Maths-
General
Easy

Question

Do the data suggest a linear . quadratic or an exponential function ? Use regression to find a model for each data set.

hintHint:

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 (output values/ y values) for a given change in independent variable (input values/ x values).
Linear Equation using regression can be represented as-
Y = a + bX, where-
a = fraction numerator left square bracket left parenthesis capital sigma y right parenthesis left parenthesis capital sigma x squared right parenthesis space minus space left parenthesis capital sigma x right parenthesis space left parenthesis capital sigma x y right parenthesis right square bracket space over denominator left square bracket n left parenthesis capital sigma x squared right parenthesis space minus space left parenthesis capital sigma x right parenthesis squared right square bracket end fraction
b = fraction numerator left square bracket n left parenthesis capital sigma x y right parenthesis space minus space left parenthesis capital sigma x right parenthesis space left parenthesis capital sigma y right parenthesis right square bracket over denominator space left square bracket n left parenthesis capital sigma x squared right parenthesis space minus space left parenthesis capital sigma x right parenthesis squared right square bracket end fraction

The correct answer is: The given function is a quadratic function and using Regression, the given function can be modelled into the equation- Y = - X2 + 20 X - 103.


    Step-by-step solution:-
    From the given table, we observe the following readings-

    x1 = 6, y1 = -19;
    x2 = 7, y2 = -12;
    x3 = 8, y3 = -7;
    x4 = 9, y4 = -4;
    x5 = 10, y5 = -3
    a). Difference between 2 consecutive x values-
                                                                                                    dx1 = x2 - x1 = 7 - 6 = 1
                                                                                                    dx2 = x3 - x2 = 8 - 7 = 1
                                                                                                    dx3 = x4 - x3 = 9 - 8 = 1
                                                                                                   dx4 = x5 - x4 = 10 - 9 = 1
    Difference between 2 consecutive y values-
                                                                                       dy1 = y2 - y1 = -12 - (-19) = -12 + 19 = 7
                                                                                         dy2 = y3 - y2 = -7 - (-12) = -7 + 12 = 5
                                                                                           dy3 = y4 - y3 = -4 - (-7) = -4 + 7 = 3
                                                                                           dy4 = y5 - y4 = -3 - (-4) = -3 + 4 = 1
    We observe that the difference for every consecutive x values is constant i.e. 1 but for y values the difference is not constant.
    Hence, the given function is not a linear function.
    b). Now, difference between 2 consecutive differences for y values-
                                                                                                        dy2 - dy1 = 7 - 5 = 2
                                                                                                        dy3 - dy2 = 5 - 3 = 2
                                                                                                        dy4 - dy3 = 3 - 1 = 2
    We observe that the difference of differences of 2 consecutive y values are constant i.e. 2.
    Hence, the given function is a quadratic function.
    Using Quadratic Regression formula and values from the adjacent table-
                                                                                                      Y = aX2 + bX + c, where-
                                                                                                      Σy = nc + b(Σx) + a(Σx2)
                                                                                                   ∴ -45 = 5c + b(40) + a(330)
                                                                                                    ∴ -45 = 5c + 40b + 330a .................................................. (Equation i)
    Σxy = c(Σx) + b(Σx2) + a(Σx3)
                                                                                             ∴ -320 = c(40) + b(330) + a(2,800)
                                                                                                ∴ -320 = 40c + 330b + 2,800a ....................................... (Equation ii)
    Σx2y = c(Σx2) + b(Σx3) + a(Σx4)
                                                                                        ∴ -2,344 = c(330) + b(2,800) + a(24,354)
                                                                                            ∴ -2,344 = 330c + 2,800b + 24,354a ....................... (Equation iii)
    Dividing Equation 2 by 8, we get-
                                                                                                      350a + 41.25b + 5c = -40 …............................................... (Equation iv)
    Subtracting Equation I from Equation iv, we get-
                                                                                                      350a + 41.25b + 5c = -40 …............................................... (Equation iv)
                                                                                                       - 330a + 40b + 5c = -45 …............................................... (Equation i)
                                                                                                              20a + 1.25b = 5 .................................................. (Equation v)
    Multiplying Equation ii with 8.25, we get-
                                                                                            23,100a + 2,722.5b + 330c = -2,640 ......................... (Equation vi)
    Subtracting Equation vi from Equation iii, we get-
                                                                                                24,354a + 2,800b + 330c = -2,344 ......................... (Equation iii)
                                                                                             - 23,100a + 2,722.5b + 330c = -2,640 ......................... (Equation vi)
                                                                                                          1,254a + 77.5b = 296 ......................... (Equation vii)
    Multiplying Equation v with 62, we get-
                                                                                                          1,240a + 77.5b = 310 ............................................... (Equation viii)
    Subtracting Equation vii from Equation viii, we get-
                                                                                                          1,240a + 77.5b = 310 ............................................... (Equation viii)
                                                                                                        - 1,254a + 77.5b = 296 ............................................... (Equation vii)
                                                                                                                   -14a = 14
    i.e. -14a = 14
                                                                                                                ∴ a = 14/ -14 ................................... (Dividing both sides by -14)
                                                                                                                    ∴ a = -1
    Substituting a = -1 in Equation v, we get-
                                                                                                                20a + 1.25b = 5 .................................................. (Equation v)
                                                                                                           ∴ 20(-1) + 1.25b = 5
                                                                                                             ∴ -20 + 1.25b = 5
                                                                                                             ∴ 1.25b = 5 + 20 ........................................ (Taking all constants together)
                                                                                                                ∴ 1.25b = 25
                                                                                                                ∴ b = 25/1.25 ............................................ (Dividing both sides by 1.25)
                                                                                                                    ∴ b = 20
    Substituting a = -1 and b = 20 in Equation i, we get-
                                                                                                          330a + 40b + 5c = -45 .............................. (Equation i)
                                                                                                   ∴ 330 (-1) + 40 (20) + 5c = -45
                                                                                                       ∴ -330 + 800 + 5c = -45
                                                                                                            ∴ 470 + 5c = -45
                                                                                                             ∴ 5c = -45 - 470 ..................... (Taking all constants together)
                                                                                                                ∴ 5c = -515
                                                                                                                ∴ c = -515/5 ........................... (Dividing both sides by 5)
                                                                                                                 ∴ c = -103
    ∴ The Quadratic Equation is-
                                                                                                             Y = aX2 + bX + c
                                                                                                     ∴ Y = -1 X2 + 20 X + (-103)
                                                                                                        ∴ Y = - X2 + 20 X - 103
    Final Answer:-
    ∴ The given function is a quadratic function and using Regression, the given function can be modelled into the equation- Y = - X2 + 20 X - 103.

