Past Examination Papers Quantitative Methods B

Examination

Case Questions

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Case Question 1

A fashion magazine wishes to construct a price index for men's clothing. The following table shows the prices and quantities bought for the years 1995 to 1997 by a typical consumer:


  1995 1996 1997
  Price () Quantity Price () Quantity Price () Quantity

Shoes 52.00 3.2 56.00 3.8 61.00 3.6
Trousers 33.00 4.4 37.70 4.7 39.10 5.1
Shirts 18.00 6.1 20.45 6.4 23.50 6.2
Underwear 4.20 8.7 4.85 8.5 4.70 7.9

Using 1995 as the base year:

  1. Construct a simple aggregate index for the years 1995 to 1997 for the price of clothing.

    (6 marks)

  2. Construct a Laspeyres Index for the years 1995 to 1997 for the price of clothing.

    (8 marks)

  3. Discuss which of these index numbers is more suitable, giving your recommendation.

    (5 marks)

  4. Discuss critically the alternative methods that could be used to construct an appropriate index.

    (6 marks)

(Total 25 marks)

Answer

 

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Case Question 2

A company produces cans of soft drinks. At its Edinburgh site there are four similar production lines A, B, C and D for the canning process. The company operates a shift system with employees working the early shift, the late shift or the night shift. Production is carefully monitored and statistics are kept on the number of sub-standard cans produced during a shift. The table below shows the relevant data for production on Monday 26 January.

Number of sub-standard cans by shift and production line, Monday 26 January

Shift Production Line
  A B C D

Early 24 24 18 22
Late 30 20 27 27
Night 30 31 36 35

  1. Carry out a two-way analysis of variance. Produce an appropriate summary table.

    (10 marks)

  2. Test whether there is a significant difference in the average number of sub-standard cans for the four production lines?

    (5 marks)

  3. Test whether there is a significant difference in the average number of sub-standard cans for the three shifts?

    (5 marks)

  4. Write a report on your findings. Indicate any other information you would require. Detail your recommendations for further research and analyses.

    (5 marks)

(Total 25 marks)

Answer

 

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Case Question 3

A bank has carried out a study to determine the efficiency of its transaction handling. The bank uses three different methods for handling customers' transactions: human tellers, automated tellers with keyboard and voice activated automatic tellers. The bank has collected information on the value of the transaction (in sterling), the type of teller used and the time taken for the transaction.

The following notation has been used:

The information was run through a multiple regression model and the following print-out was generated:

Analysis of variance

  Degrees of Freedom Sums of Squares
Regression 2 78.26
Residual 60 64.57
Total 62 142.83
     
Variable Coefficient Standard Error
Constant 1.4279  
X1 0.0028 0.0012
X2 0.2141 0.1162

  1. What is the regression equation generated by this model?

    (2 marks)

  2. How long would you expect a transaction for 1200 to take if it was handled by the voice activated teller? Explain your reasoning.

    (2 marks)

  3. What statistic could be used to measure the proportion of the total variability in the data that is explained by the regression relationship? Calculate the statistic and comment on your findings.

    (3 marks)

  4. Test whether the multiple regression model is statistically significant, and explain what this means.

    (5 marks)

  5. Which, if any, of the independent variables are statistically significant? Explain your reasoning. What implications do these findings have?

    (4 marks)

  6. What additional analyses would you recommend and why?

    (4 marks)

  7. Write a report on your findings. Discuss the usefulness of this model for the bank and make recommendations for its possible improvement.

    (5 marks)

(Total 25 marks)

Answer

 

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Case Question 4

The Weight- watchers Association encourages its members to come to weekly meetings to discuss diets and healthy eating. At each meeting members are weighed to determine the change from the previous meeting and their overall progress.

