks Standard +0.3
1 A teacher believes that the calculator paper in a GCSE Mathematics examination was easier than the non-calculator paper. The marks of a random sample of ten students are shown in the table.
StudentABCDEFGHIJ
Mark on paper 1 (non-calculator)66795887675575625084
Mark on paper 2 (calculator)57847090754282726582
  1. Use a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test the teacher's belief.
  2. State the assumption necessary for this test to be applied.
OCR S4 2014 June Q2
9 marks Standard +0.3
2 During an outbreak of a disease, it is known that \(68 \%\) of people do not have the disease. Of people with the disease, \(96 \%\) react positively to a test for diagnosing it, as do \(m \%\) of people who do not have the disease.
  1. In the case \(m = 8\), find the probability that a randomly chosen person has the disease, given that the person reacts positively to the test.
  2. What value of \(m\) would be required for the answer to part (i) to be 0.95 ?
OCR S4 2014 June Q3
9 marks Challenging +1.2
3 The discrete random variable \(X\) has probability generating function \(\frac { t } { a - b t }\), where \(a\) and \(b\) are constants.
  1. Find a relationship between \(a\) and \(b\).
  2. Use the probability generating function to find \(\mathrm { E } ( X )\) in terms of \(a\), giving your answer as simply as possible.
  3. Expand the probability generating function as a power series, as far as the term in \(t ^ { 3 }\), giving the coefficients in terms of \(a\) and \(b\).
  4. Name the distribution for which \(\frac { t } { a - b t }\) is the probability generating function, and state its parameter(s) in terms of \(a\).
OCR S4 2014 June Q4
13 marks Challenging +1.2
4 The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } x & 0 \leqslant x \leqslant 1 \\ 2 - x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that the moment generating function of \(X\) is \(\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }\). \(Y _ { 1 }\) and \(Y _ { 2 }\) are independent observations of a random variable \(Y\). The moment generating function of \(Y _ { 1 } + Y _ { 2 }\) is \(\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }\).
  2. Write down the moment generating function of \(Y\).
  3. Use the expansion of \(\mathrm { e } ^ { t }\) to find \(\operatorname { Var } ( Y )\).
  4. Deduce the value of \(\operatorname { Var } ( X )\).
OCR S4 2014 June Q5
13 marks Standard +0.8
5 Two discrete random variables \(X\) and \(Y\) have a joint probability distribution defined by $$\mathrm { P } ( X = x , Y = y ) = a ( x + y + 1 ) \quad \text { for } x = 0,1,2 \text { and } y = 0,1,2 ,$$ where \(a\) is a constant.
  1. Show that \(a = \frac { 1 } { 27 }\).
  2. Find \(\mathrm { E } ( X )\).
  3. Find \(\operatorname { Cov } ( X , Y )\).
  4. Are \(X\) and \(Y\) independent? Give a reason for your answer.
  5. Find \(\mathrm { P } ( X = 1 \mid Y = 2 )\).
OCR S4 2014 June Q6
8 marks Standard +0.3
6 A Wilcoxon rank-sum test with samples of sizes 11 and 12 is carried out.
  1. What is the least possible value of the test statistic \(W\) ?
  2. The null hypothesis is that the two samples came from identical populations. Given that the null hypothesis was rejected at the \(1 \%\) level using a 2 -tail test, find the set of possible values of \(W\).
OCR S4 2014 June Q7
12 marks Standard +0.3
7 The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { k } { ( x + \theta ) ^ { 5 } } & \text { for } x \geqslant 0 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant and \(\theta\) is a parameter taking positive values.
  1. Find an expression for \(k\) in terms of \(\theta\).
  2. Show that \(\mathrm { E } ( X ) = \frac { 1 } { 3 } \theta\). You are given that \(\operatorname { Var } ( X ) = \frac { 2 } { 9 } \theta ^ { 2 }\). A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) of \(n\) observations of \(X\) is obtained. The estimator \(T _ { 1 }\) is defined as \(T _ { 1 } = \frac { 3 } { n } \sum _ { i = 1 } ^ { n } X _ { i }\).
