Linear Scatter Plots Notes Worksheet

Let's solve the problem step by step based on the worksheet you've provided.

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Part 1: Fill in the Blanks

Correlation – the relationship between two variables in a data set.

Correlation coefficient (r) – measures the strength and direction of the relationship between two variables in a data set.
*r* must be between -1 and 1.

> ✔ These are standard definitions:
> - Correlation: The degree to which two variables are related.
> - Correlation coefficient (r): A number from -1 to 1 that quantifies both the strength (how closely the points follow a line) and direction (positive or negative) of the linear relationship.
> - The range is always -1 ≤ r ≤ 1.

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Part 2: Analyze Each Scatter Plot

We are asked to determine the type of correlation and estimate the r value for each of the four scatter plots.

Let’s go through them one by one.

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#### 1. First Scatter Plot
- Pattern: Points trend upward from left to right.
- Tightness: Points are fairly close to a straight line, but not perfectly aligned.
- Conclusion: Positive correlation.
- Estimate of r: Since it's strong but not perfect, we estimate r ≈ 0.8.

> ✔ Reasoning: The points show a clear positive trend with only minor scatter → strong positive correlation.

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#### 2. Second Scatter Plot
- Pattern: Points appear randomly scattered with no clear trend.
- Tightness: No discernible direction or pattern.
- Conclusion: No correlation.
- Estimate of r: r ≈ 0.

> ✔ Reasoning: There is no visible linear relationship; points are spread out without any consistent direction.

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#### 3. Third Scatter Plot
- Pattern: Points form a downward-sloping line from top-left to bottom-right.
- Tightness: Very tight clustering around a straight line.
- Conclusion: Negative correlation.
- Estimate of r: Very strong, so r ≈ -0.95.

> ✔ Reasoning: Nearly all points lie along a straight line with a negative slope → very strong negative correlation.

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#### 4. Fourth Scatter Plot
- Pattern: Points trend upward from left to right.
- Tightness: Points are tightly clustered around a straight line.
- Conclusion: Positive correlation.
- Estimate of r: Strong and nearly perfect → r ≈ 0.95.

> ✔ Reasoning: Almost all points lie close to a straight line with a positive slope → very strong positive correlation.

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✔ Final Answers:

| Example | Correlation Type | Estimated r Value |
|--------|-----------------------|--------------------|
| 1 | Positive | r ≈ 0.8 |
| 2 | No correlation | r ≈ 0 |
| 3 | Negative | r ≈ -0.95 |
| 4 | Positive | r ≈ 0.95 |

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🔍 Summary of Key Concepts

- Positive correlation: As x increases, y tends to increase → r > 0
- Negative correlation: As x increases, y tends to decrease → r < 0
- No correlation: No clear pattern → r ≈ 0
- Strength: Closer to ±1 = stronger correlation; closer to 0 = weaker correlation

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