Question

In: Statistics and Probability

AmeriPlas, Inc., produces 20-ounce plastic drinking cups that are embossed with the names of prominent beers...

  1. AmeriPlas, Inc., produces 20-ounce plastic drinking cups that are embossed with the names of prominent beers and soft drinks. The sales data are:

Date

Sales

Jan-13

40,358

Feb-13

45,002

Mar-13

63,165

Apr-13

57,479

May-13

52,308

Jun-13

60,062

Jul-13

51,694

Aug-13

54,469

Sep-13

48,284

Oct-13

45,239

Nov-13

40,665

Dec-13

47,968

Jan-14

37,255

Feb-14

38,521

Mar-14

55,110

Apr-14

51,389

May-14

58,068

Jun-14

64,028

Jul-14

52,873

Aug-14

62,584

Sep-14

53,373

Oct-14

52,060

Nov-14

51,727

Dec-14

51,455

Jan-15

47,906

Feb-15

53,570

Mar-15

69,189

Apr-15

64,346

May-15

77,267

Jun-15

75,787

Jul-15

74,052

Aug-15

79,756

Sep-15

73,292

Oct-15

77,207

Nov-15

68,423

Dec-15

67,274

Jan-16

65,711

Feb-16

68,005

Mar-16

78,029

Apr-16

92,764

May-16

97,175

Jun-16

86,255

Jul-16

90,496

Aug-16

87,602

Sep-16

83,577

Oct-16

92,610

Nov-16

73,949

Dec-16

77,711

Page 277

(c5p12)

  1. Prepare a time-series plot of the sales data. Does there appear to be a regular pattern of movement in the data that may be seasonal? Ronnie Mills, the product manager for this product line, believes that her brief review of sales data for the four-year period indicates that sales are slowest in November, December, January, and February than in other months. Do you agree?

It seems that the sales are the lowest in November, December, January, and February than in other months. Therefore, I agree with this statement.

  1. Since production is closely related to orders for current shipment, Ronnie would like to have a monthly sales forecast that incorporates monthly fluctuations. She has asked you to develop a trend model that includes a time index and dummy variables for all but the above mentioned four months. Do these results support Ronnie’s observations? Explain.

The trend model given with this equation is 45518.48+927.7418*t-15281.1*x2-12738.8*x2-13645*x11-12158.7*x12. They support Ronnie’s observations because

  1. Ronnie believes that there has been some increase in the rate of sales growth. To test this and to include such a possibility in the forecasting effort, she has asked that you add the square of the time index (T) to your model (call this new term T2). Is there any evidence of increase of sales growth? Compare the results of this model with those found in part (b).
  1. Use the model in part (c) to forecast sales for 2017. Calculate the mean absolute percentage error (MAPE) for the first six months of 2017. Actual sales for those six months were:

Jan-2017

87327

Feb-2017

84772

Mar-2017

112499

Apr-2017

102633

May-2017

112996

Jun-2017

119807

Solutions

Expert Solution

Solution:

  1. Prepare a time-series plot of the sales data. Does there appear to be a regular pattern of movement in the data that may be seasonal? Ronnie Mills, the product manager for this product line, believes that her brief review of sales data for the four-year period indicates that sales are slowest in November, December, January, and February than in other months. Do you agree?

The time-series plot is:

Yes, sales data for the four-year period indicates that sales are slowest in November, December, January, and February than in other months.

  1. Since production is closely related to orders for current shipment, Ronnie would like to have a monthly sales forecast that incorporates monthly fluctuations. She has asked you to develop a trend model that includes a time index and dummy variables for all but the above mentioned four months. Do these results support Ronnie’s observations? Explain.

