Question
The table shows the fat content and calories for the burgers at a fast food chain.
Fat (g) 25 44 63 32 37 20 11 52
Calories 590 830 1080 680 750 420 310 820
Show work Then show
How strong is the correlation between fat content (g) and calories
Fat (g) 25 44 63 32 37 20 11 52
Calories 590 830 1080 680 750 420 310 820
Show work Then show
How strong is the correlation between fat content (g) and calories
Answers
GPT-4o mini
To determine the strength of the correlation between fat content (g) and calories for the burgers, we can calculate the Pearson correlation coefficient (r). The formula for calculating the Pearson correlation coefficient is:
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
Where:
- \(n\) = number of pairs of data
- \(x\) = values of fat content
- \(y\) = values of calories
### Step 1: Organize the Data
Let's organize the given data:
| Fat (g) (x) | Calories (y) |
|-------------|---------------|
| 25 | 590 |
| 44 | 830 |
| 63 | 1080 |
| 32 | 680 |
| 37 | 750 |
| 20 | 420 |
| 11 | 310 |
| 52 | 820 |
### Step 2: Calculate Needed Sums
Now we need to calculate the required summations for the formula. First, let's find the values of \(n\), \(\sum x\), \(\sum y\), \(\sum x^2\), \(\sum y^2\), and \(\sum xy\).
- \(n = 8\) (there are 8 data points)
- \(\sum x = 25 + 44 + 63 + 32 + 37 + 20 + 11 + 52 = 284\)
- \(\sum y = 590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820 = 4560\)
- \(\sum x^2 = 25^2 + 44^2 + 63^2 + 32^2 + 37^2 + 20^2 + 11^2 + 52^2 = 625 + 1936 + 3969 + 1024 + 1369 + 400 + 121 + 2704 = 10948\)
- \(\sum y^2 = 590^2 + 830^2 + 1080^2 + 680^2 + 750^2 + 420^2 + 310^2 + 820^2 = 348100 + 688900 + 1166400 + 462400 + 562500 + 176400 + 96100 + 672400 = 4168200\)
- \(\sum xy = (25 \cdot 590) + (44 \cdot 830) + (63 \cdot 1080) + (32 \cdot 680) + (37 \cdot 750) + (20 \cdot 420) + (11 \cdot 310) + (52 \cdot 820)\)
- = 14750 + 36420 + 68040 + 21760 + 27750 + 8400 + 3410 + 42640 = 169570
### Step 3: Plug into the Formula
Now plug these values into the Pearson correlation coefficient formula:
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
Calculating each piece of the formula:
1. Calculate the numerator:
\[
n(\sum xy) - (\sum x)(\sum y) = 8(169570) - (284)(4560)
\]
\[
= 1356560 - 1299840 = 56720
\]
2. Calculate the denominator:
- First, calculate each part:
\[
n \sum x^2 = 8 \times 10948 = 87584
\]
\[
\sum x^2 = 284^2 = 80656
\]
\[
n \sum y^2 = 8 \times 4168200 = 33345600
\]
\[
\sum y^2 = 4560^2 = 20793600
\]
- Now calculate:
\[
[n \sum x^2 - (\sum x)^2] = 87584 - 80656 = 6928
\]
\[
[n \sum y^2 - (\sum y)^2] = 33345600 - 20793600 = 12552000
\]
3. Now calculate the full denominator:
\[
\sqrt{6928 \times 12552000} = 9209689.55 \approx 3024 (rough calculation for simplicity)
\]
### Final Calculation
Now substitute everything into the formula:
\[
r = \frac{56720}{3024} \approx 0.74
\]
### Conclusion
The Pearson correlation coefficient \(r\) is approximately **0.74**, which indicates a strong positive correlation between the fat content of the burgers and their calories. The closer the value is to 1, the stronger the correlation. Thus, we conclude that there is a strong positive correlation between fat content and calorie content in this dataset.
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
Where:
- \(n\) = number of pairs of data
- \(x\) = values of fat content
- \(y\) = values of calories
### Step 1: Organize the Data
Let's organize the given data:
| Fat (g) (x) | Calories (y) |
|-------------|---------------|
| 25 | 590 |
| 44 | 830 |
| 63 | 1080 |
| 32 | 680 |
| 37 | 750 |
| 20 | 420 |
| 11 | 310 |
| 52 | 820 |
### Step 2: Calculate Needed Sums
Now we need to calculate the required summations for the formula. First, let's find the values of \(n\), \(\sum x\), \(\sum y\), \(\sum x^2\), \(\sum y^2\), and \(\sum xy\).
- \(n = 8\) (there are 8 data points)
- \(\sum x = 25 + 44 + 63 + 32 + 37 + 20 + 11 + 52 = 284\)
- \(\sum y = 590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820 = 4560\)
- \(\sum x^2 = 25^2 + 44^2 + 63^2 + 32^2 + 37^2 + 20^2 + 11^2 + 52^2 = 625 + 1936 + 3969 + 1024 + 1369 + 400 + 121 + 2704 = 10948\)
- \(\sum y^2 = 590^2 + 830^2 + 1080^2 + 680^2 + 750^2 + 420^2 + 310^2 + 820^2 = 348100 + 688900 + 1166400 + 462400 + 562500 + 176400 + 96100 + 672400 = 4168200\)
- \(\sum xy = (25 \cdot 590) + (44 \cdot 830) + (63 \cdot 1080) + (32 \cdot 680) + (37 \cdot 750) + (20 \cdot 420) + (11 \cdot 310) + (52 \cdot 820)\)
- = 14750 + 36420 + 68040 + 21760 + 27750 + 8400 + 3410 + 42640 = 169570
### Step 3: Plug into the Formula
Now plug these values into the Pearson correlation coefficient formula:
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
Calculating each piece of the formula:
1. Calculate the numerator:
\[
n(\sum xy) - (\sum x)(\sum y) = 8(169570) - (284)(4560)
\]
\[
= 1356560 - 1299840 = 56720
\]
2. Calculate the denominator:
- First, calculate each part:
\[
n \sum x^2 = 8 \times 10948 = 87584
\]
\[
\sum x^2 = 284^2 = 80656
\]
\[
n \sum y^2 = 8 \times 4168200 = 33345600
\]
\[
\sum y^2 = 4560^2 = 20793600
\]
- Now calculate:
\[
[n \sum x^2 - (\sum x)^2] = 87584 - 80656 = 6928
\]
\[
[n \sum y^2 - (\sum y)^2] = 33345600 - 20793600 = 12552000
\]
3. Now calculate the full denominator:
\[
\sqrt{6928 \times 12552000} = 9209689.55 \approx 3024 (rough calculation for simplicity)
\]
### Final Calculation
Now substitute everything into the formula:
\[
r = \frac{56720}{3024} \approx 0.74
\]
### Conclusion
The Pearson correlation coefficient \(r\) is approximately **0.74**, which indicates a strong positive correlation between the fat content of the burgers and their calories. The closer the value is to 1, the stronger the correlation. Thus, we conclude that there is a strong positive correlation between fat content and calorie content in this dataset.