Explain the transformation needed to convert the following data to a linear data set.
{(1,0.98), (2,1.39), (3,1.71), (4, 1.98), (5,2.22), (6,2.43)
To convert the given data into a linear dataset, we need to find a linear relationship between the x-values and the corresponding y-values.
One way to do this is by graphing the points and looking for a straight line pattern. However, an easier and more systematic approach is to calculate the slope and y-intercept of the line that best fits the data, also known as the least-squares line.
To find the slope (m) and y-intercept (b) of the least-squares line, we can use the following formulas:
m = (n * ∑(xi * yi) - ∑xi * ∑yi) / (n * ∑(xi^2) - (∑xi)^2)
b = (∑yi - m * ∑xi) / n
Where n is the number of data points, xi and yi are the x and y-values of each point, and ∑ refers to the sum of the values.
Let's calculate the slope and y-intercept for the given data and express the linear data in the form of y = mx + b:
n = 6
∑xi = 1 + 2 + 3 + 4 + 5 + 6 = 21
∑yi = 0.98 + 1.39 + 1.71 + 1.98 + 2.22 + 2.43 = 10.71
∑(xi * yi) = (1 * 0.98) + (2 * 1.39) + (3 * 1.71) + (4 * 1.98) + (5 * 2.22) + (6 * 2.43) = 31.79
∑(xi^2) = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91
m = (6 * 31.79 - 21 * 10.71) / (6 * 91 - 21^2)
= (190.74 - 226.91) / (546 - 441)
= -36.17 / 105
= -0.344
b = (10.71 - (-0.344 * 21)) / 6
= (10.71 + 7.224) / 6
= 17.934 / 6
= 2.989
The equation for the linear dataset is therefore y = -0.344x + 2.989.