Question text 1Mark A coffee chain tracks quarterly sales (thousand dollars) of its new premium blend from 2020 to 2024. They identify both trend and seasonal components in the time series using a multiplicative time series model. The trend equation is estimated as: What is the missing seasonal index for quarter 3? State your answer to two decimal places. Answer: Answer 1[input]Notes Report question issue Question 11 Notes

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BlackTom AI · Accepted Answer

The problem describes a multiplicative time series model with seasonal indices for four quarters. In a multiplicative seasonal model, the seasonal indices are typically scaled so that their average (or sum, depending on convention) over a full cycle equals 1 (or 4 if the indices are expressed in terms of a sum to 4). Here, the provided seasonal indices are:
- Quarter 1: 0.96
- Quarter 2: 1.05
- Quarter 3: ?
- Quarter 4: 0.90

If we assume the sum of the four seasonal indices equals 4 (a common convention when indices are anchored around 1 on average), we can solve for the missing Quarter 3 index as follows:
0.96 + 1.05 + ? + 0.90 = 4
First, add the known values: 0.96 + 1.05 + 0.90 = 2.91
Then, ? = 4 − 2.91 = 1.09
Thus, the missing seasonal index for Quarter 3 is 1.09. To express the result to two decimal places, we keep 1.09.

Key concepts to recall: in multiplicative seasonal models, the seasonal effect multiplies the trend component, and the seasonal indices across a complete cycle are typically normalized so their average is 1 (or their sum matches the cycle length, here 4). This ensures the model decomposes cleanly into Trend × Seasonal factors. Potential pitfalls include treating the indices as independent or assuming they must all be greater than 1; here the given values show Quarter 1 and Quarter 4 are below 1 while Quarter 2 is above 1, which is consistent with a balanced seasonal pattern once Quarter 3 is correctly computed.

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