Free Collection of Line of Best Fit Worksheets for Students - Free Printable

Let's solve each problem step-by-step using linear regression to find the line of best fit, which is typically in the form:

$$
y = mx + b
$$

Where:
- $ y $ is the dependent variable (e.g., grade, height, tip),
- $ x $ is the independent variable (e.g., study hours, shoe size, meal cost),
- $ m $ is the slope,
- $ b $ is the y-intercept.

We'll use formulas for linear regression:

$$
m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}
$$
$$
b = \frac{\sum y - m\sum x}{n}
$$

---

Problem 1: Study Hours vs. Exam Grade

#### Data:
| Study Hours (x) | Grade (y) |
|------------------|-----------|
| 2 | 77 |
| 5 | 92 |
| 1 | 70 |
| 0 | 63 |
| 4 | 90 |
| 2 | 75 |
| 3 | 84 |

Total: $ n = 7 $

Let’s compute the required sums:

| x | y | xy | x² |
|---|----|-----|-----|
| 2 | 77 | 154 | 4 |
| 5 | 92 | 460 | 25 |
| 1 | 70 | 70 | 1 |
| 0 | 63 | 0 | 0 |
| 4 | 90 | 360 | 16 |
| 2 | 75 | 150 | 4 |
| 3 | 84 | 252 | 9 |
| Sum | 17 | 551 | 1036 | 61 |

Now calculate:

- $ \sum x = 17 $
- $ \sum y = 551 $
- $ \sum xy = 1036 $
- $ \sum x^2 = 61 $
- $ n = 7 $

#### a) Scatterplot:
Plot points: (2,77), (5,92), (1,70), (0,63), (4,90), (2,75), (3,84)

#### b) Equation of line of best fit:

$$
m = \frac{7(1036) - (17)(551)}{7(61) - (17)^2} = \frac{7252 - 9367}{427 - 289} = \frac{-2115}{138} \approx -15.326
$$

Wait! That can't be right — grades go up with more study time. Let's check calculations.

Wait:
$ 7 \times 1036 = 7252 $?
Actually:
$ 7 \times 1036 = 7 \times 1000 + 7 \times 36 = 7000 + 252 = 7252 $ ✔
$ 17 \times 551 = ? $

Compute $ 17 \times 551 $:
- $ 10 \times 551 = 5510 $
- $ 7 \times 551 = 3857 $
- Total = $ 5510 + 3857 = 9367 $ ✔

So numerator: $ 7252 - 9367 = -2115 $? But that gives negative slope — but data shows increasing trend.

Wait — this must be wrong. Let’s double-check xy sum:

Recompute $ xy $:

- 2×77 = 154
- 5×92 = 460
- 1×70 = 70
- 0×63 = 0
- 4×90 = 360
- 2×75 = 150
- 3×84 = 252

Add:
154 + 460 = 614
+70 = 684
+0 = 684
+360 = 1044
+150 = 1194
+252 = 1446

Ah! I made a mistake earlier: ∑xy = 1446, not 1036.

Similarly, ∑x²:
- 4 + 25 + 1 + 0 + 16 + 4 + 9 = 61 ✔

∑x = 2+5+1+0+4+2+3 = 17 ✔
∑y = 77+92+70+63+90+75+84 =
77+92=169; +70=239; +63=302; +90=392; +75=467; +84=551 ✔

So corrected:

- $ \sum xy = 1446 $
- $ \sum x^2 = 61 $
- $ \sum x = 17 $
- $ \sum y = 551 $
- $ n = 7 $

Now:

$$
m = \frac{7(1446) - (17)(551)}{7(61) - (17)^2} = \frac{10122 - 9367}{427 - 289} = \frac{755}{138} \approx 5.471
$$

Now calculate $ b $:

$$
b = \frac{\sum y - m\sum x}{n} = \frac{551 - 5.471 \times 17}{7} = \frac{551 - 93.007}{7} = \frac{457.993}{7} \approx 65.427
$$

So equation:

$$
\boxed{y = 5.471x + 65.427}
$$

#### c) Predict grade for 6 hours:

$$
y = 5.471(6) + 65.427 = 32.826 + 65.427 = \boxed{98.253} \approx \boxed{98}
$$

#### d) Could this line go on forever?

No. While the model suggests higher study time leads to higher grades, there are practical limits:
- Grades cannot exceed 100.
- Studying too much may lead to fatigue or diminishing returns.
- The model assumes a linear relationship, which may not hold beyond the data range.

