When analyzing data using a , there are three general outcomes for the of the data: positive correlation, negative correlation, or no correlation.
| Observation
|
Type of Correlation
|
| As x increases, y increases.
|
Positive
|
| As x increases, y decreases.
|
Negative
|
| There is no visible pattern.
|
No
|
By treating the table as a set of points, let's graph the years versus the number of inventors applying for patents on a scatter plot. We will restrict our graph to because we cannot have a negative number of people or a negative year. Do you see any trends?
It looks like there is some kind of correlation. Let's draw a , or trend line, to help us identify the type of correlation. To do so, we will draw a line that appears to fit the data closely.
Notice that if we choose any other pair of the points to draw the line, it will not be as close to all the points as the line above is.
Finding the Equation of the Trend Line
To start, let's find the of the trend line using the .
m= y_2- y_1/x_2- x_1
In this formula, ( x_1, y_1) and ( x_2, y_2) are two points on the line. From the graph above and the given table, we can see that the trend line passes through the points ( 1999, 22 052) and ( 2007, 13 748). We can substitute them into the formula to find our slope m.
m = y_2-y_1/x_2-x_1
m= 13 748- 22 052/2007- 1999
m= - 8304/8
m= - 1038
Now that we know the slope of the trend line, we can find its equation by using the of a line.
y- y_1= m(x- x_1)
Let's substitute m= - 1038 and ( x_1, y_1)=( 1999, 22 052) into the equation above.
y-y_1=m(x-x_1)
y- 22 052= -1038(x- 1999)
y-22 052=-1038x+2 074 962
y=-1038x+2 097 014
Number of Inventors in 2006
We will use to estimate the number of inventors applying for patents in 2006. Interpolation is used when you are estimating a value that falls within the known data set. To do that, we will replace x=2006 into the equation of the trend line.
y=-1038x-2 097 014
y=-1038( 2006)+2 097 014
y=- 2 082 228+2 097 014
y=14 786
Therefore, according to the given data, we can say that in 2006, approximately 14 786 inventors applied for patents.
Number of Inventors in 2015
We will use to predict the number of inventors applying for patents in 2015. Extrapolation is used when the value is outside the range of known values. As before, we have to substitute x=2015 into the equation of the trend line.
y=-1038x-2 097 014
y=-1038( 2015)+2 097 014
y=- 2 091 570 +2 097 014
y=5444
We could say that according to the given data, approximately 5444 inventors applied for patents in 2015. Note that we can pick different trend lines for the same set of data, so the estimates and predictions may vary.
