Questions — OCR (4619 questions)

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OCR S4 2011 June Q7
7 The continuous random variable \(U\) has unknown mean \(\mu\) and known variance \(\sigma ^ { 2 }\). In order to estimate \(\mu\), two random samples, one of 4 observations of \(U\) and the other of 6 observations of \(U\), are taken. The sample means are denoted by \(\bar { U } _ { 4 }\) and \(\bar { U } _ { 6 }\) respectively. One estimator \(S\), given by \(S = \frac { 1 } { 2 } \left( \bar { U } _ { 4 } + \bar { U } _ { 6 } \right)\), is proposed.
  1. Show that \(S\) is unbiased and find \(\operatorname { Var } ( S )\) in terms of \(\sigma ^ { 2 }\). A second estimator \(T\) of the form \(a \bar { U } _ { 4 } + b \bar { U } _ { 6 }\) is proposed, where \(a\) and \(b\) are chosen such that \(T\) is an unbiased estimator for \(\mu\) with the smallest possible variance.
  2. Find the values of \(a\) and \(b\) and the corresponding variance of \(T\).
  3. State, giving a reason, which of \(S\) and \(T\) is the better estimator.
  4. Compare the efficiencies of this preferred estimator and the mean of all 10 observations.
OCR S4 2012 June Q1
1 Independent random variables \(X\) and \(Y\) have distributions \(\mathrm { B } ( 7 , p )\) and \(\mathrm { B } ( 8 , p )\) respectively.
  1. Explain why \(X + Y \sim \mathrm {~B} ( 15 , p )\).
  2. Find \(\mathrm { P } ( X = 2 \mid X + Y = 5 )\).
OCR S4 2012 June Q2
2 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 4 x e ^ { - 2 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$
  1. Show that the moment generating function ( mgf ) of \(X\) is $$\frac { 4 } { ( 2 - t ) ^ { 2 } } , \text { where } | t | < 2$$
  2. Explain why the \(\operatorname { mgf }\) of \(- X\) is \(\frac { 4 } { ( 2 + t ) ^ { 2 } }\).
  3. Two random observations of \(X\) are denoted by \(X _ { 1 }\) and \(X _ { 2 }\). What is the \(\operatorname { mgf }\) of \(X _ { 1 } - X _ { 2 }\) ?
OCR S4 2012 June Q3
3 Because of the large number of students enrolled for a university geography course and the limited accommodation in the lecture theatre, the department provides a filmed lecture. Students are randomly assigned to two groups, one to attend the lecture theatre and the other the film. At the end of term the two groups are given the same examination. The geography professor wishes to test whether there is a difference in the performance of the two groups and selects the marks of two random samples of students, 6 from the group attending the lecture theatre and 7 from the group attending the films. The marks for the two samples, ordered for convenience, are shown below.
Lecture theatre:303648515962
Filmed lecture:40495256636468
  1. Stating a necessary assumption, carry out a suitable non-parametric test, at the \(10 \%\) significance level, for a difference between the median marks of the two groups.
  2. State conditions under which a two-sample \(t\)-test could have been used.
  3. Assuming that the tests in parts (i) and (ii) are both valid, state with a reason which test would be preferable.
OCR S4 2012 June Q4
4 The random variable \(U\) has the distribution \(\operatorname { Geo } ( p )\).
  1. Show, from the definition, that the probability generating function ( pgf ) of \(U\) is given by $$G _ { U } ( t ) = \frac { p t } { 1 - q t } , \text { for } | t | < \frac { 1 } { q } ,$$ where \(q = 1 - p\).
  2. Explain why the condition \(| t | < \frac { 1 } { q }\) is necessary.
  3. Use the pgf to obtain \(\mathrm { E } ( U )\). Each packet of Corn Crisp cereal contains a voucher and \(20 \%\) of the vouchers have a gold star. When 4 gold stars have been collected a gift can be claimed. Let \(X\) denote the number of packets bought by a family up to and including the one from which the \(4 ^ { \text {th } }\) gold star is obtained.
  4. Obtain the pgf of \(X\).
  5. Find \(\mathrm { P } ( X = 6 )\).
OCR S4 2012 June Q5
5 A one-tail sign test of a population median is to be carried out at the \(5 \%\) significance level using a sample of size \(n\).
  1. Show by calculation that the test can never result in rejection of the null hypothesis when \(n = 4\). The coach of a college swimming team expects Elena, the best 50 m freestyle swimmer, to have a median time less than 30 seconds. Elena found from records of her previous 72 swims that 44 were less than 30 seconds and 28 were greater than 30 seconds.
  2. Stating a necessary assumption, test at the \(5 \%\) significance level whether Elena's median time for the 50 m freestyle is less than 30 seconds.
OCR S4 2012 June Q6
6 The random variables \(S\) and \(T\) are independent and have joint probability distribution given in the table.
\(S\)
\cline { 2 - 5 }012
\cline { 2 - 5 }1\(a\)0.18\(b\)
20.080.120.20
\cline { 2 - 5 }
\cline { 2 - 5 }
  1. Show that \(a = 0.12\) and find the value of \(b\).
  2. Find \(\mathrm { P } ( T - S = 1 )\).
  3. Find \(\operatorname { Var } ( T - S )\).
OCR S4 2012 June Q7
7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 1 + a x ) & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that \(| a | \leqslant \frac { 1 } { 2 }\).
  2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
  3. Construct an unbiased estimator \(T _ { 1 }\) of \(a\) based on one observation \(X _ { 1 }\) of \(X\).
  4. A second observation \(X _ { 2 }\) is taken. Show that \(T _ { 2 }\), where \(T _ { 2 } = \frac { 3 } { 8 } \left( X _ { 1 } + X _ { 2 } \right)\), is also an unbiased estimator of a.
  5. Given that \(\operatorname { Var } ( X ) = \sigma ^ { 2 }\), determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is the better estimator.
OCR S4 2012 June Q8
8 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3\) and \(\mathrm { P } ( A \mid B ) = 0.6\).
  1. Show that \(\mathrm { P } ( B ) \leqslant 0.5\).
  2. Given also that \(\mathrm { P } ( A \cup B ) = x\), find \(\mathrm { P } ( B )\) in terms of \(x\).
OCR S4 2013 June Q1
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
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
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
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
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
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
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
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
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
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
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
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
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
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
1 Simplify \(( 3 + 2 x ) ^ { 3 } - ( 3 - 2 x ) ^ { 3 }\).
OCR C2 2005 January Q2
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.