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Homework answers / question archive / ACST8040 Quantitative Research Methods Assignment 2    Question 1 [20 marks] An insurance company has installed a new measure to reduce the cost on a line of business

ACST8040 Quantitative Research Methods Assignment 2    Question 1 [20 marks] An insurance company has installed a new measure to reduce the cost on a line of business

Statistics

ACST8040 Quantitative Research Methods

Assignment 2   

Question 1 [20 marks]

An insurance company has installed a new measure to reduce the cost on a line of business. The costs (in $m) of such a business line in 10 branches of the company before and after the new measure is taken over a certain period of time are recorded in the table below: 

Branch

1

2

3

4

5

6

7

8

9

10

Before

13

36

16

15

33

32

7

25

41

35

After

19

18

28

12

8

14

15

10

20

23

Let X and Y represent the costs before and after the new measure takes effect, respectively, and ? the median of the cost reduction.  

Based on the data provided in the above table, perform the following analyses:

  1. Calculate the value of the Wilcoxon signed rank test statistic T? based on the difference

Y?X between the costs before and after the new measure takes effect.           [4]

  1. Determine if there is sufficient evidence at the 5% level that the new measure is effective to reduce the cost by the Wilcoxon signed rank test using the exact p-value conditional on any ties by enumeration.           [7]
  2. Repeat the question in part (b) by the normal approximation.  [3]
  3. Estimate the median ? and obtain its exact confidence interval with a target at least 95% confidence level based on the Wilcoxon signed ranks. [6]

 

Question 2 [15 marks]

Prove the following results on the Wilcoxon signed-rank test statistic T? for one-sample location and the Ansari-Bradley rank test statistic C for two-sample dispersion:

  1. Let X1 and X2 be two independent continuous random variables. If X1 ~?X2 with a common median 0, then T??S?I{X1?0}?2I{X2?0},  where “~” represents “having the same distribution”. You may use the following equation with density f1(x) of X1

?

Pr(0? X1??X2)?? Pr(?X2 ? x) f1 (x)dx   [8]

0

  1. If the total size N of two samples combined is even and assume no ties, then C has a symmetric distribution about E0[C] under H0 :?2 ?1.  [7]

Hint: For each (r1,?,rn) drawn from {a1,?,aN}??1,2,?,N 2?1,N 2,N 2?1,?,2,1? as Y-scores, consider (r?1,?,r?n) with r?i ?N 2?1?ri , i ?1,?,n. 

          

Question 3 [25 marks]

The following X and Y represent two independent samples on the profits (in $m) of two banks in different areas during a financial year:

X?(35,57,39,30,52,42,38,49,24,36,32,44) 

Y?(47,40,61,80,28,89,54,74,45,50,21)

Carry out the following statistical analyses based on the samples X and Y :

  1. Under the location-shift model, use the Wilcoxon rank sum test to determine whether there is sufficient of evidence for sample Y to have a greater location parameter than sample X by the normal approximation at the 5% level of significance. [4]
  2. Under the location-scale parameter model, test the null hypothesis H0:Var(X)?Var(Y) against the alternative H1:Var(X)?Var(Y) by the Ansari-Bradley rank test with the normal approximation at the 10% level of significance.  [4]
  3. Calculate the values of A1,A2,?,Am and B1,B2,?,Bn for the Miller’s Jackknife test. Then find the approximate p-value for Var(X)?Var(Y) by the Miller’s Jackknife test and draw a conclusion.        [5]
  4. Let X*?3X?(3X1,?,3Xm) and Y*?Y?66?(Y1?66,?,Yn?66)

Find the empirical distribution functions F12* (t) of X* and G11* (t) of Y* at the ordered values Z(1) ???Z(23) of ?X*,Y*?. Then calculate the value of the test statistic J of the two-sample Kolmogorov-Smirnov test for general differences between X* and Y*.

Verify the value of J and find the p-value of the test by R.     [5]

  1. Based on the results in parts (a) – (d), answer the following questions with reasoning:
    1. What conclusions can be drawn from part (d)?
    2. What relations in the locations and dispersions between X and Y can be inferred from the analyses?
    3. Is the location-shift model appropriate in part (a)?
    4. Is the location-scale parameter model justified in parts (b) and (c)?
    5. Are the results in parts (a) – (c) justified?
    6. What observations can be made from the analyses for comparing the profitability and profit stability of the two banks?       [7] Question 4 [20 marks]

A set of data {Xij} in a one-way layout with 5 treatments are listed below:

 

Treatment

17 

22 

20 

86 

68 

28 

15 

39 

54 

73 

18 

43 

61 

32 

25 

Denote the effects of treatments 1,?,5 by ?1,?,?5 respectively.

  1. Test the null hypothesis H0 :?1????5 against general alternatives at the 5% level of significance by the Kruskal-Wallis test with the approximate rejection rule. [5] 
  2. Test H0 :?1????5 against ordered alternatives H1:?1????5 with at least one strict inequality at the 1% level of significance by the Jonckheere-Terpstra test with the normal approximation.  [6]
  3. Determine whether ?u ??1 or ?u ??1 for u? 2,3,4,5 by the Nemenyi-Damico-Wolfe one-sided treatments-versus-control multiple comparison procedure at ??10%. You can use R to find the critical point for making the decisions. [4] 
  4. Comment on the differences in parts (a) – (c) regarding the decisions on treatment effects

?1,?,?5 . Are they consistent or contradictive? Explain why or why not.       [5]

 

Question 5 [20 marks]

A sample of data in a one-way layout with four treatments are presented below:

 

Treatment

 

 1

    2

      3

       4

13.6 

16.8 

13.2 

3.3 

5.2 

21.6 

20.4 

9.3 

15.5 

3.8 

28.4 

8.1 

2.4 

25.2 

8.6 

18.9 

11.0 

14.5 

18.3 

6.7 

 

Let ?1,?,?4 denote the effects of the 4 treatments. 

  1. Test H0 :?1????4 against the alternative H1:?1??2 ??3??4 with at least one strict inequality by the Mack-Wolfe test with known peak p? 2 at the 5% level using the large-sample normal approximation.          [7]
  2. If the peak p is unknown, find an estimate of p and calculate the Mack-Wolfe test statistic A* with unknown peak.     [5]
  3. Test H0 :?1????4 against umbrella alternatives with unknown peak p at the 5% level of significance using R. [4]
  4. If the estimate of p in part (b) were assumed known, what would be the result of the Mack-Wolfe test with known peak ? If the test result is different from that of part (c), explain the main reason for the difference.  [4]

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