Answer: Linear correlation equation
Step-by-step explanation:
1) The procedure is very easy. I don't currently have the graph data (Points x, y) but I use Photoshop to draw a rule between the points and approximate values that I show in the table below.
These values refer to each of the blue dots in your graph.
X Y
45 39.4
46 41.2
46 42.4
56.8 52
61.6 58.4
63.6 63
65 62
65 64.4
65.6 63
69 65
Once the table is obtained, we proceed to do the following.
Once this table is obtained, it is necessary to find two additional values from it [tex]x^{2}[/tex] and x * y. At the end of obtaining the values we will make a sum of each of the columns: Sum of X, Y, [tex]x^{2}[/tex] and x *y and at the bottom of our table, so this is obtained:
X Y x2 x*y
45 39.4 2025 1773
46 41.2 2116 1895.2
46 42.4 2116 1950.4
56.8 52 3226.24 2953.6
61.6 58.4 3794.56 3597.44
63.6 63 4044.96 4006.8
65 62 4225 4030
65 64.4 4225 4186
65.6 63 4303.36 4132.8
69 65 4761 4485
583.6 550.8 34837.12 33010.24*
*Values in italics are the sum of each of the columns.
** It is difficult to add this table here, I will add it as an attachment.
The approximation equation is first degree, so we apply the formula:
Y = mx + b
Where m is given by the equation:
[tex]m= \frac{n\sum{xy}-\sum{x}*\sum{y}}{n\sum{x^{2}} -(\sum{x})^{2}} }[/tex]
Where n is the total number of data, In our case n=10,
So we replace the values we have and get the following value from m:
[tex]m= \frac{10*33010.24-(583.6)*(550.8)}{10*34837.12-(583.6)^{2} }[/tex]
So,
[tex]m= 1.112[/tex]
On the other hand to find the value of b, we apply the following formula:
[tex]b=\frac{\sum{y-m\sum{x}}}{n}[/tex]
So,
[tex]b=\frac{550.8-1.112*583.6}{10} \\[/tex]
Then,
[tex]b=-9.828[/tex]
Thus once m and b are obtained, we have the equation of the line, where:
[tex]y = 1.12x-9,828[/tex]
2) To perform the Residual Test we simply take two values from the Table in X and replace
- [tex]x = 45[/tex]
Thus [tex]y = 1.12 (39.4) -9.828[/tex]
[tex]y = 40.5[/tex] (an intermediate value and quite well approximate for any of the first two values of our table)
- [tex]X = 65.6[/tex]
So [tex]Y = 1.12 (65.6) -9.828[/tex]
[tex]Y = 63.5[/tex] (A value very close to that of our penultimate table data)
3) With the high reliability of our equation we can solve this point, thus our new value for
[tex]x = 66[/tex]
So that
[tex]y = 1.12 (66) -9.828 = 64.1[/tex]
Our person will have a height of 64.1
4) We now perform the replacement for 74 inch, but for the value of Y,
So that
[tex]74 = 1.12x-9,828[/tex]
[tex]x = (74 + 9.828) /1.12[/tex]
[tex]x = 74.84[/tex]
The length of his arms is 74.84 inch
