Math Problem Statement

Use a graphing calculator to find a linear regression model for the men’s 100-meter freestyle data given in the table, where x is years since 1992 and y is the winning time (in seconds). Do the same for the women’s 100-meter freestyle data. Do these models indicate that the women will eventually catch up with the men?

Solution

The image presents a problem where you need to find a linear regression model for the men's and women's 100-meter freestyle times based on the provided data. The men's regression model is already given as:

y=−0.145x+50.523y = -0.145x + 50.523

where yy represents the men's time, and xx is the number of years since 1992.

To complete the task, let's perform the following steps for the women's dataThe image presents a problem where you need to find a linear regression model for the men's and women's 100-meter freestyle times based on the provided data. The men's regression model is already given as:

y=−0.145x+50.523y = -0.145x + 50.523

where yy represents the men's time, and xx is the number of years since 1992.

To complete the task, let's perform the following steps for the women's

  1. Extract the data points for the women’s 100-meter freestyle times.
  2. Fit a linear regression model to the women's data.
  3. Compare the two models and check whether they suggest the women will eventually catch up with the men.

Here is the women's data extracted from the table:

\hline \text{Year (x)} & \text{Women’s Time (y)} \\ \hline 1992 & 54.79 \\ 1996 & 55.92 \\ 2000 & 54.93 \\ 2004 & 54.65 \\ 2008 & 54.50 \\ 2012 & 53.83 \\ 2016 & 53.84 \\ 2020 & 53.12 \\ \hline \end{array}$$ Let me calculate the linear regression equation for the women's times.The linear regression model for the women's 100-meter freestyle times is: $$y = -0.076x + 55.512$$ where: - $$y$$ represents the women's time in seconds. - $$x$$ is the number of years since 1992. ### Comparison of Models: - **Men’s equation**: $$y = -0.145x + 50.523$$ - **Women’s equation**: $$y = -0.076x + 55.512$$ Both models indicate that times are decreasing over the years, but the men’s times are improving faster (with a steeper slope of -0.145) than the women’s times (with a slope of -0.076). This suggests that, while both are improving, the women are not catching up to the men at this rate based on these models. Would you like to explore this further or ask any other questions? ### Here are some related questions you could consider: 1. How do the models predict future performance for men and women? 2. How would the graphs of these two models compare visually? 3. What does the slope represent in the context of the problem? 4. What are potential limitations of using linear regression for this data? 5. Could the women eventually surpass the men, and if so, when? ### Tip: When interpreting a regression model, always check the slope to understand the rate of change over time. A steeper slope means faster improvement or decline.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistics
Algebra

Formulas

Linear regression equation y = mx + b

Theorems

Linear regression analysis

Suitable Grade Level

Grades 9-12

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