Question
The table shows the fat content and calories for the burgers at a fast food chain. 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
Fat (g) 25, 44, 63, 32, 37, 20, 11, 52
Calories 590, 830, 1080, 680, 750, 420, 310, 820
Answers
GPT-4o mini
To determine the strength of the correlation between fat content and calories for the burgers, we can calculate the Pearson correlation coefficient (r). This coefficient ranges from -1 to +1, where:
- +1 indicates a perfect positive correlation,
- -1 indicates a perfect negative correlation, and
- 0 indicates no correlation.
Here’s how you can calculate it step by step:
1. **Calculate the means** of the fat content and calories.
\[
\text{Mean of Fat} = \frac{25 + 44 + 63 + 32 + 37 + 20 + 11 + 52}{8} = \frac{284}{8} = 35.5
\]
\[
\text{Mean of Calories} = \frac{590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820}{8} = \frac{4580}{8} = 572.5
\]
2. **Calculate the differences from the means** for each data point, square those differences, and compute the products of the differences for each pair.
| Fat (g) | Calories | Fat - Mean Fat | Calories - Mean Calories | (Fat - Mean Fat)² | (Calories - Mean Calories)² | (Fat - Mean Fat)(Calories - Mean Calories) |
|---------|----------|----------------|-------------------------|--------------------|----------------------------|----------------------------------------------|
| 25 | 590 | -10.5 | 17.5 | 110.25 | 306.25 | -183.75 |
| 44 | 830 | 8.5 | 257.5 | 72.25 | 66406.25 | 2187.5 |
| 63 | 1080 | 27.5 | 507.5 | 756.25 | 257525.25 | 13981.25 |
| 32 | 680 | -3.5 | 107.5 | 12.25 | 11556.25 | -376.25 |
| 37 | 750 | 1.5 | 177.5 | 2.25 | 31506.25 | 265.75 |
| 20 | 420 | -15.5 | -152.5 | 240.25 | 23256.25 | 2368.75 |
| 11 | 310 | -24.5 | -262.5 | 600.25 | 68906.25 | 6431.25 |
| 52 | 820 | 16.5 | 247.5 | 272.25 | 61356.25 | 4087.5 |
3. **Sum the squared differences** and the products of the differences:
\[
\text{Sum of } (Fat - \text{Mean Fat})^2 = 110.25 + 72.25 + 756.25 + 12.25 + 2.25 + 240.25 + 600.25 + 272.25 = 2072.5
\]
\[
\text{Sum of } (Calories - \text{Mean Calories})^2 = 306.25 + 66406.25 + 257525.25 + 11556.25 + 31506.25 + 23256.25 + 68906.25 + 61356.25 = 348943.25
\]
\[
\text{Sum of } (Fat - \text{Mean Fat})(Calories - \text{Mean Calories}) = -183.75 + 2187.5 + 13981.25 - 376.25 + 265.75 + 2368.75 + 6431.25 + 4087.5 = 15567.5
\]
4. **Calculate the correlation coefficient** using the formula:
\[
r = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{\sqrt{\sum{(x - \bar{x})^2} \sum{(y - \bar{y})^2}}}
\]
Substituting in the sums:
\[
r = \frac{15567.5}{\sqrt{2072.5 \times 348943.25}}
\]
\[
r = \frac{15567.5}{\sqrt{72255575.625}} \approx \frac{15567.5}{2687.619} \approx 0.578
\]
5. Finally, interpret the value. Since \(r \approx 0.578\), this suggests a moderate positive correlation between fat content and calories.
Thus, there is a moderate positive correlation between fat content and calories in the burgers from this fast food chain.
- +1 indicates a perfect positive correlation,
- -1 indicates a perfect negative correlation, and
- 0 indicates no correlation.
Here’s how you can calculate it step by step:
1. **Calculate the means** of the fat content and calories.
\[
\text{Mean of Fat} = \frac{25 + 44 + 63 + 32 + 37 + 20 + 11 + 52}{8} = \frac{284}{8} = 35.5
\]
\[
\text{Mean of Calories} = \frac{590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820}{8} = \frac{4580}{8} = 572.5
\]
2. **Calculate the differences from the means** for each data point, square those differences, and compute the products of the differences for each pair.
| Fat (g) | Calories | Fat - Mean Fat | Calories - Mean Calories | (Fat - Mean Fat)² | (Calories - Mean Calories)² | (Fat - Mean Fat)(Calories - Mean Calories) |
|---------|----------|----------------|-------------------------|--------------------|----------------------------|----------------------------------------------|
| 25 | 590 | -10.5 | 17.5 | 110.25 | 306.25 | -183.75 |
| 44 | 830 | 8.5 | 257.5 | 72.25 | 66406.25 | 2187.5 |
| 63 | 1080 | 27.5 | 507.5 | 756.25 | 257525.25 | 13981.25 |
| 32 | 680 | -3.5 | 107.5 | 12.25 | 11556.25 | -376.25 |
| 37 | 750 | 1.5 | 177.5 | 2.25 | 31506.25 | 265.75 |
| 20 | 420 | -15.5 | -152.5 | 240.25 | 23256.25 | 2368.75 |
| 11 | 310 | -24.5 | -262.5 | 600.25 | 68906.25 | 6431.25 |
| 52 | 820 | 16.5 | 247.5 | 272.25 | 61356.25 | 4087.5 |
3. **Sum the squared differences** and the products of the differences:
\[
\text{Sum of } (Fat - \text{Mean Fat})^2 = 110.25 + 72.25 + 756.25 + 12.25 + 2.25 + 240.25 + 600.25 + 272.25 = 2072.5
\]
\[
\text{Sum of } (Calories - \text{Mean Calories})^2 = 306.25 + 66406.25 + 257525.25 + 11556.25 + 31506.25 + 23256.25 + 68906.25 + 61356.25 = 348943.25
\]
\[
\text{Sum of } (Fat - \text{Mean Fat})(Calories - \text{Mean Calories}) = -183.75 + 2187.5 + 13981.25 - 376.25 + 265.75 + 2368.75 + 6431.25 + 4087.5 = 15567.5
\]
4. **Calculate the correlation coefficient** using the formula:
\[
r = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{\sqrt{\sum{(x - \bar{x})^2} \sum{(y - \bar{y})^2}}}
\]
Substituting in the sums:
\[
r = \frac{15567.5}{\sqrt{2072.5 \times 348943.25}}
\]
\[
r = \frac{15567.5}{\sqrt{72255575.625}} \approx \frac{15567.5}{2687.619} \approx 0.578
\]
5. Finally, interpret the value. Since \(r \approx 0.578\), this suggests a moderate positive correlation between fat content and calories.
Thus, there is a moderate positive correlation between fat content and calories in the burgers from this fast food chain.