Question

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

Answer 1

Is there a shorter way to get to the y-intercept of the trend line?

Answer 2

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).

Answer 3

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.