---

Problem 2: Height vs. Shoe Size

Data:

| Height (x) | Shoe Size (y) |
|------------|---------------|
| 67 | 8.5 |
| 70 | 9.5 |
| 73.5 | 11 |
| 75 | 12 |
| 78 | 13 |
| 66 | 8 |

n = 6

Compute:

| x | y | xy | x² |
|-----|-----|--------|-------|
| 67 | 8.5 | 569.5 | 4489 |
| 70 | 9.5 | 665 | 4900 |
| 73.5| 11 | 808.5 | 5402.25|
| 75 | 12 | 900 | 5625 |
| 78 | 13 | 1014 | 6084 |
| 66 | 8 | 528 | 4356 |
| Sum | | 4485 | 30856.25|

Sums:
- $ \sum x = 67+70+73.5+75+78+66 = 429.5 $
- $ \sum y = 8.5+9.5+11+12+13+8 = 62 $
- $ \sum xy = 569.5 + 665 + 808.5 + 900 + 1014 + 528 = 4485 $
- $ \sum x^2 = 4489 + 4900 + 5402.25 + 5625 + 6084 + 4356 = 30856.25 $
- $ n = 6 $

Now:

$$
m = \frac{6(4485) - (429.5)(62)}{6(30856.25) - (429.5)^2}
$$

Numerator:
- $ 6 \times 4485 = 26910 $
- $ 429.5 \times 62 = ? $

Break down:
- $ 400 \times 62 = 24800 $
- $ 29.5 \times 62 = 29 \times 62 = 1828, 0.5 \times 62 = 31 → total 1859 $
- So $ 24800 + 1859 = 26659 $

Numerator: $ 26910 - 26659 = 251 $

Denominator:
- $ 6 \times 30856.25 = 185137.5 $
- $ 429.5^2 = ? $

$ 430^2 = 184900 $, but $ 429.5^2 = (430 - 0.5)^2 = 430^2 - 2(430)(0.5) + (0.5)^2 = 184900 - 430 + 0.25 = 184470.25 $

So denominator: $ 185137.5 - 184470.25 = 667.25 $

Now:

$$
m = \frac{251}{667.25} \approx 0.3756
$$

Now $ b $:

$$
b = \frac{\sum y - m\sum x}{n} = \frac{62 - 0.3756 \times 429.5}{6}
$$

First: $ 0.3756 \times 429.5 \approx ? $

$ 0.3756 \times 400 = 150.24 $
$ 0.3756 \times 29.5 \approx 0.3756 \times 30 = 11.268 - 0.3756 \times 0.5 ≈ 11.268 - 0.1878 = 11.0802 $

Total ≈ $ 150.24 + 11.08 = 161.32 $

Then: $ 62 - 161.32 = -99.32 $

Divide by 6: $ b ≈ -16.553 $

So equation:

$$
\boxed{y = 0.3756x - 16.553}
$$

But we need height from shoe size, so let’s switch roles.

Wait — in part (c), we're asked: *If a man has shoe size 10.5, what is his predicted height?*

So here, shoe size is x, height is y.

But in our calculation, we used height as x and shoe size as y.

So actually, we should have done it the other way around.

Let me correct.

Let:
- $ x $ = shoe size
- $ y $ = height

So new table:

| Shoe Size (x) | Height (y) |
|---------------|------------|
| 8.5 | 67 |
| 9.5 | 70 |
| 11 | 73.5 |
| 12 | 75 |
| 13 | 78 |
| 8 | 66 |

Now recompute:

| x | y | xy | x² |
|-----|-----|--------|-------|
| 8.5 | 67 | 569.5 | 72.25 |
| 9.5 | 70 | 665 | 90.25 |
| 11 | 73.5| 808.5 | 121 |
| 12 | 75 | 900 | 144 |
| 13 | 78 | 1014 | 169 |
| 8 | 66 | 528 | 64 |
| Sum | | 4485 | 556.25|

Wait — same $ \sum xy = 4485 $, $ \sum x = 8.5+9.5+11+12+13+8 = 62 $, $ \sum y = 429.5 $, $ \sum x^2 = 72.25+90.25+121+144+169+64 = 556.25 $

So now:
- $ \sum x = 62 $
- $ \sum y = 429.5 $
- $ \sum xy = 4485 $
- $ \sum x^2 = 556.25 $
- $ n = 6 $

Now:

$$
m = \frac{6(4485) - (62)(429.5)}{6(556.25) - (62)^2}
$$

Numerator:
- $ 6 \times 4485 = 26910 $
- $ 62 \times 429.5 = ? $

$ 60 \times 429.5 = 25770 $, $ 2 \times 429.5 = 859 $, total = $ 26629 $

Numerator: $ 26910 - 26629 = 281 $

Denominator:
- $ 6 \times 556.25 = 3337.5 $
- $ 62^2 = 3844 $
- Denominator: $ 3337.5 - 3844 = -506.5 $

Negative denominator? That can’t be.

Wait — $ \sum x^2 = 556.25 $, $ n \sum x^2 = 6 \times 556.25 = 3337.5 $

$ (\sum x)^2 = 62^2 = 3844 $

So denominator: $ 3337.5 - 3844 = -506.5 $ → Negative? Impossible.

But $ \sum x^2 = 556.25 $, $ (\sum x)^2 = 3844 $, and since $ \sum x^2 < (\sum x)^2 / n $, this is possible only if values are spread out.

