To address the problem in the image, we are asked to:

1. Use a graphing calculator to find a linear regression model for the men's 100-meter freestyle data where $x$ represents the years since 1992, and $y$ is the winning time in seconds.
2. Create a similar model for the women's 100-meter freestyle data and then answer whether the models indicate that women will eventually catch up with the men's performance.

Given Data from the Table:

• Men's 100-Meter Freestyle (Years since 1992):

  • 1992 (0 years): 50.40 seconds
  • 1996 (4 years): 49.39 seconds
  • 2000 (8 years): 48.63 seconds
  • 2004 (12 years): 48.17 seconds
  • 2008 (16 years): 48.74 seconds
  • 2012 (20 years): 48.30 seconds
  • 2016 (24 years): 47.58 seconds
  • 2020 (28 years): 47.21 seconds

• Women's 100-Meter Freestyle (Years since 1992):

  • 1992 (0 years): 54.79 seconds
  • 1996 (4 years): 55.92 seconds
  • 2000 (8 years): 54.63 seconds
  • 2004 (12 years): 54.58 seconds
  • 2008 (16 years): 54.74 seconds
  • 2012 (20 years): 53.83 seconds
  • 2016 (24 years): 52.97 seconds
  • 2020 (28 years): 53.12 seconds

I will now calculate the linear regression equations for both men's and women's times based on the provided data.### Linear Regression Models:

• Men's 100-Meter Freestyle:
  $y = -0.0946x + 49.8775$
  (where $y$ is the time in seconds and $x$ is the number of years since 1992)

• Women's 100-Meter Freestyle:
  $y = -0.0854x + 55.5175$
  (where $y$ is the time in seconds and $x$ is the number of years since 1992)

Interpretation:

• The negative slopes indicate that both men and women's winning times are decreasing over the years.
• The men's times are decreasing slightly faster than the women's times, as seen from the slope (-0.0946 for men vs. -0.0854 for women).
• To determine if and when the women will catch up to the men, we can analyze the trends. Based on these equations, it seems unlikely that the women's times will catch up to the men's at this rate since the men are consistently improving faster.

Would you like to calculate when or if the times will ever intersect based on these models?

Here are 5 related questions:

1. How can we predict future winning times using these regression models?
2. What is the significance of the slope in a linear regression?
3. How accurate are these models for long-term predictions?
4. Can other forms of regression (like quadratic) provide a better model for these data trends?
5. How would changes in athletic training impact these times?

Tip: Linear regression assumes a constant rate of change. For long-term predictions, data should be reassessed frequently.