Critical region or test statistic properties

A question is this type if and only if it asks to find critical regions, critical values, possible values of test statistics, or theoretical properties of test statistics.

6 questions · Standard +0.4

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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 2010 June Q2
6 marks Standard +0.3
2 The probability generating function of the discrete random variable \(X\) is \(\frac { \mathrm { e } ^ { 4 t ^ { 2 } } } { \mathrm { e } ^ { 4 } }\). Find
  1. \(\mathrm { E } ( X )\),
  2. \(\mathrm { P } ( X = 2 )\). \(3 X _ { 1 }\) and \(X _ { 2 }\) are continuous random variables. Random samples of 5 observations of \(X _ { 1 }\) and 6 observations of \(X _ { 2 }\) are taken. No two observations are equal. The 11 observations are ranked, lowest first, and the sum of the ranks of the observations of \(X _ { 1 }\) is denoted by \(R\).
OCR Further Statistics 2022 June Q8
7 marks
8 The critical region for an \(r\) \% two-tailed Wilcoxon signed-rank test, based on a large sample of size \(n\), is \(\left\{ W _ { + } \leqslant 113 \right\} \cup \left\{ W _ { + } \geqslant 415 \right\}\).
  1. Show that \(n = 32\).
  2. Using a suitable approximation, determine the value of \(r\).
OCR Further Statistics 2024 June Q4
6 marks
4
  1. Write down the number of ways of choosing 5 objects from 12 distinct objects.
  2. Each possible set of 5 different integers selected from the integers \(1,2 , \ldots , 12\) is obtained, and for each set, the sum of the 5 integers is found. The sum \(S\) can take values between 15 and 50 inclusive. Part of the frequency distribution of \(S\) is shown in the following table, together with the cumulative frequencies.
    S151617181920212223
    Frequency112357101317
    Cumulative Frequency12471219294259
    Use these numbers to determine the critical region for a 1-tail Wilcoxon rank-sum test at the \(2 \%\) significance level when \(m = 5\) and \(n = 7\).
  3. A student says that, for a Wilcoxon rank-sum test on samples of size \(m\) and \(n\), where \(m\) and \(n\) are large, the mean and variance of the test statistic \(R _ { m }\) are 200 and \(616 \frac { 2 } { 3 }\) respectively. Show that at least one of these values must be incorrect.
OCR S4 2010 June Q3
7 marks Challenging +1.8
  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.
OCR Further Statistics 2018 March Q4
9 marks Moderate -0.8
4 Sheena travels to school by bus. She records the number of minutes, \(T\), that her bus is late on each of 32 days. She believes that on average \(T\) is greater than 5, and she carries out a significance test at the \(5 \%\) level.
  1. State a condition needed for a Wilcoxon test to be valid in this case. Assume now that this condition is satisfied.
  2. State an advantage of using a Wilcoxon test rather than a sign test.
  3. Calculate the critical region for the test, in terms of a variable which should be defined.