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a.
Determine the typical seasonal patterns for the production data using the ratio-to-moving-average method.
a.
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Answer to Problem 26CE
The typical seasonal patterns for sales are 0.7549, 0.9913, 1.4043 and 0.8495, respectively.
Explanation of Solution
Calculation:
Four-Year moving average:
Centered Moving Average:
Specific seasonal index:
Year | Quarter | Board ft Millions |
Four-quarter moving average |
Centered Moving average | Specific seasonal |
2012 | Winter | 7.8 | |||
Spring | 10.2 | ||||
Summer | 14.7 | 10.3875 | 1.41516 | ||
Fall | 9.3 | 10.5 | 10.45 | 0.88995 | |
2013 | Winter | 6.9 | 10.275 | 10.975 | 0.6287 |
Spring | 11.6 | 10.625 | 11.325 | 1.02428 | |
Summer | 17.5 | 11.325 | 11.575 | 1.51188 | |
Fall | 9.3 | 11.325 | 11.5875 | 0.80259 | |
2014 | Winter | 8.9 | 11.825 | 11.075 | 0.80361 |
Spring | 9.7 | 11.35 | 10.9 | 0.88991 | |
Summer | 15.3 | 10.8 | 11.225 | 1.36303 | |
Fall | 10.1 | 11 | 11.7875 | 0.85684 | |
2015 | Winter | 10.7 | 11.45 | 12.3125 | 0.86904 |
Spring | 12.4 | 12.125 | 12.575 | 0.98608 | |
Summer | 16.8 | 12.5 | 12.4625 | 1.34804 | |
Fall | 10.7 | 12.65 | 12.425 | 0.86117 | |
2016 | Winter | 9.2 | 12.275 | 12.6125 | 0.72944 |
Spring | 13.6 | 12.575 | 12.6 | 1.07937 | |
Summer | 17.1 | 12.65 | |||
Fall | 10.3 | 12.55 |
The Quarterly indexes are,
Winter | Spring | Summer | Fall | |
2012 | 1.415162 | 0.889952 | ||
2013 | 0.628702 | 1.024283 | 1.511879 | 0.802589 |
2014 | 0.803612 | 0.889908 | 1.363029 | 0.85684 |
2015 | 0.869036 | 0.986083 | 1.348044 | 0.861167 |
2016 | 0.729435 | 1.079365 | ||
Mean | 0.757696 | 0.99491 | 1.409529 | 0.852637 |
Typical Seasonal index:
Here,
Therefore,
The typical seasonal indexes are,
Winter | Spring | Summer | Fall | |
2012 | 1.415162 | 0.889952 | ||
2013 | 0.628702 | 1.024283 | 1.511879 | 0.802589 |
2014 | 0.803612 | 0.889908 | 1.363029 | 0.85684 |
2015 | 0.869036 | 0.986083 | 1.348044 | 0.861167 |
2016 | 0.729435 | 1.079365 | ||
Mean | 0.757696 | 0.99491 | 1.409529 | 0.852637 |
Typical Index | 0.7549 | 0.9913 | 1.4043 | 0.8495 |
b.
Interpret the typical seasonal pattern.
b.
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Explanation of Solution
The typical seasonal index for the Summer quarter is 1.4043, which is largest compared with other three quarters. That is, the production is largest in the third quarter and moreover, it represent above the average quarters because seasonal index is greater than 1. The Winter, Spring and Fall quarters represent below the average quarters because seasonal indexes for the three quarters are less than 1.
c.
Determine the trend equation.
c.
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Answer to Problem 26CE
The trend equation is
Explanation of Solution
Calculation:
Deseasonalization:
Board ft Millions | Typical Seasonal Index | Deseasonalized production |
7.8 | 0.7549 | 10.33249437 |
10.2 | 0.9913 | 10.28951881 |
14.7 | 1.4043 | 10.46784875 |
9.3 | 0.8495 | 10.94761624 |
6.9 | 0.7549 | 9.140283481 |
11.6 | 0.9913 | 11.70180571 |
17.5 | 1.4043 | 12.4617247 |
9.3 | 0.8495 | 10.94761624 |
8.9 | 0.7549 | 11.78964101 |
9.7 | 0.9913 | 9.785130637 |
15.3 | 1.4043 | 10.89510788 |
10.1 | 0.8495 | 11.88934667 |
10.7 | 0.7549 | 14.17406279 |
12.4 | 0.9913 | 12.50882679 |
16.8 | 1.4043 | 11.96325571 |
10.7 | 0.8495 | 12.5956445 |
9.2 | 0.7549 | 12.18704464 |
13.6 | 0.9913 | 13.71935842 |
17.1 | 1.4043 | 12.17688528 |
10.3 | 0.8495 | 12.12477928 |
Assign t value as 1 for first quarter of 2012, 2 for the second quarter of 2012 and so on.
Step-by-step procedure to obtain the regression using the Excel:
- Enter the data for Deseasonalized production and t in Excel sheet.
- Go to Data Menu.
- Click on Data Analysis.
- Select ‘Regression’ and click on ‘OK’
- Select the column of Deseasonalized production under ‘Input Y
Range ’. - Select the column of t under ‘Input X Range’.
- Click on ‘OK’.
Output for the Regression obtained using the Excel is as follows:
From the Excel output, the regression equation is
d.
Find the seasonally adjusted production for four quarters of 2017.
d.
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Answer to Problem 26CE
The seasonally adjusted production for four quarters of 2017 are 9.8884, 13.1261, 18.7946 and 11.4903, respectively.
Explanation of Solution
Calculation:
From the output, the regression equation is
The t value for first quarter of 2017 is 21.
The t value for second quarter of 2017 is 22.
The t value for third quarter of 2017 is 23.
The t value for fourth quarter of 2017 is 24.
Seasonally Adjusted Forecast:
Estimated Visitors | Seasonal Index | |
13.0990 | 0.7549 | 9.8884 |
13.2413 | 0.9913 | 13.1261 |
13.3836 | 1.4043 | 18.7946 |
13.5259 | 0.8495 | 11.4903 |
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Chapter 18 Solutions
EBK STATISTICAL TECHNIQUES IN BUSINESS
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