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Sagot :
Sure, let's create a detailed step-by-step solution for this problem. We need to create a scatter plot and a line of best fit to represent the provided data and then describe the correlation.
### Step-by-Step Solution
#### Step 1: Organize the Data
The given data is organized as follows:
- Years at Company: 8, 4, 15, 20, 36, 17, 9, 5, 15, 19, 22, 24, 13, 10, 16, 33
- Bi-weekly Contribution to Retirement ([tex]$): 25, 40, 65, 30, 85, 70, 15, 30, 40, 15, 50, 25, 20, 5, 20, 65 #### Step 2: Plot the Scatter Plot To create the scatter plot, we need to plot each (Years at Company, Bi-weekly Contribution) pair on a graph. Let's list out the pairs: (8, 25), (4, 40), (15, 65), (20, 30), (36, 85), (17, 70), (9, 15), (5, 30), (15, 40), (19, 15), (22, 50), (24, 25), (13, 20), (10, 5), (16, 20), (33, 65) #### Step 3: Calculate the Line of Best Fit We'll use the least squares method to calculate the slope (m) and y-intercept (b) of the line of best fit. The general equation for the line of best fit is: \[ y = mx + b \] where - \( x \) is the number of years, - \( y \) is the bi-weekly contribution, - \( m \) is the slope, and - \( b \) is the y-intercept. To calculate \( m \) and \( b \), we use the following formulas: #### Slope: \[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \] #### Y-Intercept: \[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{N(\sum x^2) - (\sum x)^2} \] where \( N \) is the number of data points (16 in this case), \( \sum xy \) is the sum of the products of corresponding x and y values, \( \sum x \) is the sum of x values, \( \sum y \) is the sum of y values, and \( \sum x^2 \) is the sum of the squares of x values. First, we calculate these intermediate sums: - \( \sum x = 8 + 4 + 15 + 20 + 36 + 17 + 9 + 5 + 15 + 19 + 22 + 24 + 13 + 10 + 16 + 33 = 266 \) - \( \sum y = 25 + 40 + 65 + 30 + 85 + 70 + 15 + 30 + 40 + 15 + 50 + 25 + 20 + 5 + 20 + 65 = 600 \) - \( \sum xy = (825) + (440) + (1565) + (2030) + (3685) + (1770) + (915) + (530) + (1540) + (1915) + (2250) + (2425) + (1320) + (105) + (1620) + (3365) = 12066 \) - \( \sum x^2 = 8^2 + 4^2 + 15^2 + 20^2 + 36^2 + 17^2 + 9^2 + 5^2 + 15^2 + 19^2 + 22^2 + 24^2 + 13^2 + 10^2 + 16^2 + 33^2 = 6068 \) Now plug these values into our formulas: #### Calculating the Slope (m): \[ m = \frac{16(12066) - (266)(600)}{16(6068) - (266)^2} \] \[ m = \frac{193056 - 159600}{97088 - 70756} \] \[ m = \frac{33456}{26332} \approx 1.27 \] #### Calculating the Y-Intercept (b): \[ b = \frac{(600)(6068) - (266)(12066)}{16(6068) - (266)^2} \] \[ b = \frac{3640800 - 3215556}{26332} \] \[ b = \frac{425244}{26332} \approx 16.16 \] So, the equation of the line of best fit is: \[ y = 1.27x + 16.16 \] #### Step 4: Plotting the Line of Best Fit Now that we have the equation of the line, we can plot it on the same graph as the scatter plot. #### Step 5: Describe the Correlation The slope \( m \approx 1.27 \) indicates that on average, for each additional year at the company, the bi-weekly retirement contribution increases by about $[/tex]1.27. The y-intercept [tex]\( b \approx 16.16 \)[/tex] indicates that when the number of years at the company is 0, the expected bi-weekly contribution is about $16.16.
#### Step 6: Interpretation of the Correlation
There is a positive correlation between the number of years an employee has been at the company and their bi-weekly contribution to retirement. This positive trend suggests that employees who have been at the company longer tend to contribute more to their retirement bi-weekly.
This line of best fit is a good representation of the correlation between these two variables because it summarizes the general trend of the data points, even though there is some variability around the line. The scatter plot confirms that most points follow this upward trend, supporting the positive relationship suggested by the line of best fit.
