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ACST8040 Quantitative Research Methods


Assignment 1
(Due 6pm on Friday, 25 March 2022)


Instructions:
• This assignment consists of 4 questions of judgements and explanations.
• It is to be completed independently by each student.
• It will count for 20% of assessment.
• The available marks of each question are indicated after the question numbers.
• The full mark of the assignment is 20.
• Determine if the statements or conclusions in each part of a question are true or false.
• Justify each “true” and “false” with convincing explanations.
• Provide a counterexample to a “false” where applicable.
• Submit your answers in PDF file via Turnitin on iLearn by Friday 6pm, 25 March 2022.
• The submitted answers must be typed (not handwritten).
• Note that Turnitin requires at least 20 typed words to submit a file.


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Question 1 [6 marks]
(a) Any continuous distribution has a unique median.
(b) If 1 2 3( , , )b b b are randomly selected from {1,2,3,4,5,6} with replacement and ordered to
1 2 3b b b≤ ≤ , then the total number of outcomes 1 2 3( , , )b b b is 56.
(c) In a test of a null hypothesis 0H against an alternative 1H at the 5% significance level,
if the p-value of the test is below 0.05, then the probability to correctly accept 1H is
greater than 95%.


Question 2 [6 marks]
Let 1, , nX X be independent continuous random variables with Pr( ) 0.5iX θ< = for a
real number θ , 1, ,i n=  , and (1) ( )nX X≤ ≤ the order statistics of 1, , nX X .
(a) The critical point of the sign test for the null hypothesis 0 : 0H θ = can be determined
without knowing the specific distributions of 1, , nX X .
(b) If 1, , nX X are identically distributed, then the Wilcoxon signed-rank test is generally
more efficient than the sign test for 0 : 0H θ = .
(c) Let ( )( ),i ix x denote the observed values of ( )( ),i iX X , 1, ,i n=  , ~ ( ,0.5)B Bin n and
Pr( )B bα α≥ = . If ( ) ( )2 2( 1 ) ( ), ,n b b i jx x x xα α+ − = , then ( ) ( )2 2( 1 ) ( ), ,n b b i jX X X Xα α+ − =
and hence
2 2( 1 ) ( )Pr Pr 1( ) ( )i j n b bX X X Xα αθ θ α+ −< < = < < = − .


Question 3 [4 marks]
You are given the following two random samples:
1 2 3, , (2,7,12)( )X X X = and 1 5, , (2,4,7,7,9)( )Y Y = .
(a) Under the null hypothesis of no treatment effect, the variance of the Wilcoxon rank sum
test statistic W (conditional on ties) is equal to 1185 112 .
(b) The scores of 1 5, ,Y Y used to define the Ansari-Bradley rank test statistic C for
dispersion are given by 1.5,3,4,4,2 respectively.


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Question 4 [4 marks]
Two random samples are drawn from two populations represented by continuous random
variables X and Y . The following results are obtained from these two samples:
• The two-sample Kolmogorov-Smirnov test for general differences between 5X + and
Y produces a large p-value;
• The Miller’s Jackknife test for dispersion accepts the null hypothesis for ( , )X Y ;
• The Lepage rank test for location and dispersion rejects the null hypothesis for ( , )X Y .
From these results we can draw the following conclusions:
(a) The model assumptions for all three tests are justified.
(b) The results of the three tests are consistent.