From the provided image, we can analyze the data based on the time series plot. Here's how we tackle this problem step by step:

Part A: Type of Pattern in the Data

The data shows a linear trend, as indicated by the plot and the description in the image.

Part B: Develop the Linear Trend Equation

To find the linear trend equation, we need the following:

1. General Equation: $T_t = a + b \cdot t$, where:

   • $a$ is the intercept (value of $T_t$ when $t = 0$).
   • $b$ is the slope (change in $T_t$ per unit change in $t$).

2. Calculation of Slope ($b$): $b = \frac{\sum t \cdot T_t - \frac{\sum t \cdot \sum T_t}{n}}{\sum t^2 - \frac{(\sum t)^2}{n}}$

   Where:

   • $\sum t$: Sum of all $t$-values.
   • $\sum T_t$: Sum of all $T_t$-values.
   • $\sum t \cdot T_t$: Sum of the product of $t$ and $T_t$ for each point.
   • $\sum t^2$: Sum of the squares of $t$-values.
   • $n$: Number of data points.

3. Intercept ($a$): $a = \frac{\sum T_t}{n} - b \cdot \frac{\sum t}{n}$

We would need to extract exact data points from the graph or table (if available) to perform these calculations accurately. If you'd like, I can guide you through these steps with the data provided or approximate based on visible points.

Part C: Forecast for $t = 8$

Once the equation $T_t = a + b \cdot t$ is established, substitute $t = 8$ to compute the forecasted value.

In the image, the forecast is already given as 80.3, so this step confirms the correctness of the trend equation once determined.

Would you like help extracting data from the graph to calculate $a$ and $b$, or is there any part you'd like clarified further?

Here are 5 related questions to deepen understanding:

1. How is the slope $b$ in a linear trend equation interpreted in a time series?
2. What assumptions underlie the linear trend model in forecasting?
3. How does the choice of time period affect trend estimation in real-world applications?
4. What are potential limitations of using a linear trend model for longer forecasts?
5. Can non-linear trends be transformed into linear ones? If so, how?

Tip: Always validate the linearity assumption by checking residuals to ensure the model fits the data well