STATISTICS

FORECASTING METHODS FOR SOLUTION

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QUESTION

Question 1
A large firm employing thousands of employees has been accused of discriminating against its
female managers. The accusation is based on a random sample of 100 managers. The mean salary of
38 female managers is found to be significantly less than mean salary of 62 male managers at 1%
level of significance, which provides overwhelming evidence that male managers are paid more than
female managers.
The director of the firm asserts that the firm has a strict policy of equal pay for equal work and the
difference may be owing to other factors. To address this issue, he reports data on a number of other
variables together with dummy variable for gender as: Gender = 0 represents a female manager and =
1 a male manager.

(a)   Graph salary against each of education and experience variables, and report the kind of
relationship between these two sets of variables.
(b) Fit an appropriate regression model giving reason for the choice of your model, and report
the fitted equation.
(c) Interpret slope coefficients of variables education, experience and gender.
(d) Report value of coefficient of determination.  Is it a reasonable model?
(e) On the basis of your fitted model, can you say that this firm discriminates against the
female managers?
(f) Now split the data into two parts on the basis on gender and find the mean salary, mean
education and mean experience for male and female managers. Do you think the lower
level of education or the experience may be causing the female managers’ salaries to be
lower?  Just comment on the figures without performing any analysis.

Question 2
Refer to data sets of Question 2, Assignment 1.  To each data set fit a suitable regression model that
can be used for forecasting purposes.
Question 3
(a) Using the first 10 years of data from Question 3, Assignment 1, fit a suitable regression
model.
(b) Forecast average guest-night occupancy for each of the 4 quarters of 2008 and the first
quarter of 2009.
(c) Prepare a time plot of forecasts obtained using the decomposition method, the Holt-Winter
method (i.e., Questions 3 & 4) in Assignment 1, and forecast values from part (b) above and
the actual values for the 5 quarters of 2008 and 2009.  Which model seems to provide the
closest fit to data?
Question 4
The following table lists weekly sales of boxes of detergent sold at a local store. The

data are provided in file “Ass2-2011dat.xls”.
Weekly sales of boxes of detergent
time
index Y_t
time
index Y_t
2

time
index Y_t
time
index
Y_t
1  244  14  227  27  210  40  293
2  283  15  282  28  248  41  341
3  223  16  247  29  243  42  320
4  207  17  245  30  282  43  329
5  200  18  249  31  256  44  318
6  215  19  248  32  220  45  293
7  245  20  211  33  244  46  350
8  254  21  232  34  249  47  334
9  231  22  272  35  311  48  345
10  268  23  194 36 286 49  319
11  250  24  244 37 300 50  346
12  230  25  226 38 296 51  318
13  200  26  200 39 271 52  383
(a) Answer the following questions.
(i) Obtain time series plot and the estimated ACF for the above series.  Do the graphs
suggest that differencing is required to make the series stationary?  If so, what order of
differencing is sufficient to induce a stationary mean?
(ii) Identify an ARIMA model underlying this series and fit it.
(iii) Record in-sample residuals and obtain correlogram of these residuals to check the
adequacy of the fitted model. Does the fitted model seem a good model? Comment.
(b) Consider data of Assignment 1, Question 3 again.  Identify a suitable ARIMA
model for these data, but do not fit the model
.
SOLUTION

 

