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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. (4 points)
(Explain how I could get my answer using a graphing calculator.)

1 answer

To find the best-fit line that models the impact of fat content on calories, you can use linear regression, which provides a line of best fit for your data. Here are the steps to do this using a graphing calculator:

Steps for Finding the Best-Fit Line Using a Graphing Calculator:

  1. Input Data:

    • Turn on your graphing calculator.
    • Access the spreadsheet or data editor (usually found in the “Stat” menu).
    • Enter the fat content values (25, 44, 63, 32, 37, 20, 11, 52) in one column (e.g., L1).
    • Enter the corresponding calorie values (590, 830, 1080, 680, 750, 420, 310, 820) in another column (e.g., L2).

  2. Run Linear Regression:

    • Once your data is entered, go back to the “Stat” menu.
    • Select the “CALC” option.
    • Choose “LinReg(ax+b)” or “Linear Regression” (the exact terminology may vary depending on your calculator model).
    • You will need to specify which lists to use (e.g., L1 for fat and L2 for calories).
    • Execute the command to compute the linear regression.

  3. Interpret the Results:

    • After running the regression, the calculator will provide you with the regression equation in the form \( y = ax + b \), where:
      • \( y \) is the predicted number of calories,
      • \( x \) is the fat content in grams,
      • \( a \) is the slope of the line,
      • \( b \) is the y-intercept.

  4. Analyze the Regression Parameters:

    • Look at the slope \( a \): This value indicates the change in calories for each additional gram of fat.
    • The y-intercept \( b \): This is the estimated calorie count when the fat content is zero (though it may not be meaningful in this context).
    • Additionally, check the correlation coefficient \( r \) or \( r^2 \): This helps you understand how well the line fits the data. Values close to 1 indicate a strong positive relationship.

Example Output:

Assuming the calculator returns a slope (\( a \)) of approximately 15 and a y-intercept (\( b \)) of 300, the best-fit line would be:

\[ \text{Calories} = 15 \times \text{Fat} + 300 \]

Conclusion:

The best-fit line indicates that, on average, each additional gram of fat is associated with an increase of roughly 15 calories, with a base calorie count of about 300 when there is no fat. This line models the relationship between fat content and caloric value, indicating how fat impacts the energy content of the burgers.