The attached table shows the relevant information for the 10 members who attended the last group meeting


Member Weight (kilograms)
  05Jan98 21May98 28May98

Edna Mitchell 92.3 90.4 90.2
Mary Booth 70.2 65.3 66.2
Rosemary Wilson 64.7 67.2 67.2
Ronnie Biggs 86.7 85.1 83.9
Vivienne Black 68.4 65.6 65.4
Judy Russell 74.8 71.7 72.3
John Hunter 86.4 82.1 82.0
May West 95.2 88.4 88.3
Charlie Dawson 84.1 80.6 81.3
Margaret Ross 72.3 68.7 67.3

  1. Comparing the average weight as recorded at the beginning of January with that recorded on May 21, determine whether there is a significant decrease in the average weight of the members.

    (9 marks)

  2. Determine if there is a significant difference in the average weight as recorded at the two sessions in May 1998.

    (9 marks)

  3. Write a report on your findings. Describe any assumptions you have made. Indicate any further information you would require and any further analyses you would wish to perform.

    (7 marks)

(Total 25 marks)

Answer

 

(Total of 100 marks)

END OF PAPER

Solutions

Note from the Examiner: These answers are skeletal guidelines only. Though they contain the major points sought, the student's response should provide further insight into many of the points. Simply repeating the major bullet points does not necessarily demonstrate the required level of understanding of the subject.

Solutions to Case Questions

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Case Solution 1

  1. Construct a simple aggregate index for the years 1995 to 1997 for the price of clothing.

    (6 marks)

    See below.

    Table

    Clothing item 1995 1996 1997      
      Price () Quantity Price () Quantity Price () Quantity      
      P95 Q95 P96 Q96 P97 Q97 P95Q95 P96Q95 P96Q95

    Shoes 52.0 3.2 56.00 3.8 61.0 3.6 166.40 179.20 195.20
    Trousers 33.0 4.4 37.70 4.7 39.1 5.1 145.20 165.88 172.04
    Shirts 18.0 6.1 20.45 6.4 23.5 6.2 109.80 124.75 143.35
    Underwear 4.2 8.7 4.85 8.5 4.7 7.9 36.54 42.20 40.89
      P95 Q95 P96 Q96 P97 Q97 P95Q95 P96Q95 P96Q95
    Total 107.2   119.00   128.30 22.80 457.94 512.02 551.48

     

  2. Construct a Laspeyres Index for the years 1995 to 1997 for the price of clothing.

    (8 marks)

    A Laspeyres Index for the years 1995 to 1997 for the price of clothing:

  3. Discuss which of these index numbers is more suitable, giving your recommendation.

    (5 marks)

    Laspeyres index is most suitable since it takes account of the quantities of clothing purchased. Quantities used reflect those purchased in the base year 1995. Simple aggregate index gives equal importance (weight) to all of the items.

  4. Discuss critically the alternative methods that could be used to construct an appropriate index.

    (6 marks)

    A Paasche index could also be used. This would use current year quantities instead of base year quantities (1995). Often difficult to obtain accurate quantity figures for the current year.

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Case Solution 2

  1. Carry out a two-way analysis of variance. Produce an appropriate summary table.

    (10 marks)

    Initial calculations:

      A B C D Total Mean
    Early 24 24 18 22 88 22
    Late 30 20 27 27 104 26
    Night 30 31 36 35 132 33
                 
    Total 84 75 81 84 324  
    Mean 28 25 27 28 27  
                 

    ANOVA Table

    Source Degrees of Freedom Sums of Squares Mean Square F-ratio

    Treatments 3 18.00 6.00 0.42
    Blocks 2 248.00 124.00 8.65
    Error 6 86.00 14.33  
    Total 11 352.00    

  2. Test whether there is a significant difference in the average number of sub-standard cans for the four production lines?

    (5 marks)

    Test for a significant difference in the average number of sub-standard cans for the four production lines.:

    HO: A = B = C = D

    HA: A B C D

    Test Statistic Fcalc = 0.42 with (3,6) degrees of freedom

    From tables:

    Critical value of F at 5% = 4.76
    Critical value of F at 1% = 9.78

    Reject HA

    There is no significant difference in the average number of sub-standard cans for the four production lines.

