Regents Exam Recap: January 2012 - Mr Honner - Free Printable

It looks like you've uploaded an image containing four linear regression (Lin Reg) models, each with the form:

$$
y = a + bx
$$

Each model provides:
- $ a $: the y-intercept
- $ b $: the slope
- $ r $: the correlation coefficient (Pearson's r)

Let’s analyze and interpret each of these four regression models. Since no specific question was asked in your message, I’ll provide a comprehensive explanation of what each model means and how to interpret them.

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🔍 Understanding the Components

- $ y = a + bx $: This is the equation of a straight line used in linear regression.
- $ y $: predicted value
- $ x $: independent variable
- $ a $: y-intercept (value of $ y $ when $ x = 0 $)
- $ b $: slope (how much $ y $ changes for a unit increase in $ x $)
- $ r $: Correlation coefficient
- Measures the strength and direction of the linear relationship between $ x $ and $ y $
- Range: $ -1 \leq r \leq 1 $
- $ r \approx 1 $: strong positive linear relationship
- $ r \approx -1 $: strong negative linear relationship
- $ r \approx 0 $: weak or no linear relationship

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✔ Analysis of Each Model

#### (1)
$$
y = 59.026 + 6.767x \\
r = 0.8643
$$
- Interpretation:
- When $ x = 0 $, $ y = 59.026 $
- For every 1-unit increase in $ x $, $ y $ increases by 6.767 units
- Strong positive linear relationship ($ r = 0.8643 $ → high positive correlation)
- Conclusion: Good fit; positive trend

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#### (2)
$$
y = 2.45 + 0.95x \\
r = 0.6022
$$
- Interpretation:
- When $ x = 0 $, $ y = 2.45 $
- For every 1-unit increase in $ x $, $ y $ increases by 0.95 units
- Moderate positive linear relationship ($ r = 0.6022 $ → moderate correlation)
- Conclusion: Moderate fit; positive trend, but weaker than (1)

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#### (3)
$$
y = 0.7 + 24.2x \\
r = 0.8361
$$
- Interpretation:
- When $ x = 0 $, $ y = 0.7 $
- For every 1-unit increase in $ x $, $ y $ increases by 24.2 units (very steep slope)
- Strong positive linear relationship ($ r = 0.8361 $ → strong correlation)
- Conclusion: Very good fit; very steep positive trend

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#### (4)
$$
y = -2.9 + 24.1x \\
r = -0.8924
$$
- Interpretation:
- When $ x = 0 $, $ y = -2.9 $
- For every 1-unit increase in $ x $, $ y $ increases by 24.1 units (positive slope)
- But $ r = -0.8924 $ → strong negative correlation
- ❗️ Contradiction Alert!
- The slope $ b = 24.1 $ is positive, indicating that as $ x $ increases, $ y $ increases.
- But the correlation coefficient $ r $ is negative, which suggests that as $ x $ increases, $ y $ decreases.
- ⚠️ This is impossible unless there's a mistake in the data or labeling.

> 🛑 This is a critical inconsistency.
>
> In simple linear regression:
> - The sign of $ b $ (slope) must match the sign of $ r $.
> - If $ r $ is negative, $ b $ must be negative.
> - Here, $ b > 0 $, but $ r < 0 $ → inconsistent

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🧩 Final Evaluation

| Model | Slope Sign | Correlation Sign | Consistent? | Strength |
|-------|------------|------------------|-------------|----------|
| (1) | + | + | ✔ Yes | Strong |
| (2) | + | + | ✔ Yes | Moderate |
| (3) | + | + | ✔ Yes | Strong |
| (4) | + | – | ✘ No | Strong (but inconsistent) |

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✔ Likely Issue with Model (4)

There is likely a typo or error in one of the values:
- Either $ b $ should be negative (e.g., $ b = -24.1 $), or
- $ r $ should be positive (e.g., $ r = 0.8924 $)

Because:
- $ r $ and $ b $ must have the same sign in linear regression.
- A negative $ r $ implies a negative slope, not a positive one.

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✔ Summary

- Models (1), (2), and (3) are consistent and reasonable.
- Model (4) has a logical contradiction: positive slope but negative correlation.
- Recommendation: Double-check the values for model (4). It might be a typo.

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If you have a specific question about this (e.g., “Which model best fits the data?” or “Predict $ y $ when $ x = 5 $” for one model), please let me know and I can answer it directly!