1 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. 2 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. 3 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.