  3. Test whether there is a significant difference in the average number of sub-standard cans for the three shifts?

    (5 marks)

    Test for a significant difference in the average number of sub-standard cans for the three shifts:

    HO: EARLY = LATE = NIGHT

    HA: EARLY LATE NIGHT

    Test Statistic Fcalc = 8.65 with (2,6) degrees of freedom

    From tables:

    Critical value of F at 5% = 5.14
    Critical value of F at 1% = 10.92

    Reject HO at the 5% level of significance.

    There is a significant difference in the average number of sub-standard cans for the three shifts, at the 5% level of significance.

  4. Write a report on your findings. Indicate any other information you would require. Detail your recommendations for further research and analyses.

    (5 marks)

    Looking at the average number of defective cans it seems that there are less on production line B. However this difference is not found to be significant.

    When the average number sub-standard is compared for the three shifts it can be seen that the night shift has the highest average (33) in contrast to that of the early shift (22).

    Further information needs to be obtained relating to the total production, comparisons with other days and periods, how sub-standard is defined.

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Case Solution 3

  1. What is the regression equation generated by this model?

    (2 marks)

    Y = 1.4279 + 0.0028 X1 + 0.2141 X2

  2. How long would you expect a transaction for 1200 to take if it was handled by the voice activated teller? Explain your reasoning.

    (2 marks)

    Using the above equation.

    This assumes that the regression equation is valid for a transaction of 1200, and that X1 = 1200 and X2 = 2.

  3. What statistic could be used to measure the proportion of the total variability in the data that is explained by the regression relationship? Calculate the statistic and comment on your findings.

    (3 marks)

    The statistic R2 could be used to measure the proportion of variability that is explained by the regression relationship.

    55% of the variability in the data is explained by the regression relationship.

  4. Test whether the multiple regression model is statistically significant, and explain what this means.

    (5 marks)

    Analysis of variance

    Source Degrees of Freedom Sums of Squares Mean Square F-ratio

    Regression 2 78.26 39.13 36.23
    Residual 60 64.57 1.08  
    Total 62 142.83    

    HO:

    Regression model does not explain a significance proportion of the variance in Y.

    HA:

    Regression model explains a significance proportion of the variance in Y.

    Fcalc = 36.23 with (2,60) degrees of freedom.

    From tables the critical values are:

    F-ratio at 5% = 3.15
    F-ratio at 1% = 4.13

    Reject HO at the 1% level of significance.

    The regression model explains a significant proportion of the variation in Y, at the 1% level of significance.

  5. Which, if any, of the independent variables are statistically significant? Explain your reasoning. What implications do these findings have?

    (4 marks)

    In order to test the significance of the independent variables the relevant t-statistics needs to be calculated.

    Variable Coefficient Standard Error t-statistic
    X1 0.0028 0.0012 2.33
    X2 0.2141 0.1162 1.84

    Testing the X1 = variable

    Model Y = b0 + b1X1 + b2X2

    Reject HO at the 5% level of significance.

    The regression coefficient b1 is significantly different from 0 at the 5% level of significance.

    Retain X1 transaction value (in ), in the regression equation.

    Testing the X2 = variable

    HO: b2 = 0

    HA: b2 0

    Reject HA.

    The regression coefficient b2 is not significantly different from 0.

    Remove X2 , method of handling transaction, from the regression model. Re-run model and test again.

  6. What additional analyses would you recommend and why?

    (4 marks)

    Check the residuals. Scatter diagram. Examine residuals for serial correlation and heteroscedascity. Check for randomness using a runs test.

  7. Write a report on your findings. Discuss the usefulness of this model for the bank and make recommendations for its possible improvement.

    (5 marks)

    The analysis has shown that the proposed regression model explains 55% of the variation in the Y values.

    Testing the significance of the model, it is found to be significant at the 5% level of significance.

