Find the slope of the median median line for the dataset.
X y
2 1
3 4
4 6
5 3
A) m=-0.8889
B) m=1
C) m=1.1111
D) m=1.1481
To find the median median line, we first need to find the median of the x-values and the median of the y-values.
The median of the x-values is 3.5, and the median of the y-values is 3.5.
Next, we need to find the equation of the line that passes through the point (3.5, 3.5) and the median of the slope of the four lines that can be drawn between each pair of points in the dataset.
The four lines that can be drawn are:
- From (2,1) to (3,4)
- From (2,1) to (4,6)
- From (2,1) to (5,3)
- From (3,4) to (5,3)
The slopes of these lines are (4-1)/(3-2) = 3, (6-1)/(4-2) = 2.5, (3-1)/(5-2) = 0.67, and (3-4)/(5-3) = -0.5.
The median of these slopes is (2.5 + 0.67) / 2 = 1.585.
So, the equation of the median median line is y - 3.5 = 1.585(x - 3.5). Simplifying, we get y = 1.585x - 2.1975.
Therefore, the slope of the median median line is 1.585, which is closest to option C) m=1.1111.
To find the slope of the median median line for the dataset, follow these steps:
Step 1: Find the median of the x-coordinates and the median of the y-coordinates.
The median of the x-coordinates is (3 + 4) / 2 = 3.5.
The median of the y-coordinates is (3 + 4) / 2 = 3.5.
Step 2: Calculate the differences between each x-coordinate and the median x-coordinate, and each y-coordinate and the median y-coordinate.
For the first data point (2, 1):
Difference in x-coordinate: 2 - 3.5 = -1.5
Difference in y-coordinate: 1 - 3.5 = -2.5
For the second data point (3, 4):
Difference in x-coordinate: 3 - 3.5 = -0.5
Difference in y-coordinate: 4 - 3.5 = 0.5
For the third data point (4, 6):
Difference in x-coordinate: 4 - 3.5 = 0.5
Difference in y-coordinate: 6 - 3.5 = 2.5
For the fourth data point (5, 3):
Difference in x-coordinate: 5 - 3.5 = 1.5
Difference in y-coordinate: 3 - 3.5 = -0.5
Step 3: Calculate the slope of the median median line.
The slope is calculated by dividing the sum of the products of the differences in the x-coordinates and the differences in the y-coordinates by the sum of the squares of the differences in the x-coordinates.
Sum of the products: (-1.5 * -2.5) + (-0.5 * 0.5) + (0.5 * 2.5) + (1.5 * -0.5) = 5.625
Sum of the squares of the differences in x-coordinates: (-1.5)^2 + (-0.5)^2 + (0.5)^2 + (1.5)^2 = 6.5
Slope = Sum of the products / Sum of the squares of the differences in x-coordinates
Slope = 5.625 / 6.5 = 0.8654
Therefore, the slope of the median median line for the dataset is approximately 0.8654. None of the options A), B), C), or D) provide this exact value, so there may be an error in the options provided.