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
48.48
53.83
2016
48.17
53.84
2020
47.21
53.12


To create linear regression models for both the men's and women's 100-meter freestyle times based on the given data, you need to follow these steps:

### 1. Define Variables:
Let:
- \( x \) represent the number of years since 1992 (i.e., 1992 corresponds to \( x = 0 \)).
- \( y \) represent the winning time in seconds for each year.

For the men's and women's dataTo create linear regression models for both the men's and women's 100-meter freestyle times based on the given data, you need to follow these steps:

### 1. Define Variables:
Let:
- \( x \) represent the number of years since 1992 (i.e., 1992 corresponds to \( x = 0 \)).
- \( y \) represent the winning time in seconds for each year.

For the men's and women's 
- \( x \) values: 0, 4, 8, 12, 16, 20, 24, 28 (years since 1992).
- Men's \( y \) values (winning times in seconds): 50.40, 49.80, 48.63, 49.02, 48.74, 48.48, 48.17, 47.21.
- Women's \( y \) values (winning times in seconds): 54.79, 55.92, 55.00, 54.65, 54.50, 53.83, 53.84, 53.12.

### 2. Input the Data into a Graphing Calculator:
Using the graphing calculator, input the \( x \) and \( y \) values for both men and women.

### 3. Perform Linear Regression:
Run the linear regression calculation to find the equation of the line of best fit for both the men's and women's data.

#### For Men:
- The linear regression model for men's times is:
\[
y = -0.116x + 50.065
\]
(Rounding to three decimal places)

#### For Women:
- The linear regression model for women's times is:
\[
y = -0.058x + 55.359
\]
(Rounding to three decimal places)

### 4. Interpretation:
- The men's times decrease by approximately 0.116 seconds per year.
- The women's times decrease by approximately 0.058 seconds per year.
  
Since the men's times are decreasing at a faster rate than the women's times, the linear models suggest that women will **not** catch up with men based on this data.

Would you like more detailed steps or have any other questions?

---

### Follow-up Questions:
1. How can we calculate the correlation coefficient from this data?
2. What does the slope of the linear regression line represent in this context?
3. Could a nonlinear model provide a better fit for the data? Why or why not?
4. How can you predict future times using these linear models?
5. How would this analysis change if new data points for future years were included?

### Tip:
When analyzing regression models, always compare the slopes to understand the rate of change and check the correlation coefficient to assess how well the data fits the model.

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?

Learn how to use a graphing calculator to find linear regression models for men's and women's 100-meter freestyle times. This detailed guide explains how to analyze the data from 1992 to 2020, create regression models, and interpret whether women's times will eventually catch up with men's.

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