Which function best models the data?

Steps to Solve

1. Identify the Variables

   Let ( x ) be the number of passengers and ( y ) be the number of suitcases.

2. Plot the Data

   Represent the data points from the table as ordered pairs:
   ((75, 159), (92, 180), (115, 239), (128, 272), (143, 290), (154, 310), (178, 357), (200, 405))

3. Find Slope and Y-intercept

   Use the formula for linear regression or a calculator to find the best fit line. The slope ( m ) can be estimated using the formula:

   $$ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} $$

   And the y-intercept ( b ) can be estimated using:

   $$ b = \frac{\sum y - m(\sum x)}{N} $$

4. Calculate Necessary Sums

   Calculate the necessary sums:

   • ( \sum x )
   • ( \sum y )
   • ( \sum xy )
   • ( \sum x^2 )

5. Determine the Best Fit Function

   After calculating the slope ( m ) and the y-intercept ( b ), combine them to form the equation ( y = mx + b ) and compare it with the given functions.

6. Select the Closest Function

   Identify which of the given functions best matches the calculated function.