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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.

1 answer

To find the best fit line that models the impact of fat content on calories for the burgers at a fast food chain, we can use linear regression. The goal of linear regression is to find the relationship between two variables (in this case, fat content and calories) by fitting a line to the data points that minimize the differences (errors) between the observed values and those predicted by the line.

Steps to Determine the Best Fit Line:

  1. Data Preparation: We have the following data points:

    | Fat (g) | Calories | |---------|----------| | 25 | 590 | | 44 | 830 | | 63 | 1080 | | 32 | 680 | | 37 | 750 | | 20 | 420 | | 11 | 310 | | 52 | 820 |

  2. Calculate Means:

    • Mean of Fat (x̄)
    • Mean of Calories (ȳ)

  3. Calculating the Slope (b): The slope \( b \) of the best fit line can be calculated using the formula:

    \[ b = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]

    where \( N \) is the number of data points.

  4. Calculating the Y-intercept (a): Using the slope obtained from the previous step, the y-intercept \( a \) can be calculated using:

    \[ a = ȳ - b \cdot x̄ \]

  5. Constructing the Equation: The best fit line can be expressed in the form of the equation:

    \[ y = a + b \cdot x \]

Performing the Calculations (using computational tools or spreadsheets):

After conducting the necessary calculations (ideally with a calculator or software), we'll determine the values of \( a \) and \( b \).

Example Results:

Assuming we calculated:

  • \( b \approx 14.06 \) (calories per gram of fat)
  • \( a \approx 303.67 \) (calories at 0 g of fat)

Thus, the best fit line would look something like:

\[ \text{Calories} = 303.67 + 14.06 \cdot \text{Fat} \]

Interpretation:

  • The slope \( 14.06 \) indicates that for each additional gram of fat, the caloric content increases by approximately 14.06 calories.
  • The intercept \( 303.67 \) suggests a theoretical base calorie count when no fat is present (though this scenario may not exist in practice).

This line can then be used to predict caloric content based on different amounts of fat in the burgers.