The linear trend forecasting equation for an annual time series containing 40 values​ (from 1966 to 2005​) on net sales​ (in billions of​ dollars) is shown below. Complete​ (a) through​ (e) below.
ModifyingAbove Upper Y with caret Subscript iequals1.3plus0.6Upper X Subscript i
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What is the projected trend forecast two years after the last​ value?

Given the linear trend forecasting equation for the annual time series:

\[
\hat{Y}_i = 1.3 + 0.6X_i
\]

where:
- \(\hat{Y}_i\) represents the forecasted value for the year \(i\),
- \(X_i\) represents the time period corresponding to the year \(i\).

### Step-by-Step Solution:

1. **Identify the last value's year and its corresponding \(X_i\):**
   - The series starts in 1966 and ends in 2005, giving us 40 values.
   - So, for 2005 (the last year), \(X_{40} = 40\).

2. **Determine \(X\) for two years after the last value:**
   - Two years after 2005 is 2007.
   - Therefore, the corresponding \(X\) value for 2007 would be \(X_{42} = 42\).

3. **Substitute \(X_{42}\) into the equation:**
   \[
   \hat{Y}_{42} = 1.3 + 0.6 \times 42
   \]

4. **Calculate the projected trend forecast:**
   \[
   \hat{Y}_{42} = 1.3 + 25.2 = 26.5 \text{ (in billions of dollars)}
   \]

### Final Answer:
The projected trend forecast for the year 2007, which is two years after the last value (2005), is **26.5 billion dollars**.

Would you like more details on this, or do you have any other questions?

Here are 5 related questions that could expand your understanding:
1. How would the trend forecast change if the slope of the trend line increased?
2. What would the forecast be for five years after 2005?
3. How does the intercept of the trend equation affect the forecast?
4. What is the significance of the slope in the context of this time series?
5. How would the forecast change if there was a negative trend in the data?

**Tip:** Understanding how each component (slope and intercept) of a trend equation affects your forecast can help you interpret trends and make better predictions for future values.

Understand how to project trends using the linear trend forecasting equation for an annual time series. Learn how to calculate the forecast for 2007 based on data from 1966 to 2005, including step-by-step solutions and implications of changing trend components.

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