You take a random sample of 1000 Adult men (Age 19-65) in New York city and measured their actual weight and their Ideal weight. You find the following regression function

ideal = 52.5 + 0.7 * actual

Explain if the following inferences are correct

Your cousin weighs 70 pounds. You infer that his ideal weight must be 52.5+0.7*70=101.5 pounds

Your 52 year old Father living in Boston weighs 170 pounds. You expect that his ideal weight must be 52.5+0.7*200= 192.5 pounds.

You have 10  friends (Adult men) at Columbia each weighing close to 160 pounds. You expect that their ideal weight on average should be close to 52.5+0.7*160=164.5 pounds.

Your Uncle living in Las Vegas weighs 240 pounds. You expect his ideal weight should be close to 52.5+0.7*240= 220.5 pounds.

Let's analyze each inference based on the provided regression function:

### 1. **Your cousin weighs 70 pounds. You infer that his ideal weight must be 52.5 + 0.7 × 70 = 101.5 pounds.**

**Explanation:**  
This inference is technically correct in terms of applying the regression function. However, there is a significant issue. The sample population used to derive the regression function was adult men in New York City, with the assumption that they fall within the typical weight range of adult men (likely far above 70 pounds). Since 70 pounds is extremely low for an adult man, this inference may not be reliable because it is outside the range of the data used to create the model. This is an example of *extrapolation* beyond the range of observed data, which can lead to inaccurate predictions.

### 2. **Your 52-year-old father living in Boston weighs 170 pounds. You expect that his ideal weight must be 52.5 + 0.7 × 200 = 192.5 pounds.**

**Explanation:**  
This inference is incorrect. The calculation you provided (52.5 + 0.7 × 200) mistakenly uses the weight 200 pounds instead of the actual weight of 170 pounds. The correct calculation should be:
\[
\text{Ideal weight} = 52.5 + 0.7 \times 170 = 171.5 \text{ pounds}
\]
Moreover, the regression function was derived from data specific to adult men in New York City, so applying it to someone in Boston assumes that the relationship between actual and ideal weight is the same across different regions. While this might be a reasonable assumption, it's still a form of generalization.

### 3. **You have 10 friends (Adult men) at Columbia each weighing close to 160 pounds. You expect that their ideal weight on average should be close to 52.5 + 0.7 × 160 = 164.5 pounds.**

**Explanation:**  
This inference is correct. Since the average weight of your friends is close to 160 pounds, applying the regression function to calculate their average ideal weight is appropriate:
\[
\text{Ideal weight (average)} = 52.5 + 0.7 \times 160 = 164.5 \text{ pounds}
\]
This falls within a reasonable weight range for adult men, making this inference valid within the context of the regression model.

### 4. **Your uncle living in Las Vegas weighs 240 pounds. You expect his ideal weight should be close to 52.5 + 0.7 × 240 = 220.5 pounds.**

**Explanation:**  
This inference correctly applies the regression function:
\[
\text{Ideal weight} = 52.5 + 0.7 \times 240 = 220.5 \text{ pounds}
\]
However, like the previous regional example, it assumes that the relationship between actual and ideal weight is the same for men in Las Vegas as it is for those in New York City. Additionally, 240 pounds is on the higher end of the weight spectrum, but still within a reasonable range for the model, so this inference is likely acceptable.

### Summary:
- **Cousin (70 pounds)**: Incorrect due to extrapolation beyond the data range.
- **Father (170 pounds)**: Incorrect due to a calculation error (and some regional generalization).
- **Friends (160 pounds)**: Correct and valid within the model.
- **Uncle (240 pounds)**: Correct, but with a caution about regional generalization.

### Tip:
When applying a regression model, be cautious of extrapolation (making predictions outside the range of the observed data) and generalizing findings to different populations or regions.

### Follow-up Questions:
1. What is extrapolation in regression analysis and why can it be problematic?
2. How does the assumption of a linear relationship in a regression model affect predictions?
3. What are potential issues with applying a model derived from one population to a different population?
4. How would you check the accuracy of a regression model before using it for predictions?
5. What other factors, besides actual weight, might influence ideal weight that this regression model does not account for?

Explore the application of linear regression in predicting ideal weights based on actual weights. Assess if inferences like predicting ideal weights for specific individuals using a given regression function are valid or not. Includes detailed examples and explanations.

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