But wait — let's recalculate $ \sum x^2 $:

- $ 8.5^2 = 72.25 $
- $ 9.5^2 = 90.25 $
- $ 11^2 = 121 $
- $ 12^2 = 144 $
- $ 13^2 = 169 $
- $ 8^2 = 64 $

Sum:
72.25 + 90.25 = 162.5
+121 = 283.5
+144 = 427.5
+169 = 596.5
+64 = 660.5

Ah! I made a mistake earlier.

Correct $ \sum x^2 = 660.5 $

Now:

- $ n \sum x^2 = 6 \times 660.5 = 3963 $
- $ (\sum x)^2 = 62^2 = 3844 $
- Denominator: $ 3963 - 3844 = 119 $

Numerator: $ 26910 - 26629 = 281 $

So:

$$
m = \frac{281}{119} \approx 2.361
$$

Now $ b $:

$$
b = \frac{\sum y - m\sum x}{n} = \frac{429.5 - 2.361 \times 62}{6}
$$

$ 2.361 \times 62 = 2.361 \times 60 = 141.66, + 2.361 \times 2 = 4.722 → total 146.382 $

Then: $ 429.5 - 146.382 = 283.118 $

Divide by 6: $ b \approx 47.186 $

So equation:

$$
\boxed{y = 2.361x + 47.186}
$$

where $ y $ = height, $ x $ = shoe size

#### c) Predict height for shoe size 10.5:

$$
y = 2.361(10.5) + 47.186 = 24.7905 + 47.186 = \boxed{71.9765} \approx \boxed{72} \text{ inches}
$$

#### d) What does the slope represent?

The slope $ m = 2.361 $ means that for every 1-unit increase in shoe size, the predicted height increases by about 2.36 inches.

---

Problem 3: Meal Cost vs. Tip

Data:

| Meal Cost (x) | Tip (y) |
|---------------|---------|
| 4.75 | 0.50 |
| 6.84 | 0.90 |
| 12.52 | 1.50 |
| 20.42 | 3.00 |
| 8.97 | 1.00 |

n = 5

Compute:

| x | y | xy | x² |
|--------|------|----------|----------|
| 4.75 | 0.50 | 2.375 | 22.5625 |
| 6.84 | 0.90 | 6.156 | 46.7856 |
| 12.52 | 1.50 | 18.78 | 156.7504 |
| 20.42 | 3.00 | 61.26 | 416.6764 |
| 8.97 | 1.00 | 8.97 | 79.4609 |
| Sum | | 97.541 | 722.2359 |

Sums:
- $ \sum x = 4.75 + 6.84 + 12.52 + 20.42 + 8.97 = 53.5 $
- $ \sum y = 0.50 + 0.90 + 1.50 + 3.00 + 1.00 = 7.00 $
- $ \sum xy = 2.375 + 6.156 + 18.78 + 61.26 + 8.97 = 97.541 $
- $ \sum x^2 = 22.5625 + 46.7856 + 156.7504 + 416.6764 + 79.4609 = 722.2358 $
- $ n = 5 $

Now:

$$
m = \frac{5(97.541) - (53.5)(7.00)}{5(722.2358) - (53.5)^2}
$$

Numerator:
- $ 5 \times 97.541 = 487.705 $
- $ 53.5 \times 7 = 374.5 $
- $ 487.705 - 374.5 = 113.205 $

Denominator:
- $ 5 \times 722.2358 = 3611.179 $
- $ 53.5^2 = 2862.25 $
- $ 3611.179 - 2862.25 = 748.929 $

So:
$$
m = \frac{113.205}{748.929} \approx 0.1511
$$

Now $ b $:

$$
b = \frac{7.00 - 0.1511 \times 53.5}{5} = \frac{7.00 - 8.07885}{5} = \frac{-1.07885}{5} = -0.2158
$$

So equation:

$$
\boxed{y = 0.1511x - 0.2158}
$$

Predict tip for $10.50:

$$
y = 0.1511(10.50) - 0.2158 = 1.58655 - 0.2158 = \boxed{1.37075} \approx \boxed{\$1.37}
$$

---

Final Answers:

---

Problem 1:

a) Scatterplot: Plot the given points
b) Equation: $ \boxed{y = 5.471x + 65.427} $
c) Grade expected: $ \boxed{98} $
d) No, because grades max at 100, and studying beyond a point may not help.

---

Problem 2:

a) Scatterplot: Plot (shoe size, height)
b) Equation: $ \boxed{y = 2.361x + 47.186} $
c) Height expected: $ \boxed{72} $ inches
d) Slope: For each increase of 1 shoe size, height increases by ~2.36 inches.

---

Problem 3:

Equation: $ \boxed{y = 0.1511x - 0.2158} $
Tip expected: $ \boxed{\$1.37} $

Let me know if you'd like graphs or calculator steps!