Answer 1 (A) part
Year t No. Visits
1966 1 651
1967 2 690
1968 3 864
1969 4 767
1970 5 746
1971 6 762
1972 7 705
1973 8 736
1974 9 778
1975 10 669
1976 11 629
1977 12 627
1978 13 524
1979 14 561
1980 15 641
1981 16 687
1982 17 723
1983 18 732
1984 19 641
1985 20 642
1986 21 640
1987 22 670
1988 23 693
1989 24 780
1990 25 610
1991 26 584
1992 27 740
1993 28 743
1994 29 788
1995 30 884
1996 31 804
1997 32 788
1998 33 750
1999 34 772
2000 35 716
2001 36 925
2002 37 762
2003 38 797
2004 39 850
2005 40 852
2006 41 901
2007 42 943
2008 43 918
2009 44 867
2010 45 889
 part (B)
Time series model is used here because collection of reading is related to the different period of times.
Year(x) x=t-1988 y(visits) xy x^2 y2
1966 -22 651 -14322 484 423801
1967 -21 690 -14490 441 476100
1968 -20 864 -17280 400 746496
1969 -19 767 -14573 361 588289
1970 -18 746 -13428 324 556516
1971 -17 762 -12954 289 580644
1972 -16 705 -11280 256 497025
1973 -15 736 -11040 225 541696
1974 -14 778 -10892 196 605284
1975 -13 669 -8697 169 447561
1976 -12 629 -7548 144 395641
1977 -11 627 -6897 121 393129
1978 -10 524 -5240 100 274576
1979 -9 561 -5049 81 314721
1980 -8 641 -5128 64 410881
1981 -7 687 -4809 49 471969
1982 -6 723 -4338 36 522729
1983 -5 732 -3660 25 535824
1984 -4 641 -2564 16 410881
1985 -3 642 -1926 9 412164
1986 -2 640 -1280 4 409600
1987 -1 670 -670 1 448900
1988 0 693 0 0 480249
1989 1 780 780 1 608400
1990 2 610 1220 4 372100
1991 3 584 1752 9 341056
1992 4 740 2960 16 547600
1993 5 743 3715 25 552049
1994 6 788 4728 36 620944
1995 7 884 6188 49 781456
1996 8 804 6432 64 646416
1997 9 788 7092 81 620944
1998 10 750 7500 100 562500
1999 11 772 8492 121 595984
2000 12 716 8592 144 512656
2001 13 925 12025 169 855625
2002 14 762 10668 196 580644
2003 15 797 11955 225 635209
2004 16 850 13600 256 722500
2005 17 852 14484 289 725904
2006 18 901 16218 324 811801
2007 19 943 17917 361 889249
2008 20 918 18360 400 842724
2009 21 867 18207 441 751689
2010 22 889 19558 484 790321
Sum 33441 34378 67819 25312447
y xy x2 y2
b   = 0.506908
a   = 743.1333
According to the model
Trend Equation is y= 743.13 + 0.507X
Answer  c
co-efficient = 0.507 It means that trent is directly proportional to the approx half of the independent variable.
Answer  d
co-efficient of deermination r   = 0.194358
r2  = 0.037775
r2 shows that co-efficient of deternination is is less related with existing model.
Answer e
Firm does not discrimate aginst female manager.

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Answer 1 (A) part
Year t No. Visits
1966 1 651
1967 2 690
1968 3 864
1969 4 767
1970 5 746
1971 6 762
1972 7 705
1973 8 736
1974 9 778
1975 10 669
1976 11 629
1977 12 627
1978 13 524
1979 14 561
1980 15 641
1981 16 687
1982 17 723
1983 18 732
1984 19 641
1985 20 642
1986 21 640
1987 22 670
1988 23 693
1989 24 780
1990 25 610
1991 26 584
1992 27 740
1993 28 743
1994 29 788
1995 30 884
1996 31 804
1997 32 788
1998 33 750
1999 34 772
2000 35 716
2001 36 925
2002 37 762
2003 38 797
2004 39 850
2005 40 852
2006 41 901
2007 42 943
2008 43 918
2009 44 867
2010 45 889
 part (B)
Time series model is used here because collection of reading is related to the different period of times.
Year(x) x=t-1988 y(visits) xy x^2 y2
1966 -22 651 -14322 484 423801
1967 -21 690 -14490 441 476100
1968 -20 864 -17280 400 746496
1969 -19 767 -14573 361 588289
1970 -18 746 -13428 324 556516
1971 -17 762 -12954 289 580644
1972 -16 705 -11280 256 497025
1973 -15 736 -11040 225 541696
1974 -14 778 -10892 196 605284
1975 -13 669 -8697 169 447561
1976 -12 629 -7548 144 395641
1977 -11 627 -6897 121 393129
1978 -10 524 -5240 100 274576
1979 -9 561 -5049 81 314721
1980 -8 641 -5128 64 410881
1981 -7 687 -4809 49 471969
1982 -6 723 -4338 36 522729
1983 -5 732 -3660 25 535824
1984 -4 641 -2564 16 410881
1985 -3 642 -1926 9 412164
1986 -2 640 -1280 4 409600
1987 -1 670 -670 1 448900
1988 0 693 0 0 480249
1989 1 780 780 1 608400
1990 2 610 1220 4 372100
1991 3 584 1752 9 341056
1992 4 740 2960 16 547600
1993 5 743 3715 25 552049
1994 6 788 4728 36 620944
1995 7 884 6188 49 781456
1996 8 804 6432 64 646416
1997 9 788 7092 81 620944
1998 10 750 7500 100 562500
1999 11 772 8492 121 595984
2000 12 716 8592 144 512656
2001 13 925 12025 169 855625
2002 14 762 10668 196 580644
2003 15 797 11955 225 635209
2004 16 850 13600 256 722500
2005 17 852 14484 289 725904
2006 18 901 16218 324 811801
2007 19 943 17917 361 889249
2008 20 918 18360 400 842724
2009 21 867 18207 441 751689
2010 22 889 19558 484 790321
Sum 33441 34378 67819 25312447
y xy x2 y2
b   = 0.506908
a   = 743.1333
According to the model
Trend Equation is y= 743.13 + 0.507X
Answer  c
co-efficient = 0.507 It means that trent is directly proportional to the approx half of the independent variable.
Answer  d
co-efficient of deermination r   = 0.194358
r2  = 0.037775
r2 shows that co-efficient of deternination is is less related with existing model.
Answer e
Firm does not discrimate aginst female manager.

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