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Which of the following statements is false regarding the following scatterplot of a time series with superimposed trend line?

a. The green trend line can be specified as  yˆt=b0 + b1t + b2t2
b. A linear trend line will under-forecast future values.
c. There is one change in direction of the time series.
d. b2 < 0

To determine which statement is false regarding the scatterplot of the time series with a superimposed trend line, let's analyze each option:

a. **The green trend line can be specified as \(\hat{y}_t = b_0 + b_1t + b_2t^2\)**:
   - The given trend line is a parabola, which can be represented by a quadratic equation. This statement is likely true.

b. **A linear trend line will under-forecast future values**:
   - A linear trend line would not fit the data well due to the apparent quadratic nature of the trend. This statement is likely true.

c. **There is one change in direction of the time series**:
   - The graph shows a single peak, indicating one change in direction (from increasing to decreasing). This statement is likely true.

d. **\(b_2 < 0\)**:
   - For a parabola that opens downwards, \(b_2\) in the equation \(\hat{y}_t = b_0 + b_1t + b_2t^2\) should be negative. This statement is true.

All the statements seem correct based on the information provided, making it unclear which is false. However, let's reconsider the interpretations:

- Statement b could be misleading because while it's true that a linear trend would not fit well, the term "under-forecast" may imply a specific context (like forecasting future values beyond the observed range). If not properly defined, this statement might be misleading.

Therefore, **statement b might be considered false due to the potential ambiguity in its interpretation**.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. What are the characteristics of a quadratic trend in time series analysis?
2. How do you determine the direction of a parabola in a quadratic equation?
3. Why might a linear trend line be insufficient for forecasting in this scenario?
4. What is the significance of the \(b_2\) coefficient in the quadratic equation?
5. How can you verify if the trend line fits the data well?
6. What methods can be used to detect changes in direction in time series data?
7. How does the choice of trend line affect forecasting accuracy?
8. What are other types of trend lines that could fit this data?

**Tip:** When analyzing trend lines in time series data, it's essential to consider the mathematical form of the trend and its suitability for the data pattern.

Explore the analysis of a scatterplot of a time series with a superimposed trend line. Understand the characteristics of quadratic trends, including the formulation of trend lines using quadratic equations. Address the false statement regarding the trend line and its implications for forecasting.

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