    Related Questions to study

    General
    Maths-

    The body surface area (BSA) of a human being is used to determine doses of medication. The formula for finding BSA is BSA = square root of fraction numerator H cross times M over denominator 3600 end fraction end root ,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?

    The body surface area (BSA) of a human being is used to determine doses of medication. The formula for finding BSA is BSA = square root of fraction numerator H cross times M over denominator 3600 end fraction end root ,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?

    Maths-General
    General
    Maths-

    A Saving account has a balance of $1 . Savings plan A will add $1000 to an account each month , and plan B will double the amount each month ?
    a. Which plan is better in the short run ? for How long , Explain.
    b. Which plan is better in the long run ? Explain.

    A Saving account has a balance of $1 . Savings plan A will add $1000 to an account each month , and plan B will double the amount each month ?
    a. Which plan is better in the short run ? for How long , Explain.
    b. Which plan is better in the long run ? Explain.

    Maths-General
    General
    Maths-

    How can you determine whether a linear , exponential or quadratic function best models the data ?

    How can you determine whether a linear , exponential or quadratic function best models the data ?

    Maths-General
    parallel
    General
    Maths-

    Solve the radical equation 13 minus fourth root of x equals 10 Check for extraneous solutions

    Solve the radical equation 13 minus fourth root of x equals 10 Check for extraneous solutions

    Maths-General
    General
    Maths-

    What is the error in the student’s reasoning below ? Describe how to correct the statement ?

    What is the error in the student’s reasoning below ? Describe how to correct the statement ?

    Maths-General
    General
    Maths-

    Compare the functions f(x)= 3X+2 , g(x)= 2x2+3 and h(x)= 2x . Show that as x increases , h(x) will eventually exceed f(x) and g(x).

    Compare the functions f(x)= 3X+2 , g(x)= 2x2+3 and h(x)= 2x . Show that as x increases , h(x) will eventually exceed f(x) and g(x).

    Maths-General
    parallel
    General
    Maths-

    Solve the radical equation square root of 15 minus x end root minus square root of 6 x end root equals negative 3

    Solve the radical equation square root of 15 minus x end root minus square root of 6 x end root equals negative 3

    Maths-General
    General
    Maths-

    Calculate the second difference for data in the table. Use a graphing calculator to find the quadratic regression for each data set. Make a conjecture about the relationship between the a values in the quadratic models and the second difference of the data.

    Calculate the second difference for data in the table. Use a graphing calculator to find the quadratic regression for each data set. Make a conjecture about the relationship between the a values in the quadratic models and the second difference of the data.

    Maths-General
    General
    Maths-

    Solve the radical equationsquare root of 6 x minus 20 end root minus x equals negative 6. Check for extraneous solutions

    Solve the radical equationsquare root of 6 x minus 20 end root minus x equals negative 6. Check for extraneous solutions

    Maths-General
    parallel
    General
    Maths-

    Use the functions shown :
    a. Evaluate each function for x=6 , x=8 and x=12
    b. When will function h exceed function f and function g ?

    Use the functions shown :
    a. Evaluate each function for x=6 , x=8 and x=12
    b. When will function h exceed function f and function g ?

    Maths-General
    General
    Maths-

    The graph shows population models for three cities , based on data over a five year period. If the populations continue to increase in the same ways , when will the population of city C exceed the populations of the other two cities ?

    The graph shows population models for three cities , based on data over a five year period. If the populations continue to increase in the same ways , when will the population of city C exceed the populations of the other two cities ?