calsulus help pleaseee :((

Question

Enter the data shown below from the U.S Census Bureau and answer the questions related to this data. Round values to nearest Hundredth for regression equation. 

WORLD POPULATION

YEAR	POPULATION ( billions )
1960	3.04
1965	3.35
1970	3.71
1975	4.09
1980	4.45
1985	4.85
1990	5.28
1995	5.69
2000	6.08
2005	6.51
2010	6.92

a) write the linear regression equation for this data

b) Write the quadratic regression equation for this data.

c) Write the cubic regression equation for this data.

d) Which equation appears to give the best fit for the data? Use this equation to predict the world population for the years 2050 and 2100. Do you thinks these predictions are realistic? explain why or why not? PLEASE HELP ME WITH THIS :)

e) The following is a graph provided by the united Nation that shows 3 different prediction for the world population until the year 2100.The united Nations has made three projections:Low,Medium, and high. Based on your regression analysis, which of the three projection would seems to be most likely? please Justify.

 

Thank you sooooo much :)

Answer 1

(a)

The equation for  linear regression is  y = a + b x .

Now we have to calculate values for a and b .

The  U.S Census  can be tabulated as shown below .

S.no	Year (x)	Population in billions (y)	xy	x²	y²
1	0 (1960)	3.04	0	0	9.24
2	5 (1965)	3.35	16.75	25	11.22
3	10 (1970)	3.71	37.1	100	13.76
4	15 (1975)	4.09	61.35	225	16.72
5	20 (1980)	4.45	89	400	19.80
6	25 (1985)	4.85	121.25	625	23.52
7	30 (1990)	5.28	158.4	900	27.87
8	35 (1995)	5.69	199.15	1225	32.37
9	40 (2000)	6.08	243.2	1600	36.96
10	45 (2005)	6.51	292.95	2025	42.38
11	50 (2010)	6.92	346	2500	47.88
∑	275	53.97	1565.15	9625	281.72

From the above table, Σx = 275, Σy = 53.97, Σxy = 1565.15, Σx² = 9625, Σy² = 281.72 ,

number of samples n = 11 .

 

 

So the linear regression is  y = 2.945 + 0.078 x .

Answer 2

(b)

We make use of Quadratic Expression.

The Quadratic Equation is in the form of P = ax² + bx + c

Where P is the population after x years  and x is the number of years after 1960 .

Now consider the three points from the table, (5, 3.35) , (20, 4.45) and (40, 6.08).
 

Consider the Point (5, 3.35) substitute  P = 3.35 and x = 5 in the Quadratic Equation .

 P = ax² + bx + c ⇒  a(5)² + 5b + c  ⇒ 25a + 5b + c

25a + 5b + c = 3.35                   ......................Equation (1)

Consider the Point (20, 4.45) substitute P = 4.45 and x = 20 in the Quadratic Equation .

P = ax² + bx + c ⇒ a(20)² + 20b + c ⇒ 400a + 20b + c

400a + 20b + c = 4.45                      .................Equation (2)
 

Consider the Point (40, 6.08) substitute  P = 6.08 and x = 40 in the Quadratic Equation .

P = ax² + bx + c ⇒ a(40)² + 40b + c ⇒ 1600a + 40b + c

1600a + 40b + c = 6.08                   ...................Equation (3)

Solving Equations (1), (2) and (3), we get

a = 0.00023 , b = 0.0675 and c = 3.006

The Quadratic Regression P = 0.00023 x² + 0.0675x + 3.006 .

Answer 3

(c)

We make use of cubic regression equation .

The cubic regression equation is in the form of  P = ax³ + bx² + cx + d .

Where P is the population after x years  and x is the number of years after 1960 .

Now consider the four points from the table, (5, 3.35) , (20, 4.45) ,(30, 5.28) and (40, 6.08).

 

Consider the Point (5, 3.35) substitute  P = 3.35 and x = 5 in the cubic regression equation .

 P = ax³ + bx² + cx + d  ⇒  a(5)³ +b(5)² + 5c+ d  ⇒ 125a + 25b + 5c + d 

125a + 25b + 5c + d  = 3.35                   ......................Equation (1)

 

Consider the Point (20, 4.45) substitute P = 4.45 and x = 20 in the cubic regression equation .

P = ax³ + bx² + cx + d  ⇒  a(20)³ +b(20)² + 20c+ d  ⇒ 8000a + 400b + 20c + d 

8000a + 400b + 20c + d   = 4.45                   ......................Equation (2)

 

Consider the Point (30, 5.28) substitute  P = 5.28 and x = 30 in the cubic regression equation .

P = ax³ + bx² + cx + d  ⇒  a(30)³ +b(30)² + 30c+ d  ⇒ 27000a + 900b + 30c + d 

27000a + 900b + 30c + d   = 5.28               ......................Equation (3)

 

Consider the Point (40, 6.08) substitute  P = 6.08 and x = 40 in the cubic regression equation .

P = ax³ + bx² + cx + d  ⇒  a(40)³ +b(40)² + 40c+ d  ⇒ 64000a + 1600b + 40c + d 

64000a + 1600b + 40c + d  = 6.08                   ......................Equation (4)

 

Solving Equations (1), (2) ,(3) and (4), we get
a = -0.0000153 , b =  0.00123 , c = ​ 0.05063 and d =  3.068

The cubic regression equation P = -0.0000153x³ +  0.00123x² +  0.05063x +  3.068 .

Answer 4

(d)

Check for best fit of the data .

Consider the population in 2010 , substitute x = 50 in  linear regression is  y = 2.945 + 0.078 x .

 y = 2.945 + 0.078 (50) ⇒  2.945 + 3.9 = 6.845 billions .

 

Consider the population in 2010 , substitute x = 50 in  Quadratic Regression P = 0.00023 x² + 0.0675x + 3.006 .

 P = 0.00023 (50)² + 0.0675(50) + 3.006⇒  6.956 billions .

 

Consider the population in 2010 , substitute x = 50 in  cubic regression equation P = -0.0000153x³ +  0.00123x² +  0.05063x +  3.068 .

 P = -0.0000153 (50)³ +  0.00123 (50)² +  0.05063 (50) +  3.068 ⇒  6.696 billions .

 

The actual population in 2010 is 6.92 billions , hence the Quadratic Regression equation is the best fit of the data .

So we make use of Quadratic Regression equation , to evaluate population in 2050 and 2100 .

 

The population in 2050 , substitute x = 90 in  Quadratic Regression P = 0.00023 x² + 0.0675x + 3.006 .

P = 0.00023 (90)² + 0.0675(90) + 3.006⇒  10.944 billions .

So the population in 2050 is 10.944 billions .

 

The population in 2100 , substitute x = 140 in  Quadratic Regression P = 0.00023 x² + 0.0675x + 3.006 .

P = 0.00023 (140)² + 0.0675(140) + 3.006⇒  16.964 billions .

So the population in 2100 is 16.964  billions .

Answer 5

(e)

The Quadratic Regression equation is the best fit for the data .

Hence we consider Quadratic Regression equation , to evalute the population in 2100 .

The population in 2100 , substitute x = 140 in quadratic regression P = 0.00023 x² + 0.0675x + 3.006 .

P = 0.00023 (140)² + 0.0675(140) + 3.006⇒  16.964 billions .

So the population in 2100 is 16.964  billions .

Hence the projection high by UN (marked red in color) is the most likely projection from given graph .