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Determine a seasonal index for each of the four quarters.
Find the number of visitors for each quarter of 2017 if 10% increase in the total number of visitors in 2016.
Determine the trend equation.
Project the number of visitors for 2017.
Find the seasonally adjusted forecasts.
Identify the best forecast.
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Answer to Problem 31CE
The seasonal indexes for the four quarters are 1.2046, 1.0206, 0.6297 and 01.1451, respectively.
The number of visitors for each quarter of 2017 if 10% increase in the total number of visitors in 2016 is 255.25 visitors per quarter.
The trend equation is
The number of visitors for 2017 are 242.0171, 248.634, 255.2509 and 261.8678.
The seasonally adjusted forecasts are 291.5338, 253.7559, 160.7315 and 299.8648.
The best forecast is the fourth quarter of 2017.
Explanation of Solution
Calculation:
Four-Year moving average:
Centered Moving Average:
Specific seasonal index:
Year | Quarter | Visitors |
Four-quarter moving average |
Centered Moving average | Specific seasonal |
2010 | 1 | 86 | |||
2 | 62 | ||||
3 | 28 | 70 | 0.4 | ||
4 | 94 | 67.5 | 75 | 1.253333 | |
2011 | 1 | 106 | 72.5 | 80 | 1.325 |
2 | 82 | 77.5 | 85 | 0.964706 | |
3 | 48 | 82.5 | 91.75 | 0.523161 | |
4 | 114 | 87.5 | 100.75 | 1.131514 | |
2012 | 1 | 140 | 96 | 109.75 | 1.275626 |
2 | 120 | 105.5 | 119 | 1.008403 | |
3 | 82 | 114 | 126.75 | 0.646943 | |
4 | 154 | 124 | 132 | 1.166667 | |
2013 | 1 | 162 | 129.5 | 136.75 | 1.184644 |
2 | 140 | 134.5 | 141.5 | 0.989399 | |
3 | 100 | 139 | 147.25 | 0.679117 | |
4 | 174 | 144 | 154.5 | 1.126214 | |
2014 | 1 | 188 | 150.5 | 162 | 1.160494 |
2 | 172 | 158.5 | 168.5 | 1.020772 | |
3 | 128 | 165.5 | 174 | 0.735632 | |
4 | 198 | 171.5 | 180.25 | 1.098474 | |
2015 | 1 | 208 | 176.5 | 187.25 | 1.110814 |
2 | 202 | 184 | 193.25 | 1.045278 | |
3 | 154 | 190.5 | 200.75 | 0.767123 | |
4 | 220 | 196 | 210.25 | 1.046373 | |
2016 | 1 | 246 | 205.5 | 219.5 | 1.120729 |
2 | 240 | 215 | 228 | 1.052632 | |
3 | 190 | 224 | |||
4 | 252 | 232 |
The Quarterly indexes are,
I | II | III | IV | |
2010 | 0.4 | 1.253333 | ||
2011 | 1.325 | 0.964706 | 0.523161 | 1.131514 |
2012 | 1.275626 | 1.008403 | 0.646943 | 1.166667 |
2013 | 1.184644 | 0.989399 | 0.679117 | 1.126214 |
2014 | 1.160494 | 1.020772 | 0.735632 | 1.098474 |
2015 | 1.110814 | 1.045278 | 0.767123 | 1.046373 |
2016 | 1.120729 | 1.052632 | ||
Mean | 1.1962 | 1.0135 | 0.6253 | 1.1371 |
Seasonal index:
Here,
Therefore,
The seasonal indexes are,
I | II | III | IV | |
2010 | 0.4 | 1.253333 | ||
2011 | 1.325 | 0.964706 | 0.523161 | 1.131514 |
2012 | 1.275626 | 1.008403 | 0.646943 | 1.166667 |
2013 | 1.184644 | 0.989399 | 0.679117 | 1.126214 |
2014 | 1.160494 | 1.020772 | 0.735632 | 1.098474 |
2015 | 1.110814 | 1.045278 | 0.767123 | 1.046373 |
2016 | 1.120729 | 1.052632 | ||
Mean | 1.1962 | 1.0135 | 0.6253 | 1.1371 |
Seasonal Index |
The total number of visitors in year 2016 is
The 10% of 928 visitors is
The number of visitors in 2017 is
Therefore, the number of visitors in each quarter of 2017 is
Trend Equation:
Step-by-step procedure to obtain the regression using the Excel:
- Enter the data for Year, Visitors and t in Excel sheet.
- Go to Data Menu.
- Click on Data Analysis.
- Select ‘Regression’ and click on ‘OK’
- Select the column of Visitors 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 output, the regression equation is
Projection of the number of visitors for 2017:
The t value for first quarter of 2017 is 29.
The t value for second quarter of 2017 is 30.
The t value for third quarter of 2017 is 31.
The t value for third quarter of 2017 is 32.
Seasonally Adjusted Forecast:
Estimated Visitors | Seasonal Index | |
242.0171 | 1.2046 | 291.5338 |
248.634 | 1.0206 | 253.7559 |
255.2509 | 0.6297 | 160.7315 |
261.8678 | 1.1451 | 299.8648 |
The seasonal index for the fourth quarter is high compared with remaining three quarters. Hence, the forecast for the fourth quarter is best.
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Chapter 18 Solutions
EBK STATISTICAL TECHNIQUES IN BUSINESS
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