Question
Do the data suggest a linear . quadratic or an exponential function ? Use regression to find a model for each data set.
Hint:
1. When the difference between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant. i.e. y(n)- y(n-1) is constant for any value of n, the function is known as a linear function.
2. Regression is a statistical tool used to find a model that can represent the relation between a given change in dependent variable (
) for a given change in independent variable (
).
Linear Equation using regression can be represented as-
Y = a + bX, where-
a =
b =
The correct answer is: The given function is a linear function and using Regression, the given function can be modelled into the equation- Y = 2.48X - 19.96.
Step-by-step solution:-
From the given table, we observe the following readings-
x1 = 0, y1 = 100;
x2 = 1, y2 = 89.5;
x3 = 2, y3 = 78.9;
x4 = 3, y4 = 68.4;
x5 = 4, y5 = 57.8
a). Difference between 2 consecutive x values-
dx1 = x2 - x1 = 1 - 0 = 1
dx2 = x3 - x2 = 2 - 1 = 1
dx3 = x4 - x3 = 3 - 2 = 1
dx4 = x5 - x4 = 4 - 3 = 1
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 89.5 - 100 = -10.5
dy2 = y3 - y2 = 78.9 - 89.5 = -10.6
dy3 = y4 - y3 = 68.4 - 78.9 = -10.5
dy4 = y5 - y4 = 57.8 - 68.4 = -10.6
We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is almost constant i.e. -10.5.
Hence, the given function is a linear function.
Using Linear Regression formula-
Y = a + bX, where-
a = 
∴ a = 
.......................... (As per adjacent table)
∴ a = 
∴ a =
∴ a = -19.96
b = 
∴ b =
∴ b = (-626 + 750)/ (150 - 100)
∴ b = 
∴ b = 2.48
∴ The Linear Equation is-
Y = a + bX
∴ Y = -19.96 + 2.48X
∴ Y = 2.48X - 19.96
Final Answer:-
∴ The given function is a linear function and using Regression, the given function can be modelled into the equation- Y = 2.48X - 19.96.
x2 = 1, y2 = 89.5;
x3 = 2, y3 = 78.9;
x4 = 3, y4 = 68.4;
x5 = 4, y5 = 57.8
a). Difference between 2 consecutive x values-
dx1 = x2 - x1 = 1 - 0 = 1
dx2 = x3 - x2 = 2 - 1 = 1
dx3 = x4 - x3 = 3 - 2 = 1
dx4 = x5 - x4 = 4 - 3 = 1
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 89.5 - 100 = -10.5
dy2 = y3 - y2 = 78.9 - 89.5 = -10.6
dy3 = y4 - y3 = 68.4 - 78.9 = -10.5
dy4 = y5 - y4 = 57.8 - 68.4 = -10.6
We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is almost constant i.e. -10.5.
Hence, the given function is a linear function.
Using Linear Regression formula-
Y = a + bX, where-
a =
∴ a =
.......................... (As per adjacent table)∴ a =
∴ a =
∴ a = -19.96
b =
∴ b =
∴ b = (-626 + 750)/ (150 - 100)
∴ b =
∴ b = 2.48
∴ The Linear Equation is-
Y = a + bX
∴ Y = -19.96 + 2.48X
∴ Y = 2.48X - 19.96
Final Answer:-
∴ The given function is a linear function and using Regression, the given function can be modelled into the equation- Y = 2.48X - 19.96.
Check for extraneous solutions
,where H is the height in centimeters and M is the mass in kilograms. A doctor calculates a particular dose of medicine for a patient whose BSA is less than 1.9. If the patient is 160 cm tall, what must the mass of the person be for the dose to be appropriate?
