Use a graphing calculator to find a linear regression model for an athletic​ conference's men's​ 100-meter freestyle data given in the accompanying​ table, where x is years since 1992 and y is 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?
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Part 1
The linear regression model for the​ men's times is given by the following equation.
yequals
  
enter your response herexplus
  
enter your response here
​(Round to three decimal places as​ needed.)
.
.
.
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Part 1
100-Meter Freestyle
Men
Women
 
1992
50.40
54.79
1996
49.80
55.92
2000
48.63
55.00
2004
49.02
54.65
2008
48.74
54.50
2012
48.30
53.83
2016
48.17
53.84
2020
47.38
53.12


The linear regression models for the 100-meter freestyle times are as follows:

For the men's times:
\[
y = -0.091x + 50.078
\]
where \( x \) is the number of years since 1992, and \( y \) is the winning time in seconds.

For the women's times:
\[
y = -0.077x + 55.529
\]

The slope of the men's model (\(-0.091\)) indicates a faster rate of improvement compared to the women's model (\(-0.077\)). This suggests that, while both men and women are improving their times, the men are improving slightly faster. Based on these models, it does not seem that the women will catch up to the men in the near future.

Would you like more details on how the calculations were made, or do you have any questions?

Here are some related questions you could explore:
1. How can we predict the times for future years, such as 2030, using these models?
2. What is the correlation coefficient for these linear regressions?
3. At what year do these models predict the times will reach a certain threshold (e.g., 45 seconds for men)?
4. What are the residuals for each model, and how can we interpret them?
5. How would using a polynomial regression change the results?

**Tip**: The slope of a regression line gives the rate of change over time — a negative slope indicates improvement in time!

Use a graphing calculator to find a linear regression model for an athletic conference's men's 100-meter freestyle data given in the accompanying table, where x is years since 1992 and y is 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?

Explore linear regression models for men's and women's 100-meter freestyle data from 1992 to 2020. Analyze trends and compare the rates of improvement for both genders. The solution provides key mathematical concepts such as linear regression and data analysis. Suitable for high school students in Grades 10-12.

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