  3. Show that \(T _ { 1 }\) is an unbiased estimator of \(\theta\), and find the variance of \(T _ { 1 }\).
  4. A second unbiased estimator \(T _ { 2 }\) is defined by \(T _ { 2 } = \frac { 1 } { 3 } \left( X _ { 1 } + 3 X _ { 2 } + 5 X _ { 3 } \right)\). For the case \(n = 3\), which of \(T _ { 1 }\) and \(T _ { 2 }\) is more efficient? \section*{OCR}
OCR C2 2005 January Q1
5 marks Moderate -0.5
1 Simplify \(( 3 + 2 x ) ^ { 3 } - ( 3 - 2 x ) ^ { 3 }\).
OCR C2 2005 January Q2
7 marks Moderate -0.3
2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 \quad \text { and } \quad u _ { n + 1 } = \frac { 1 } { 1 - u _ { n } } \text { for } n \geqslant 1 .$$
  1. Write down the values of \(u _ { 2 } , u _ { 3 } , u _ { 4 }\) and \(u _ { 5 }\).
  2. Deduce the value of \(u _ { 200 }\), showing your reasoning.
OCR C2 2005 January Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_488_604_895_769} A landmark \(L\) is observed by a surveyor from three points \(A , B\) and \(C\) on a straight horizontal road, where \(A B = B C = 200 \mathrm {~m}\). Angles \(L A B\) and \(L B A\) are \(65 ^ { \circ }\) and \(80 ^ { \circ }\) respectively (see diagram). Calculate
  1. the shortest distance from \(L\) to the road,
  2. the distance \(L C\).
OCR C2 2005 January Q4
8 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_547_511_1813_817} The diagram shows a sketch of parts of the curves \(y = \frac { 16 } { x ^ { 2 } }\) and \(y = 17 - x ^ { 2 }\).
  1. Verify that these curves intersect at the points \(( 1,16 )\) and \(( 4,1 )\).
  2. Calculate the exact area of the shaded region between the curves.
OCR C2 2005 January Q5
8 marks Moderate -0.3
5
  1. Prove that the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ can be expressed in the form $$2 \cos ^ { 2 } \theta + \cos \theta - 1 = 0$$
  2. Hence solve the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR C2 2005 January Q6
9 marks Moderate -0.8
6
  1. Find \(\int x \left( x ^ { 2 } + 2 \right) \mathrm { d } x\).
    1. Find \(\int \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).
    2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { } x }\). Find the equation of the curve, given that it passes through the point \(( 4,0 )\).
OCR C2 2005 January Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-3_563_639_1379_753} The diagram shows an equilateral triangle \(A B C\) with sides of length 12 cm . The mid-point of \(B C\) is \(O\), and a circular arc with centre \(O\) joins \(D\) and \(E\), the mid-points of \(A B\) and \(A C\).
  1. Find the length of the arc \(D E\), and show that the area of the sector \(O D E\) is \(6 \pi \mathrm {~cm} ^ { 2 }\).
  2. Find the exact area of the shaded region.
OCR C2 2005 January Q8
9 marks Standard +0.3
8
  1. On a single diagram, sketch the curves with the following equations. In each case state the coordinates of any points of intersection with the axes.
    1. \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
    2. \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
    3. The curves in part (i) intersect at the point \(P\). Prove that the \(x\)-coordinate of \(P\) is $$\frac { 1 } { \log _ { 2 } a - \log _ { 2 } b } .$$
OCR C2 2005 January Q9
11 marks Standard +0.3
9 A geometric progression has first term \(a\), where \(a \neq 0\), and common ratio \(r\), where \(r \neq 1\). The difference between the fourth term and the first term is equal to four times the difference between the third term and the second term.
  1. Show that \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1 = 0\).
  2. Show that \(r - 1\) is a factor of \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\). Hence factorise \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\).
  3. Hence find the two possible values for the ratio of the geometric progression. Give your answers in an exact form.
  4. For the value of \(r\) for which the progression is convergent, prove that the sum to infinity is \(\frac { 1 } { 2 } a ( 1 + \sqrt { } 5 )\).
OCR C2 2006 January Q1
6 marks Moderate -0.8
1 The 20th term of an arithmetic progression is 10 and the 50th term is 70 .
  1. Find the first term and the common difference.
  2. Show that the sum of the first 29 terms is zero.
OCR C2 2006 January Q2
6 marks Moderate -0.8
2 Triangle \(A B C\) has \(A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and angle \(B = 80 ^ { \circ }\). Calculate
  1. the area of the triangle,
  2. the length of \(C A\),
  3. the size of angle \(C\).