Answer :
Let's break down the solution step-by-step:
### 1. Create a Scatter Plot
To create a scatter plot representing the data:
#### Data Information:
- Years: [2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014]
- Best Times (in "minutes:seconds.milliseconds" format):
- [2:33.42, 2:24.81, 2:10.93, 2:03.45, 1:58.67, 1:59.17, 1:55.06, 1:55.82, 1:54.81, 2:00.03]
#### Conversion of Best Times to Seconds:
To create a scatter plot, we need to convert the recorded best times from the "minutes:seconds.milliseconds" format into total seconds.
[tex]\[ \text{Converted Time (in seconds)} = \text{Minutes} \times 60 + \text{Seconds} \][/tex]
Using this formula, we convert each time:
- 2005: [tex]$2 \times 60 + 33.42 = 153.42$[/tex] seconds
- 2006: [tex]$2 \times 60 + 24.81 = 144.81$[/tex] seconds
- 2007: [tex]$2 \times 60 + 10.93 = 130.93$[/tex] seconds
- 2008: [tex]$2 \times 60 + 3.45 = 123.45$[/tex] seconds
- 2009: [tex]$1 \times 60 + 58.67 = 118.67$[/tex] seconds
- 2010: [tex]$1 \times 60 + 59.17 = 119.17$[/tex] seconds
- 2011: [tex]$1 \times 60 + 55.06 = 115.06$[/tex] seconds
- 2012: [tex]$1 \times 60 + 55.82 = 115.82$[/tex] seconds
- 2013: [tex]$1 \times 60 + 54.81 = 114.81$[/tex] seconds
- 2014: [tex]$2 \times 60 + 0.03 = 120.03$[/tex] seconds
#### Scatter Plot:
To create the scatter plot, you plot the years on the x-axis and the corresponding converted times (in seconds) on the y-axis.
### 2. Determine Correlation
#### Observing the Scatter Plot:
By plotting the points:
- (2005, 153.42)
- (2006, 144.81)
- (2007, 130.93)
- (2008, 123.45)
- (2009, 118.67)
- (2010, 119.17)
- (2011, 115.06)
- (2012, 115.82)
- (2013, 114.81)
- (2014, 120.03)
#### Analysis:
To determine the correlation, observe the trend:
- Generally, from 2005 to 2014, the times are decreasing (i.e., getting faster).
This suggests there is a negative correlation between the year and the best time.
### 3. Conclusion Statement and Line of Best Fit
#### Slope and Equation of the Line of Best Fit:
To find the line of best fit, we use the formula:
[tex]\[ \text{Best Fit Line}: y = mx + b \][/tex]
Where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
#### Calculation:
A common method is using the least squares regression:
[tex]\[ m = \frac{N \sum (xy) - \sum x \sum y}{N \sum x^2 - (\sum x)^2} \][/tex]
[tex]\[ b = \frac{\sum y - m \sum x}{N} \][/tex]
Where:
- [tex]\( N \)[/tex] = number of data points (10)
- [tex]\( x \)[/tex] = years
- [tex]\( y \)[/tex] = converted times in seconds
(Note: These calculations often use statistical software for precision, but for illustration, this formula is provided.)
#### Conclusion:
Given that the correlation is negative, the conclusion might be:
"There is a negative correlation between the year and the recorded best times. This indicates that, generally, the swimmer’s performance has improved over the years with the recorded times getting faster."
The exact values for the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] can be determined using statistical tools, but typically, an improvement trend (negative correlation) would suggest a descent in times as years progress.
This reflects continuous improvement in the swimmer's performance over the observed period.
### 1. Create a Scatter Plot
To create a scatter plot representing the data:
#### Data Information:
- Years: [2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014]
- Best Times (in "minutes:seconds.milliseconds" format):
- [2:33.42, 2:24.81, 2:10.93, 2:03.45, 1:58.67, 1:59.17, 1:55.06, 1:55.82, 1:54.81, 2:00.03]
#### Conversion of Best Times to Seconds:
To create a scatter plot, we need to convert the recorded best times from the "minutes:seconds.milliseconds" format into total seconds.
[tex]\[ \text{Converted Time (in seconds)} = \text{Minutes} \times 60 + \text{Seconds} \][/tex]
Using this formula, we convert each time:
- 2005: [tex]$2 \times 60 + 33.42 = 153.42$[/tex] seconds
- 2006: [tex]$2 \times 60 + 24.81 = 144.81$[/tex] seconds
- 2007: [tex]$2 \times 60 + 10.93 = 130.93$[/tex] seconds
- 2008: [tex]$2 \times 60 + 3.45 = 123.45$[/tex] seconds
- 2009: [tex]$1 \times 60 + 58.67 = 118.67$[/tex] seconds
- 2010: [tex]$1 \times 60 + 59.17 = 119.17$[/tex] seconds
- 2011: [tex]$1 \times 60 + 55.06 = 115.06$[/tex] seconds
- 2012: [tex]$1 \times 60 + 55.82 = 115.82$[/tex] seconds
- 2013: [tex]$1 \times 60 + 54.81 = 114.81$[/tex] seconds
- 2014: [tex]$2 \times 60 + 0.03 = 120.03$[/tex] seconds
#### Scatter Plot:
To create the scatter plot, you plot the years on the x-axis and the corresponding converted times (in seconds) on the y-axis.
### 2. Determine Correlation
#### Observing the Scatter Plot:
By plotting the points:
- (2005, 153.42)
- (2006, 144.81)
- (2007, 130.93)
- (2008, 123.45)
- (2009, 118.67)
- (2010, 119.17)
- (2011, 115.06)
- (2012, 115.82)
- (2013, 114.81)
- (2014, 120.03)
#### Analysis:
To determine the correlation, observe the trend:
- Generally, from 2005 to 2014, the times are decreasing (i.e., getting faster).
This suggests there is a negative correlation between the year and the best time.
### 3. Conclusion Statement and Line of Best Fit
#### Slope and Equation of the Line of Best Fit:
To find the line of best fit, we use the formula:
[tex]\[ \text{Best Fit Line}: y = mx + b \][/tex]
Where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
#### Calculation:
A common method is using the least squares regression:
[tex]\[ m = \frac{N \sum (xy) - \sum x \sum y}{N \sum x^2 - (\sum x)^2} \][/tex]
[tex]\[ b = \frac{\sum y - m \sum x}{N} \][/tex]
Where:
- [tex]\( N \)[/tex] = number of data points (10)
- [tex]\( x \)[/tex] = years
- [tex]\( y \)[/tex] = converted times in seconds
(Note: These calculations often use statistical software for precision, but for illustration, this formula is provided.)
#### Conclusion:
Given that the correlation is negative, the conclusion might be:
"There is a negative correlation between the year and the recorded best times. This indicates that, generally, the swimmer’s performance has improved over the years with the recorded times getting faster."
The exact values for the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] can be determined using statistical tools, but typically, an improvement trend (negative correlation) would suggest a descent in times as years progress.
This reflects continuous improvement in the swimmer's performance over the observed period.