    When the individual regression coefficients are tested it is found that variable X1 (transaction value) makes a significant contribution to the model and should be retained.

    The variable X2, method of handling transaction, does not make a significant contribution to the model and should be excluded.

    A new model should be derived and tested for significance.

    No details are given of the way the study was carried out. Was the sample representative?

    The sample size of 63 seems relatively small for the type of study carried out.

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Case Solution 4

See the summary of calculations below.

Summary of Calculations for (i) and (ii)

  Comparing weights (kg) in Jan with 21 May Comparing weights (kg) in 21 May with 28 May
Member Weight (kilograms) d (ddbar) (ddbar)2 Weight (kilograms) d (ddbar) (ddbar)2

  05 Jan 98 21 May 98       21 May 98 28 May 98      
Edna Mitchell 92.3 90.4 1.9 -1.1 1.21 90.4 90.2 0.2 0.1 0.01
Mary Booth 70.2 65.3 4.9 1.9 3.61 65.3 66.2 -0.9 -1.0 1.00
Rosemary Wilson 64.7 67.2 -2.5 -5.5 30.25 67.2 67.2 0.0 -0.1 0.01
Ronnie Biggs 86.7 85.1 1.6 -1.4 1.96 85.1 83.9 1.2 1.1 1.21
Vivienne Black 68.4 65.6 2.8 -0.2 0.04 65.6 65.4 0.2 0.1 0.01
Judy Russell 74.8 71.7 3.1 0.1 0.01 71.7 72.3 -0.6 -0.7 0.49
John Hunter 86.4 82.1 4.3 1.3 1.69 82.1 82.0 0.1 0.0 0.00
May West 95.2 88.4 6.8 3.8 14.44 88.4 88.3 0.1 0.0 0.00
Charlie Dawson 84.1 80.6 3.5 0.5 0.25 80.6 81.3 -0.7 -0.8 0.64
Margaret Ross 72.3 68.7 3.6 0.6 0.36 68.7 67.3 1.4 1.3 1.69
                     
    Total 30.00 0.00 53.82   Total 1.00 0.00 5.06
    Mean 3       Mean 0.1    
    Standard Deviation 2.45       Standard Deviation 0.75    
    Standard Error 0.77       Standard Error 0.24    
    t-statistic 3.88 with 9 df       t-statistic 0.42 with 9 df    

  1. Comparing the average weight as recorded at the beginning of January with that recorded on May 21, determine whether there is a significant decrease in the average weight of the members.

    (9 marks)

    In this case study paired differences should be used in the hypothesis tests.

    Comparing January with May 21st:

    HO: d 0   Since testing a decrease
    HA: d > 0    

    Reject HO at the 1% level of significance.

    There is a significant decrease in the average weight between January and May 21st at the 1% level of significance.

  2. Determine if there is a significant difference in the average weight as recorded at the two sessions in May 1998.

    (9 marks)

    Comparing May 21st with May 28th:

    HO: d = 0   Since testing a decrease
    HA: d 0    

    Reject HA.

    There is no significant difference in the average weights for May 21st and May 28th.

  3. Write a report on your findings. Describe any assumptions you have made. Indicate any further information you would require and any further analyses you would wish to perform.

    (7 marks)

    The data show that there is a significant decrease in the average weight recorded in Jan and on May 21st at the 1% level of significance.

    The greatest weight loss was recorded by May West. Only one member Rosemary Wilson showed a weight gain.

    When the average weights are compared for the two meetings in May it is found that there is no significant difference.

    The tests assume that the sample was taken at random and that it comes from a normally distributed population. However the members included are only those who turned up at the last meeting and who were at the previous two sessions. There may be another group who decided not to come because they had put on some weight.

    The data should also be examined for differences in the recorded weights for males and females. There is also likely to be a relationship to the initial weight when the member joined and to the length of membership of the Association.

    Information on the accuracy of the weighing process and on the type of clothing worn should also be obtained e.g. winter clothes rather than summer wear.

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