The output is:

Regression Statistics
Multiple R 0.93343
R Square 0.871291
Adjusted R Square 0.822079
Standard Error 6688.479
Observations 48
ANOVA
df SS MS F Significance F
Regression 13 1.03E+10 7.92E+08 17.70482 1.88E-11
Residual 34 1.52E+09 44735747
Total 47 1.18E+10
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 45518.48 2108.71 21.58594 2E-21 41233.07 49803.9 41233.07 49803.9
t 924.7418 71.26704 12.97573 1E-14 779.9097 1069.574 779.9097 1069.574
x1 -15281.1 3568.693 -4.28198 0.000143 -22533.5 -8028.62 -22533.5 -8028.62
x2 -12738.8 3561.57 -3.57674 0.001069 -19976.8 -5500.84 -19976.8 -5500.84
x3 0 0 65535 #NUM! 0 0 0 0
x4 0 0 65535 #NUM! 0 0 0 0
x5 0 0 65535 #NUM! 0 0 0 0
x6 0 0 65535 #NUM! 0 0 0 0
x7 0 0 65535 #NUM! 0 0 0 0
x8 0 0 65535 #NUM! 0 0 0 0
x9 0 0 65535 #NUM! 0 0 0 0
x10 0 0 65535 #NUM! 0 0 0 0
x11 -13645 3561.57 -3.83117 #NUM! -20883 -6407.01 -20883 -6407.01
x12 -12158.7 3568.693 -3.40706 0.001703 -19411.2 -4906.28 -19411.2 -4906.28

The trend model that includes a time index and dummy variables is:

Sales = 45518.48 + 924.7418*t - 15281.1*x2 - 12738.8*x2 - 13645*x11 - 12158.7*x12

These results support Ronnie’s observations because the four-year period indicates that sales are slowest in November, December, January, and February than in other months.

  1. Ronnie believes that there has been some increase in the rate of sales growth. To test this and to include such a possibility in the forecasting effort, she has asked that you add the square of the time index (T) to your model (call this new term T2). Is there any evidence of increase of sales growth? Compare the results of this model with those found in part (b).

The output is:

Regression Statistics
Multiple R 0.952126
R Square 0.906544
Adjusted R Square 0.866896
Standard Error 5785.089
Observations 48
ANOVA
df SS MS F Significance F
Regression 14 1.07E+10 7.65E+08 22.86478 5.18E-13
Residual 33 1.1E+09 33467250
Total 47 1.18E+10
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 53731.77 2724.102 19.72459 7.81E-20 48189.55 59274 48189.55 59274
t -45.0203 246.7313 -0.18247 0.856333 -546.999 456.9583 -546.999 456.9583
t² 19.79106 4.875659 4.059156 0.000284 9.87146 29.71067 9.87146 29.71067
x1 -15775.9 3089.087 -5.10696 1.35E-05 -22060.6 -9491.06 -22060.6 -9491.06
x2 -13035.7 3081.389 -4.23046 0.000174 -19304.8 -6766.55 -19304.8 -6766.55
x3 0 0 65535 #NUM! 0 0 0 0
x4 0 0 65535 #NUM! 0 0 0 0
x5 0 0 65535 #NUM! 0 0 0 0
x6 0 0 65535 #NUM! 0 0 0 0
x7 0 0 65535 #NUM! 0 0 0 0
x8 0 0 65535 #NUM! 0 0 0 0
x9 0 0 65535 #NUM! 0 0 0 0
x10 0 0 65535 #NUM! 0 0 0 0
x11 -13941.9 3081.389 -4.52454 #NUM! -20211 -7672.73 -20211 -7672.73
x12 -12653.5 3089.087 -4.0962 0.000256 -18938.3 -6368.72 -18938.3 -6368.72

There is no evidence of the increase in sales growth.

  1. Use the model in part (c) to forecast sales for 2017. Calculate the mean absolute percentage error (MAPE) for the first six months of 2017. Actual sales for those six months were:
Actual Forecasted Error
87327 83268.27 4058.732 5%
84772 87922.73 3150.732 4%
112499 102912.3 9586.707 9%
102633 104905.8 2272.752 2%
112996 106938.8 6057.206 5%
119807 109011.4 10795.58 9%
MAPE 6%

orchestra answered 2 years ago
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