Scatter Plots: Line of Best Fit Worksheet | 8th Grade PDF Worksheets - Free Printable

Let's solve this step-by-step based on the worksheet titled "Scatter Plots: Line of Best Fit".

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✔ Section A: Analyze Scatter Plots

We are given four scatter plots and asked to:
- Draw a line of best fit if possible.
- Write the type and strength of correlation under each.

#### 🔍 How to interpret:
- Type of correlation: Positive, negative, or no correlation.
- *Positive*: As one variable increases, the other tends to increase.
- *Negative*: As one increases, the other decreases.
- *None*: No clear pattern.
- Strength: Strong, moderate, weak, or none.
- *Strong*: Points closely follow a line.
- *Weak*: Points are scattered widely.

Let’s analyze each:

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Plot 1 (Leftmost):
- Points trend upward from bottom-left to top-right.
- Most points follow a general direction.
- Answer: Positive, strong correlation

Plot 2 (Second from left):
- Points are scattered with no clear trend.
- No consistent direction.
- Answer: No correlation (or very weak)

Plot 3 (Third from left):
- Points trend downward from top-left to bottom-right.
- Moderate clustering around a line.
- Answer: Negative, moderate correlation

Plot 4 (Rightmost):
- Points are scattered randomly; no pattern.
- Answer: No correlation

> ⬇️ Write these under each diagram.

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✔ Section B: Height of Students vs. Fathers

We have data for 9 students:

| Student | Height of Student (cm) | Height of Father (cm) |
|--------|-------------------------|------------------------|
| A | 138 | 151 |
| B | 141 | 155 |
| C | 145 | 153 |
| D | 148 | 170 |
| E | 149 | 161 |
| F | 154 | 176 |
| G | 155 | 185 |
| H | 161 | 186 |
| I | 162 | 192 |

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#### a. Draw a scatter plot

We already have axes:
- x-axis: Height of student (130–170 cm)
- y-axis: Height of father (140–190 cm)

We're told the first three points (A, B, C) are plotted:
- A: (138, 151)
- B: (141, 155)
- C: (145, 153)

Now plot the rest:
- D: (148, 170)
- E: (149, 161)
- F: (154, 176)
- G: (155, 185)
- H: (161, 186)
- I: (162, 192)

📌 Plot all as X marks on the grid.

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#### b. Describe the correlation between the two sets of heights

Looking at the data:
- As student height increases, father’s height also tends to increase.
- The relationship is positive.
- Points generally follow an upward trend but not perfectly — some variation.

✔ Answer:
There is a positive correlation between the height of the student and the height of their father. The correlation appears to be moderate to strong, as most points follow a general upward trend.

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#### c. Draw a line of best fit

To draw the line of best fit:
- It should go through the middle of the data points.
- Balance equal numbers of points above and below.
- Not necessarily passing through any specific point.

📌 Steps:
1. Look at the overall trend.
2. Draw a straight line that best represents the direction of the points.
3. It should pass near most points, especially in the center.

This line will help us predict values.

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#### d. Estimate father’s height for a new student who is 151 cm tall

We use the line of best fit to estimate.

Let’s find where x = 151 cm on the x-axis, then go up to the line and read the y-value (father’s height).

But since we don’t have the actual graph here, let’s estimate using the trend.

Let’s look at known points near 151 cm:

- Student C: 145 cm → father 153 cm
- Student E: 149 cm → father 161 cm
- Student F: 154 cm → father 176 cm

So at 151 cm, we’re between 149 and 154.

Let’s interpolate:

From 149 to 154: +5 cm in student height
From 161 to 176: +15 cm in father height

So per cm increase in student height:
→ 15 / 5 = +3 cm in father height

From 149 cm student → 151 cm is +2 cm
So estimated father height:
161 + (2 × 3) = 167 cm

Alternatively, check the trend visually:
- At 150 cm student → father ~165–170?
- Our line of best fit might suggest about 165–170 cm

But let’s do a rough linear regression estimate.

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🔢 Optional: Use Linear Regression (for accuracy)

Let’s calculate approximate line of best fit: $ y = mx + b $

Where:
- $ x $ = student height
- $ y $ = father height

We can estimate slope $ m $ using two points:
Use C (145, 153) and I (162, 192)

$$
m = \frac{192 - 153}{162 - 145} = \frac{39}{17} ≈ 2.29
$$

Now use point (145, 153):
$$
y - 153 = 2.29(x - 145)
$$

At $ x = 151 $:
$$
y = 2.29(151 - 145) + 153 = 2.29(6) + 153 = 13.74 + 153 = 166.74
$$

So estimated father height ≈ 167 cm

✔ Answer for d: The estimated height of the father is approximately 167 cm.

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✔ Final Answers Summary

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#### Section A:
1. Positive, strong correlation
2. No correlation
3. Negative, moderate correlation
4. No correlation

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#### Section B:
a. Scatter plot: Plot all 9 points (first 3 given).
b. There is a positive, moderate to strong correlation between student height and father height.
c. Draw a line of best fit that passes through the center of the data points, showing the upward trend.
d. Estimated father’s height for a 151 cm student: ≈ 167 cm

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📌 Notes:

- On paper, you’d draw the scatter plot and line by hand.
- The line of best fit should be drawn so it balances the points above and below.
- Estimation using the line is acceptable even without exact calculations.

Let me know if you'd like a visual sketch description or help plotting!