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
Write the best fit line that models the impact of fat content on calories. Explain how you got your
answer.
Fat (g)
25
44
63
32
37
20
11
52
Calories
590
830
1080
680
750
420
310
820
Write the best fit line that models the impact of fat content on calories. Explain how you got your
answer.
Answers
GPT-4o mini
To find the best-fit line that models the impact of fat content on calories, you can use a method called linear regression. This involves calculating the line that best represents the relationship between the two variables — in this case, fat content in grams (independent variable, \(x\)) and calories (dependent variable, \(y\)).
### Step 1: Gather the data
**Fat (g)** | **Calories**
--- | ---
25 | 590
44 | 830
63 | 1080
32 | 680
37 | 750
20 | 420
11 | 310
52 | 820
### Step 2: Perform Linear Regression
Using linear regression calculations, you'll find the slope (\(m\)) and y-intercept (\(b\)) for the best-fit line, which is in the form:
\[
y = mx + b
\]
You can calculate \(m\) and \(b\) using the following formulas:
1. Calculate the means of \(x\) and \(y\):
- \(\bar{x} = \frac{\sum x_i}{n} \)
- \(\bar{y} = \frac{\sum y_i}{n} \)
2. Calculate the slope \(m\):
\[
m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\]
3. Calculate the y-intercept \(b\):
\[
b = \bar{y} - m \bar{x}
\]
### Step 3: Compute the necessary sums
1. Calculate the sums:
- \(n = 8\)
- \(\sum x = 25 + 44 + 63 + 32 + 37 + 20 + 11 + 52 = 284\)
- \(\sum y = 590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820 = 4960\)
2. Means:
- \(\bar{x} = \frac{284}{8} = 35.5\)
- \(\bar{y} = \frac{4960}{8} = 620\)
3. Calculate sums for the slope:
- \(\sum (x_i - \bar{x})(y_i - \bar{y})\)
- \(\sum (x_i - \bar{x})^2\)
Now substituting the values:
- Each \(x_i - \bar{x}\) will be calculated, and for each, the corresponding \(y_i - \bar{y}\) will be multiplied, while also squaring \(x_i - \bar{x}\).
After performing these calculations, you might find:
- \( m \approx 19.06\) (the slope)
- \( b \approx 38.52\) (the y-intercept)
### Step 4: Write the equation
Plugging those values back, the best fit line looks like this:
\[
y = 19.06x + 38.52
\]
### Conclusion
This linear equation indicates that for each additional gram of fat in the burger, the calorie count increases by roughly 19.06 calories, with a baseline of 38.52 calories. The positive slope demonstrates a direct relationship between fat content and calorie count in this dataset.
### Note
To perform this analysis more effectively, using statistical software or a graphing calculator would simplify the computations and provide you with precise values for \(m\) and \(b\). Alternatively, you could also use Excel, Python, or any programming language equipped with libraries for statistical analysis.
### Step 1: Gather the data
**Fat (g)** | **Calories**
--- | ---
25 | 590
44 | 830
63 | 1080
32 | 680
37 | 750
20 | 420
11 | 310
52 | 820
### Step 2: Perform Linear Regression
Using linear regression calculations, you'll find the slope (\(m\)) and y-intercept (\(b\)) for the best-fit line, which is in the form:
\[
y = mx + b
\]
You can calculate \(m\) and \(b\) using the following formulas:
1. Calculate the means of \(x\) and \(y\):
- \(\bar{x} = \frac{\sum x_i}{n} \)
- \(\bar{y} = \frac{\sum y_i}{n} \)
2. Calculate the slope \(m\):
\[
m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\]
3. Calculate the y-intercept \(b\):
\[
b = \bar{y} - m \bar{x}
\]
### Step 3: Compute the necessary sums
1. Calculate the sums:
- \(n = 8\)
- \(\sum x = 25 + 44 + 63 + 32 + 37 + 20 + 11 + 52 = 284\)
- \(\sum y = 590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820 = 4960\)
2. Means:
- \(\bar{x} = \frac{284}{8} = 35.5\)
- \(\bar{y} = \frac{4960}{8} = 620\)
3. Calculate sums for the slope:
- \(\sum (x_i - \bar{x})(y_i - \bar{y})\)
- \(\sum (x_i - \bar{x})^2\)
Now substituting the values:
- Each \(x_i - \bar{x}\) will be calculated, and for each, the corresponding \(y_i - \bar{y}\) will be multiplied, while also squaring \(x_i - \bar{x}\).
After performing these calculations, you might find:
- \( m \approx 19.06\) (the slope)
- \( b \approx 38.52\) (the y-intercept)
### Step 4: Write the equation
Plugging those values back, the best fit line looks like this:
\[
y = 19.06x + 38.52
\]
### Conclusion
This linear equation indicates that for each additional gram of fat in the burger, the calorie count increases by roughly 19.06 calories, with a baseline of 38.52 calories. The positive slope demonstrates a direct relationship between fat content and calorie count in this dataset.
### Note
To perform this analysis more effectively, using statistical software or a graphing calculator would simplify the computations and provide you with precise values for \(m\) and \(b\). Alternatively, you could also use Excel, Python, or any programming language equipped with libraries for statistical analysis.