To find the best-fit line that models the relationship between fat content (in grams) and calories for the burgers listed, we can use linear regression to determine the line of best fit.
We can use the provided data to calculate the coefficients of the linear equation \( y = mx + b \), where:
- \( y \) is the calorie count,
- \( x \) is the fat content,
- \( m \) is the slope of the line, and
- \( b \) is the y-intercept.
The data given is:
- (25, 590)
- (44, 830)
- (63, 1080)
- (32, 680)
- (37, 750)
- (20, 420)
- (11, 310)
- (52, 820)
Step 1: Calculate the necessary sums
- Calculate the number of data points \( n \).
- Find the sums:
- \( \sum x \) (sum of fat content),
- \( \sum y \) (sum of calories),
- \( \sum x^2 \) (sum of squares of fat content),
- \( \sum xy \) (sum of the product of fat content and calories).
\[ \begin{array}{c|c} \text{Fat (g)} (x) & \text{Calories} (y) \ \hline 25 & 590 \ 44 & 830 \ 63 & 1080 \ 32 & 680 \ 37 & 750 \ 20 & 420 \ 11 & 310 \ 52 & 820 \ \end{array} \]
Step 2: Calculate these sums
- \( n = 8 \)
- \( \sum x = 25 + 44 + 63 + 32 + 37 + 20 + 11 + 52 = 284 \)
- \( \sum y = 590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820 = 4360 \)
- \( \sum xy = (25 \times 590) + (44 \times 830) + (63 \times 1080) + (32 \times 680) + (37 \times 750) + (20 \times 420) + (11 \times 310) + (52 \times 820) = 14750 + 36520 + 68040 + 21760 + 27750 + 8400 + 3410 + 42640 = 224480 \)
- \( \sum x^2 = 25^2 + 44^2 + 63^2 + 32^2 + 37^2 + 20^2 + 11^2 + 52^2 = 625 + 1936 + 3969 + 1024 + 1369 + 400 + 121 + 2704 = 12950 \)
Step 3: Use the regression formulas
Using the formulas for the slope \( m \) and intercept \( b \): \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ b = \frac{(\sum y) - m(\sum x)}{n} \]
Let's plug in the calculated sums:
Calculate \( m \):
\[ m = \frac{8(224480) - (284)(4360)}{8(12950) - (284)^2} \] \[ = \frac{1795840 - 1231840}{103600 - 80656} \] \[ = \frac{564000}{22944} \approx 24.6 \]
Calculate \( b \):
\[ b = \frac{4360 - 24.6(284)}{8} \] \[ = \frac{4360 - 6984.8}{8} \approx \frac{-2624.8}{8} \approx -328.1 \]
Step 4: Write the best-fit line
The best-fit line is: \[ y \approx 24.6x - 328.1 \]
Explanation:
The slope \( m \) indicates that for every additional gram of fat, the calorie count increases by approximately 24.6 calories. The intercept \( b \) suggests that if a burger had 0 grams of fat, it would provide about -328.1 calories, which is not practically meaningful but can be used to describe the line mathematically in this context.
When constructing this model, we calculated the necessary values to find the best-fit equation using linear regression, which allowed us to quantify the relationship between fat content and calories effectively.