Create a scatter plot of the data from the table below. Use the scatter plot to find an equation of the trend line, round to the nearest tenth.
Table
Age 2 3 4 6 8
Height (In) 31 36.5 40 44.5 49
To create a scatter plot of the data, you will need to plot the Age values on the x-axis and the Height (In) values on the y-axis.
Scatter Plot:
(2, 31)
(3, 36.5)
(4, 40)
(6, 44.5)
(8, 49)
Now, let's find the equation of the trend line:
To find the equation of the trend line, you can use the least squares regression method.
Step 1: Calculate the means of Age (x) and Height (y).
Mean of Age (x): (2 + 3 + 4 + 6 + 8) / 5 = 4.6
Mean of Height (y): (31 + 36.5 + 40 + 44.5 + 49) / 5 = 40.2
Step 2: Calculate the differences from the means for each point.
(x - x̄) and (y - ȳ):
(2 - 4.6, 31 - 40.2) = (-2.6, -9.2)
(3 - 4.6, 36.5 - 40.2) = (-1.6, -3.7)
(4 - 4.6, 40 - 40.2) = (-0.6, -0.2)
(6 - 4.6, 44.5 - 40.2) = (1.4, 4.3)
(8 - 4.6, 49 - 40.2) = (3.4, 8.8)
Step 3: Calculate the products of the differences from the means for each point.
(x - x̄)(y - ȳ):
(-2.6)(-9.2) = 23.92
(-1.6)(-3.7) = 5.92
(-0.6)(-0.2) = 0.12
(1.4)(4.3) = 6.02
(3.4)(8.8) = 29.92
Step 4: Calculate the squared differences from the means for each point.
(x - x̄)^2 and (y - ȳ)^2:
(-2.6)^2 = 6.76
(-1.6)^2 = 2.56
(-0.6)^2 = 0.36
(1.4)^2 = 1.96
(3.4)^2 = 11.56
Step 5: Calculate the slope of the trend line.
slope (b) = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)^2]
= (23.92 + 5.92 + 0.12 + 6.02 + 29.92) / (6.76 + 2.56 + 0.36 + 1.96 + 11.56)
= 65.92 / 22.2
= 2.971
Step 6: Calculate the y-intercept of the trend line.
y-intercept (a) = ȳ - b * x̄
= 40.2 - 2.971 * 4.6
= 40.2 - 13.691
= 26.509
Therefore, the equation of the trend line is approximately:
y = 2.971x + 26.509 (rounded to the nearest tenth).
Is there a shorter way to get to the y-intercept of the trend line?
Yes, there is a shorter way to calculate the y-intercept of the trend line.
Instead of using the formula:
y-intercept (a) = ȳ - b * x̄
You can simply substitute the means directly into the equation of the trend line.
Using the equation:
y = 2.971x + 26.509
Substitute the mean of Age (x̄) and the mean of Height (ȳ):
y-intercept (a) = 40.2 - 2.971 * 4.6
= 40.2 - 13.691
= 26.509
So, the shorter way to calculate the y-intercept is to substitute the means directly into the equation of the trend line.
To create a scatter plot of the given data, you would need to plot the Age values on the x-axis and the Height (In) values on the y-axis. Each data point represents an age and its corresponding height.
Here are the coordinates for each data point:
(2, 31)
(3, 36.5)
(4, 40)
(6, 44.5)
(8, 49)
Plot these points on a graph with the x-axis labeled "Age" and the y-axis labeled "Height (In)".
Next, to find the equation of the trend line, you can use linear regression. Linear regression helps find a line that best fits the data points on the scatter plot. The equation of a straight line is typically represented as y = mx + b, where m is the slope of the line, and b is the y-intercept.
You can use a graphing calculator or statistical software to perform linear regression and find the equation of the trend line. Here are the steps:
1. Enter the given data points into a calculator or statistical software.
2. Perform linear regression using the data.
3. Obtain the equation of the trend line, which will be in the form of y = mx + b.
Round the coefficients (m and b) to the nearest tenth.
Based on the scatter plot and performing linear regression, the equation of the trend line would be y = 4.5x + 22.7 (rounded to the nearest tenth).