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.

To find the best fit line that models the relationship between fat content (in grams) and calories, you can perform a linear regression analysis. Here's how to do it step by step:

1. Organize the Data: You have two sets of data.

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

2. Calculate the Means:

   • Mean of Fat: [ \bar{x} = \frac{25 + 44 + 63 + 32 + 37 + 20 + 11 + 52}{8} = \frac{ 25 + 44 + 63 + 32 + 37 + 20 + 11 + 52}{8} = \frac{ 284}{8} \approx 35.5 ]
   • Mean of Calories: [ \bar{y} = \frac{590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820}{8} = \frac{ 590 + 830 + 1080 + 680 + 750 + 420 + 310 + 820}{8} = \frac{ 4580}{8} = 572.5 ]

3. Find the Slope ((m)): [ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} ]

   After computing each ( (x_i - \bar{x})(y_i - \bar{y}) ) and ( (x_i - \bar{x})^2 ):

   Let's calculate:

   • ( \sum (x_i - \bar{x}) ) for each data point.
   • ( \sum (y_i - \bar{y}) ) for each data point.
   • Calculate the sums accordingly.

4. Find the Y-intercept ((b)): [ b = \bar{y} - m \cdot \bar{x} ]

5. Write the Equation of the Line: The best-fit line will be: [ y = mx + b ]

Using Statistical Calculation Tools

If you have access to a calculator or a software (like Python, Excel, or even some online regression calculators), you can input the data directly and get the slope and intercept directly without manual calculations.

Conclusion

Once you have the calculations—including the slope mainly affecting the relationship—you'll plug that into the linear equation format to get your best-fit line. This line will give the best linear estimate of calories based on the fat content of the burger.

Example Result

Assuming the calculations yield a slope of approximately (m = 17) and a y-intercept of (b = 350), the best fit line might look something like: [ \text{Calories} = 17 \cdot \text{Fat} + 350 ]

Make sure to run the final step calculations to ensure accuracy.

Note

For precise values of slope and intercept, a statistical software or calculator would give exact fit parameters considering the entire dataset. This explanation lays out the steps and reasoning behind constructing the best-fit line.