Question 1178847: The human resources manager for a medium-sized business is interested in predicting the dollar
value of medical expenditures filed by employees of her company for the year 2019. From her
company's database she has collected the following information showing the dollar value of
medical expenditures made by employees for the previous four years: Consider the following
data, given in millions of dollars:
2015 2016
1st quarter: 152 1st quarter: 217
2nd quarter: 62 2nd quarter: 209
3rd quarter: 157 3rd quarter: 202
4th quarter: 167 4th quarter: 221
2017 2018
pt quarter: 182 1st quarter: 236
2nd quarter: 192 2nd quarter: 242
3rd quarter: 191 3rd quarter: 231
4th quarter: 197 4th quarter: 224
1. Plot these data. Based on your visual observations, what time-series components
are present in the data?
2. Explain whether the additive or multiplicative decomposition model would be more
appropriate.
3.Determine the seasonal index for each quarter (based on the above selected
method).
4.Use the seasonal index values computed in part iii to provide seasonal adjusted
forecasts for each quarter of 2019.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's analyze this time-series data:
**1. Plot the Data and Identify Time-Series Components:**
* **Plotting:**
* Create a graph with the quarters (2015-Q1, 2015-Q2, ..., 2018-Q4) on the x-axis and the medical expenditures (in millions of dollars) on the y-axis.
* Connect the data points with lines to show the trend.
* **Observations:**
* **Trend:** There is a clear upward trend in medical expenditures over the years.
* **Seasonality:** There appears to be a seasonal pattern, with higher expenditures in the first and fourth quarters and lower expenditures in the second and third quarters.
* **Random Variation:** There are some fluctuations that cannot be explained by the trend or seasonality, indicating random variation.
**2. Additive vs. Multiplicative Decomposition Model:**
* **Additive Model:** Assumes the components (trend, seasonality, random) are added together: Y = Trend + Seasonality + Random.
* **Multiplicative Model:** Assumes the components are multiplied together: Y = Trend * Seasonality * Random.
* **Choice:**
* In this case, the **additive model** seems more appropriate. The seasonal fluctuations appear to be relatively constant in magnitude over time, rather than increasing proportionally with the trend. If the seasonal fluctuations were growing proportionally with the trend, then the multiplicative model would be more appropriate.
**3. Determine the Seasonal Index for Each Quarter (Additive Model):**
1. **Calculate the Quarterly Averages:**
* Q1: (152 + 217 + 182 + 236) / 4 = 196.75
* Q2: (62 + 209 + 192 + 242) / 4 = 176.25
* Q3: (157 + 202 + 191 + 231) / 4 = 195.25
* Q4: (167 + 221 + 197 + 224) / 4 = 202.25
2. **Calculate the Overall Average:**
* (196.75 + 176.25 + 195.25 + 202.25) / 4 = 192.625
3. **Calculate the Seasonal Index:**
* Q1: 196.75 - 192.625 = 4.125
* Q2: 176.25 - 192.625 = -16.375
* Q3: 195.25 - 192.625 = 2.625
* Q4: 202.25 - 192.625 = 9.625
**4. Seasonally Adjusted Forecasts for 2019:**
1. **Estimate the Trend for 2019:**
* We can use a simple linear trend or a more complex method. For simplicity, let's use the average increase per year.
* Average increase per year: ((236+242+231+224) - (152+62+157+167)) / 4 = 63.5
* Estimated 2019 average: 233.25 + 63.5 = 296.75.
2. **Add the Seasonal Indexes:**
* 2019-Q1: 296.75 + 4.125 = 300.875
* 2019-Q2: 296.75 - 16.375 = 280.375
* 2019-Q3: 296.75 + 2.625 = 299.375
* 2019-Q4: 296.75 + 9.625 = 306.375
**Therefore, the seasonally adjusted forecasts for 2019 (in millions of dollars) are:**
* 2019-Q1: 300.875
* 2019-Q2: 280.375
* 2019-Q3: 299.375
* 2019-Q4: 306.375
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