Let's break this down step-by-step.

(a) Linear Regression and Correlation Coefficient:

Given the data points:

• $t$ (years since 1994): $0, 2, 4, 6, 8, 10$
• $p$ (price in million dollars): $0.68, 0.70, 0.90, 1.25, 1.50, 1.90$

We have already done the linear regression, and based on your provided results:

• The linear regression line is $p(t) = 0.14t + 0.69$.
• The correlation coefficient $r = 0.97$, indicating a strong positive linear relationship.

(b) Graph Suggestion for Nonlinear Relationship:

From the provided description, the graph does not strongly suggest a nonlinear relationship because:

• The points are mostly scattered around the regression line.
• There’s no clear curving pattern that would indicate that the relationship between $t$ and $p$ is nonlinear.

Thus, the correct conclusion would be: No, the points appear to be randomly scattered about the regression line, which does not imply a nonlinear relationship.

(c) Residuals:

The residuals are the differences between the observed values and the predicted values from the regression line.

The residual for each year $t$ is calculated as: $\text{Residual} = p_{\text{observed}} - p_{\text{predicted}}$ For each year, after applying the regression equation, the residuals are calculated as:

• For $t = 0$: Predicted $p(0) = 0.69$, so Residual = $0.68 - 0.69 = -0.01$
• For $t = 2$: Predicted $p(2) = 0.97$, so Residual = $0.70 - 0.97 = -0.27$
• For $t = 4$: Predicted $p(4) = 1.25$, so Residual = $0.90 - 1.25 = -0.35$
• For $t = 6$: Predicted $p(6) = 1.53$, so Residual = $1.25 - 1.53 = -0.28$
• For $t = 8$: Predicted $p(8) = 1.81$, so Residual = $1.50 - 1.81 = -0.31$
• For $t = 10$: Predicted $p(10) = 2.09$, so Residual = $1.90 - 2.09 = -0.19$

Conclusion on Residuals:

• The residuals do not display a simple increasing or decreasing pattern, which suggests that while the linear model provides a good fit, there may be some fluctuations indicating that a nonlinear model could better fit the data in some cases.
• However, the residuals do not show any strong indication of curvature, so a nonlinear model is not necessarily required.

Final Answer:

The residuals are negative across the board, showing some deviation, but not suggesting a strong nonlinear pattern.

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Let me know if you need more clarification or details on any part! Here are some additional questions that could help expand on this topic:

1. How do you interpret the value of the correlation coefficient in a real-world context?
2. What would be the implication if the correlation coefficient was much lower (e.g., 0.2)?
3. How would the residuals change if a different regression model (e.g., quadratic) was used?
4. What does it mean for a model to be "a good fit" in terms of residuals?
5. How might you visually determine the appropriateness of a nonlinear model by looking at the residual plot?

Tip: A residual plot is often the best way to check for nonlinearity. If the points in the residual plot form a clear pattern (e.g., curve), then a nonlinear model might be needed.