### Step-by-Step Solution
#### Step 1: Organize the Data
The given data is organized as follows:
- Years at Company: 8, 4, 15, 20, 36, 17, 9, 5, 15, 19, 22, 24, 13, 10, 16, 33
- Bi-weekly Contribution to Retirement ([tex]$): 25, 40, 65, 30, 85, 70, 15, 30, 40, 15, 50, 25, 20, 5, 20, 65 #### Step 2: Plot the Scatter Plot To create the scatter plot, we need to plot each (Years at Company, Bi-weekly Contribution) pair on a graph. Let's list out the pairs: (8, 25), (4, 40), (15, 65), (20, 30), (36, 85), (17, 70), (9, 15), (5, 30), (15, 40), (19, 15), (22, 50), (24, 25), (13, 20), (10, 5), (16, 20), (33, 65) #### Step 3: Calculate the Line of Best Fit We'll use the least squares method to calculate the slope (m) and y-intercept (b) of the line of best fit. The general equation for the line of best fit is: \[ y = mx + b \] where - \( x \) is the number of years, - \( y \) is the bi-weekly contribution, - \( m \) is the slope, and - \( b \) is the y-intercept. To calculate \( m \) and \( b \), we use the following formulas: #### Slope: \[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \] #### Y-Intercept: \[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{N(\sum x^2) - (\sum x)^2} \] where \( N \) is the number of data points (16 in this case), \( \sum xy \) is the sum of the products of corresponding x and y values, \( \sum x \) is the sum of x values, \( \sum y \) is the sum of y values, and \( \sum x^2 \) is the sum of the squares of x values. First, we calculate these intermediate sums: - \( \sum x = 8 + 4 + 15 + 20 + 36 + 17 + 9 + 5 + 15 + 19 + 22 + 24 + 13 + 10 + 16 + 33 = 266 \) - \( \sum y = 25 + 40 + 65 + 30 + 85 + 70 + 15 + 30 + 40 + 15 + 50 + 25 + 20 + 5 + 20 + 65 = 600 \) - \( \sum xy = (825) + (440) + (1565) + (2030) + (3685) + (1770) + (915) + (530) + (1540) + (1915) + (2250) + (2425) + (1320) + (105) + (1620) + (3365) = 12066 \) - \( \sum x^2 = 8^2 + 4^2 + 15^2 + 20^2 + 36^2 + 17^2 + 9^2 + 5^2 + 15^2 + 19^2 + 22^2 + 24^2 + 13^2 + 10^2 + 16^2 + 33^2 = 6068 \) Now plug these values into our formulas: #### Calculating the Slope (m): \[ m = \frac{16(12066) - (266)(600)}{16(6068) - (266)^2} \] \[ m = \frac{193056 - 159600}{97088 - 70756} \] \[ m = \frac{33456}{26332} \approx 1.27 \] #### Calculating the Y-Intercept (b): \[ b = \frac{(600)(6068) - (266)(12066)}{16(6068) - (266)^2} \] \[ b = \frac{3640800 - 3215556}{26332} \] \[ b = \frac{425244}{26332} \approx 16.16 \] So, the equation of the line of best fit is: \[ y = 1.27x + 16.16 \] #### Step 4: Plotting the Line of Best Fit Now that we have the equation of the line, we can plot it on the same graph as the scatter plot. #### Step 5: Describe the Correlation The slope \( m \approx 1.27 \) indicates that on average, for each additional year at the company, the bi-weekly retirement contribution increases by about $[/tex]1.27. The y-intercept [tex]\( b \approx 16.16 \)[/tex] indicates that when the number of years at the company is 0, the expected bi-weekly contribution is about $16.16.
#### Step 6: Interpretation of the Correlation
There is a positive correlation between the number of years an employee has been at the company and their bi-weekly contribution to retirement. This positive trend suggests that employees who have been at the company longer tend to contribute more to their retirement bi-weekly.
This line of best fit is a good representation of the correlation between these two variables because it summarizes the general trend of the data points, even though there is some variability around the line. The scatter plot confirms that most points follow this upward trend, supporting the positive relationship suggested by the line of best fit.
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