Questions — OCR S4 (86 questions)

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OCR S4 2015 June Q6
6 In a two-tail Wilcoxon rank-sum test, the sample sizes are 13 and 15. The sum of the ranks for the sample of size 13 is 135 . Carry out the test at the \(5 \%\) level of significance.
OCR S4 2015 June Q7
7 The discrete random variable \(X\) can take the values 0,1 and 2 with equal probabilities.
The random variables \(X _ { 1 }\) and \(X _ { 2 }\) are independent observations of \(X\), and the random variables \(Y\) and \(Z\) are defined as follows:
\(Y\) is the smaller of \(X _ { 1 }\) and \(X _ { 2 }\), or their common value if they are equal; \(Z = \left| X _ { 1 } - X _ { 2 } \right|\).
  1. Draw up a table giving the joint distribution of \(Y\) and \(Z\).
  2. Find \(P ( Y = 0 \mid Z = 0 )\).
  3. Find \(\operatorname { Cov } ( Y , Z )\).
OCR S4 2015 June Q8
8 The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) have the distributions \(\mathrm { B } \left( n _ { 1 } , \theta \right)\) and \(\mathrm { B } \left( n _ { 2 } , \theta \right)\) respectively. Two possible estimators for \(\theta\) are $$T _ { 1 } = \frac { 1 } { 2 } \left( \frac { X _ { 1 } } { n _ { 1 } } + \frac { X _ { 2 } } { n _ { 2 } } \right) \text { and } T _ { 2 } = \frac { X _ { 1 } + X _ { 2 } } { n _ { 1 } + n _ { 2 } } .$$
  1. Show that \(T _ { 1 }\) and \(T _ { 2 }\) are both unbiased estimators, and calculate their variances.
  2. Find \(\frac { \operatorname { Var } \left( T _ { 1 } \right) } { \operatorname { Var } \left( T _ { 2 } \right) }\). Given that \(n _ { 1 } \neq n _ { 2 }\), use the inequality \(\left( n _ { 1 } - n _ { 2 } \right) ^ { 2 } > 0\) to find which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.
OCR S4 2018 June Q1
1 A Wilcoxon signed-rank test is carried out at the \(5 \%\) level of significance on a random sample of size 32 . The hypotheses are \(\mathrm { H } _ { 0 } : m = m _ { 0 } , \mathrm { H } _ { 1 } : m < m _ { 0 }\) where \(m\) is the population median and \(m _ { 0 }\) is a specific numerical value. The value obtained for the test statistic \(T\) is 162 . Find the outcome of the test.
OCR S4 2018 June Q2
2 The distances from home to work, in km , of 8 men and 5 women were recorded and are given below. The workers were chosen at random.
Men47101316172021
Women12141822
Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether there is a significant difference between the distances from home to work between men and women.
OCR S4 2018 June Q3
3 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup B ) = 0.8\).
  1. Find \(\mathrm { P } ( A \cap B )\).
  2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
  3. Find \(\mathrm { P } ( A \mid B )\). Events \(A\) and \(B\) are as above and a third event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1 , \mathrm { P } ( A \cap B \cap C ) = 0.05\), \(\mathrm { P } ( A \cap C ) = \mathrm { P } ( B \cap C )\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 3 \mathrm { P } \left( A ^ { \prime } \cap B \cap C ^ { \prime } \right)\).
  4. Find \(\mathrm { P } ( C )\).
OCR S4 2018 June Q4
4 The random variable \(X\) has a \(\chi ^ { 2 }\) distribution with \(v\) degrees of freedom. The moment generating function of \(X\) is $$\mathrm { M } _ { X } ( t ) = ( 1 - 2 t ) ^ { - \frac { 1 } { 2 } v }$$
  1. Show that \(\mathrm { E } ( X ) = v\).
  2. Find \(\operatorname { Var } ( X )\).
  3. Obtain the moment generating function of the sum \(Y\) of two independent \(\chi ^ { 2 }\) random variables, one with 6 degrees of freedom and the other with 8 degrees of freedom.
  4. Identify the distribution of \(Y\).
OCR S4 2018 June Q5
5 The independent discrete random variables \(U\) and \(V\) can each take the values 1, 2 and 3, all with probability \(\frac { 1 } { 3 }\). The random variables \(X\) and \(Y\) are defined as follows: $$X = | U - V | , Y = U + V .$$
  1. In the Printed Answer Book complete the table showing the joint probability distribution of \(X\) and \(Y\).
  2. Find \(\operatorname { Cov } ( X , Y )\).
  3. State with a reason whether \(X\) and \(Y\) are independent.
  4. Find \(\mathrm { P } ( Y = 3 \mid X = 1 )\).
OCR S4 2018 June Q6
6 In each round of a quiz a contestant can answer up to three questions. Each correct answer scores 1 point and allows the contestant to go on to the next question. A wrong answer scores 0 points and the contestant is allowed no further question in that round. If all 3 questions are answered correctly 1 bonus point is scored, making a total score of 4 for the round. For a certain contestant, \(A\), the probability of giving a correct answer is \(\frac { 3 } { 4 }\), independently of any other question. The random variable \(X _ { r }\) is the number of points scored by \(A\) during the \(r ^ { \text {th } }\) round.
  1. Find the probability generating function of \(X _ { r }\).
  2. Use the probability generating function found in part (i) to find the mean and variance of \(X _ { r }\).
  3. Write down an expression for the probability generating function of \(X _ { 1 } + X _ { 2 }\) and find the probability that \(A\) has a total score of 4 at the end of two rounds.
OCR S4 2018 June Q7
7 Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of a continuous random variable with probability density function $$f ( x ) = \begin{cases} \frac { 1 } { \theta } & 0 \leqslant x \leqslant \theta
0 & \text { otherwise } \end{cases}$$ where \(\theta\) is a parameter whose value is to be estimated.
  1. Find \(\mathrm { E } ( X )\).
  2. Show that \(S _ { 1 } = X _ { 1 } + X _ { 2 }\) is an unbiased estimator of \(\theta\).
    \(L\) is the larger of \(X _ { 1 }\) and \(X _ { 2 }\), or their common value if they are equal.
  3. Show that the probability density function of \(L\) is \(\frac { 2 l } { \theta ^ { 2 } }\) for \(0 \leqslant l \leqslant \theta\).
  4. Find \(\mathrm { E } ( L )\).
  5. Find an unbiased estimator \(S _ { 2 }\) of \(\theta\), based on \(L\).
  6. Determine which of the two estimators \(S _ { 1 }\) and \(S _ { 2 }\) is the more efficient.
OCR S4 2010 June Q3
  1. Assuming that all rankings are equally likely, show that \(\mathrm { P } ( R \leqslant 17 ) = \frac { 2 } { 231 }\). The marks of 5 randomly chosen students from School \(A\) and 6 randomly chosen students from School \(B\), who took the same examination, achieving different marks, were ranked. The rankings are shown in the table.
    Rank1234567891011
    School\(A\)\(A\)\(A\)\(B\)\(A\)\(A\)\(B\)\(B\)\(B\)\(B\)\(B\)
  2. For a Wilcoxon rank-sum test, obtain the exact smallest significance level for which there is evidence of a difference in performance at the two schools.