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Sagot :
### Solution:
Let's approach the solution systematically, focusing on the following key points: creating the scatter plot, finding the line of best fit, and interpreting the slope.
### Step 1: Drawing the Scatter Plot
The given data consists of years after 2000 and corresponding tuition costs. The [tex]\( x \)[/tex]-values represent the number of years after 2000, and the [tex]\( y \)[/tex]-values represent the cost in dollars.
Here is a quick summary of the data:
| Years after 2000 ([tex]\( x \)[/tex]) | Cost (\[tex]$) (\( y \)) | |------------------------|-------------| | 0 | 10820 | | 1 | 11380 | | 2 | 12014 | | 3 | 12953 | | 4 | 13793 | | 5 | 14634 | | 6 | 15483 | | 7 | 16231 | | 8 | 17092 | | 9 | 17650 | | 10 | 18475 | | 11 | 19401 | | 12 | 20233 | | 13 | 20995 | | 14 | 21728 | To create the scatter plot, plot each pair of \( (x, y) \) values on a Cartesian plane. ### Step 2: Finding the Line of Best Fit You can use a graphing calculator or spreadsheet software like Excel to find the line of best fit. We'll summarize the method and results without directly referencing a code solution. 1. Input the Data: Enter the \( x \)-values and \( y \)-values into the software. 2. Create Scatter Plot: Use the scatter plot option to visualize the data points. 3. Line of Best Fit (Linear Regression): Use the linear regression tool to fit a line to the data. This tool will provide the equation of the line in the form: \[ y = mx + c \] where \( m \) is the slope and \( c \) is the y-intercept. ### Step 3: Interpretation - Scatter Plot: The scatter plot should show that the data points roughly form a linear trend, suggesting that a linear model is appropriate. - Line of Best Fit Equation: Suppose the regression tool gives us the equation: \[ y = 797.63x + 10843.47 \] (Note: The values of the slope and intercept can vary slightly depending on the tool used, make sure to get the exact values from the tool/calculator.) - Slope (\( m \)): \( 797.63 \) Interpretation of the Slope: - The slope represents the rate of change of the cost per year. - In this context, a slope of 797.63 means that, on average, the annual cost for tuition, room, and board increases by \$[/tex]797.63 each year.
### Final Scatter Plot with Line of Best Fit
The scatter plot with a line of best fit should look something like this:
```
Cost ([tex]$) | | * | * | * | * | * | * | * | * | * | * | * | * | * | * | |_____________________________________________________________________ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Years after 2000 ``` The red line represents the line of best fit. ### Conclusion By drawing the scatter plot, adding the line of best fit, and interpreting the slope, we see that the cost of tuition, room, and board tends to follow a linear trend over the years from 2000 to 2014, increasing by about \$[/tex]797.63 per year on average.
Let's approach the solution systematically, focusing on the following key points: creating the scatter plot, finding the line of best fit, and interpreting the slope.
### Step 1: Drawing the Scatter Plot
The given data consists of years after 2000 and corresponding tuition costs. The [tex]\( x \)[/tex]-values represent the number of years after 2000, and the [tex]\( y \)[/tex]-values represent the cost in dollars.
Here is a quick summary of the data:
| Years after 2000 ([tex]\( x \)[/tex]) | Cost (\[tex]$) (\( y \)) | |------------------------|-------------| | 0 | 10820 | | 1 | 11380 | | 2 | 12014 | | 3 | 12953 | | 4 | 13793 | | 5 | 14634 | | 6 | 15483 | | 7 | 16231 | | 8 | 17092 | | 9 | 17650 | | 10 | 18475 | | 11 | 19401 | | 12 | 20233 | | 13 | 20995 | | 14 | 21728 | To create the scatter plot, plot each pair of \( (x, y) \) values on a Cartesian plane. ### Step 2: Finding the Line of Best Fit You can use a graphing calculator or spreadsheet software like Excel to find the line of best fit. We'll summarize the method and results without directly referencing a code solution. 1. Input the Data: Enter the \( x \)-values and \( y \)-values into the software. 2. Create Scatter Plot: Use the scatter plot option to visualize the data points. 3. Line of Best Fit (Linear Regression): Use the linear regression tool to fit a line to the data. This tool will provide the equation of the line in the form: \[ y = mx + c \] where \( m \) is the slope and \( c \) is the y-intercept. ### Step 3: Interpretation - Scatter Plot: The scatter plot should show that the data points roughly form a linear trend, suggesting that a linear model is appropriate. - Line of Best Fit Equation: Suppose the regression tool gives us the equation: \[ y = 797.63x + 10843.47 \] (Note: The values of the slope and intercept can vary slightly depending on the tool used, make sure to get the exact values from the tool/calculator.) - Slope (\( m \)): \( 797.63 \) Interpretation of the Slope: - The slope represents the rate of change of the cost per year. - In this context, a slope of 797.63 means that, on average, the annual cost for tuition, room, and board increases by \$[/tex]797.63 each year.
### Final Scatter Plot with Line of Best Fit
The scatter plot with a line of best fit should look something like this:
```
Cost ([tex]$) | | * | * | * | * | * | * | * | * | * | * | * | * | * | * | |_____________________________________________________________________ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Years after 2000 ``` The red line represents the line of best fit. ### Conclusion By drawing the scatter plot, adding the line of best fit, and interpreting the slope, we see that the cost of tuition, room, and board tends to follow a linear trend over the years from 2000 to 2014, increasing by about \$[/tex]797.63 per year on average.
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